Ideas in Geometry - Illinois Math

Ideas in Geometry

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Preface

The aim of these notes is to convey the spirit of mathematical thinking asdemonstrated through topics mainly from geometry. The reader must be carefulnot to forget this emphasis on deduction and visual reasoning. To this end,many questions are asked in the text that follows. Sometimes these questionsare answered, other times the questions are left for the reader to ponder. To letthe reader know which questions are left for cogitation, a large question markis displayed:

?The instructor of the course will address some of these questions. If a questionis not discussed to the reader’s satisfaction, then I encourage the reader to puton a thinking-cap and think, think, think! If the question is still unresolved, goto the World Wide Web and search, search, search!

This document is open-source. It is licensed under the Creative CommonsAttribution-ShareAlike (CC BY-SA) License. Loosely speaking, this means thatthis document is available for free. Anyone can get a free copy of this document(source or PDF) from the following site:

http://www.math.uiuc.edu/Courses/math119/

Please report corrections, suggestions, gripes, complaints, and criticisms to:[email protected] or [email protected]

Thanks and Acknowledgments

This document is based on a set of lectures originally given by Bart Snapp atthe University of Illinois at Urbana-Champaign in the Fall of 2005 and Springof 2006. Each semester since, these notes have been revised and modified. Agrowing number of instructors have made contributions, including Tom Cooney,Melissa Dennison, Jesse Miller, and Bart Snapp.

Thanks to Alison Ahlgren, the Quantitative Reasoning Coordinator at theUniversity of Illinois at Urbana-Champaign, for developing this course and forworking with me and all the other instructors during the continual developmentof this document. Also thanks to Harry Calkins for help with Mathematica

graphics and acting as a sounding-board for some of the ideas expressed in thisdocument.

In 2009, Greg Williams, a Master of Arts in Teaching student at Coastal Car-olina University, worked with Bart Snapp to produce the chapter on geometrictransformations.

A number of students have also contributed to this document by eithertyping or suggesting problems. They are: Camille Brooks, Michelle Bruno,Marissa Colatosti, Katie Colby, Anthony ‘Tino’ Forneris, Amanda Genovise,Melissa Peterson, Nicole Petschenko, Jason Reczek, Christina Reincke, DavidSeo, Adam Shalzi, Allice Son, Katie Strle, Beth Vaughn.

Contents

1 Beginnings, Axioms, and Viewpoints 1

1.1 Euclid and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The Most Successful Textbook Ever Written . . . . . . . 11.1.2 The Parallel Postulate . . . . . . . . . . . . . . . . . . . . 5

1.2 Points of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1 Synthetic Geometry . . . . . . . . . . . . . . . . . . . . . 131.2.2 Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . 141.2.3 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . 17

1.3 City Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.1 Getting Work Done . . . . . . . . . . . . . . . . . . . . . 251.3.2 (Un)Common Structures . . . . . . . . . . . . . . . . . . . 28

2 Proof by Picture 38

2.1 Basic Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.1 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.3 Complement . . . . . . . . . . . . . . . . . . . . . . . . . 402.1.4 Putting Things Together . . . . . . . . . . . . . . . . . . . 41

2.2 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.3 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.3.1 Tessellations and Art . . . . . . . . . . . . . . . . . . . . . 522.4 Proof by Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4.1 Proofs Involving Right Triangles . . . . . . . . . . . . . . 562.4.2 Proofs Involving Boxy Things . . . . . . . . . . . . . . . . 602.4.3 Proofs Involving Sums . . . . . . . . . . . . . . . . . . . . 632.4.4 Proofs Involving Sequences . . . . . . . . . . . . . . . . . 672.4.5 Thinking Outside the Box . . . . . . . . . . . . . . . . . . 68

3 Topics in Plane Geometry 83

3.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.1 Centers in Triangles . . . . . . . . . . . . . . . . . . . . . 833.1.2 Theorems about Triangles . . . . . . . . . . . . . . . . . . 87

3.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2.1 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2.2 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.2.3 Combining Areas and Ratios—Probability . . . . . . . . . 102

4 Compass and Straightedge Constructions 115

4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Trickier Constructions . . . . . . . . . . . . . . . . . . . . . . . . 124

4.2.1 Challenge Constructions . . . . . . . . . . . . . . . . . . . 1254.2.2 Problem Solving Strategies . . . . . . . . . . . . . . . . . 129

4.3 Constructible Numbers . . . . . . . . . . . . . . . . . . . . . . . . 1324.4 Impossibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.4.1 Doubling the Cube . . . . . . . . . . . . . . . . . . . . . . 1424.4.2 Squaring the Circle . . . . . . . . . . . . . . . . . . . . . . 1424.4.3 Trisecting the Angle . . . . . . . . . . . . . . . . . . . . . 143

5 Isometries 146

5.1 Matrices as Functions . . . . . . . . . . . . . . . . . . . . . . . . 1465.1.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.1.2 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.1.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.2 The Algebra of Matrices . . . . . . . . . . . . . . . . . . . . . . . 1595.2.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . 1595.2.2 Compositions of Matrices . . . . . . . . . . . . . . . . . . 1605.2.3 Mixing and Matching . . . . . . . . . . . . . . . . . . . . 163

5.3 The Theory of Groups . . . . . . . . . . . . . . . . . . . . . . . . 1675.3.1 Groups of Reflections . . . . . . . . . . . . . . . . . . . . 1675.3.2 Groups of Rotations . . . . . . . . . . . . . . . . . . . . . 1685.3.3 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . 169

6 Convex Sets 173

6.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.1.1 An Application . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2 Convex Sets in Three Dimensions . . . . . . . . . . . . . . . . . . 1806.2.1 Analogies to Two Dimensions . . . . . . . . . . . . . . . . 1806.2.2 Platonic Solids . . . . . . . . . . . . . . . . . . . . . . . . 180

6.3 Ideas Related to Convexity . . . . . . . . . . . . . . . . . . . . . 1856.3.1 The Convex Hull . . . . . . . . . . . . . . . . . . . . . . . 1856.3.2 Sets of Constant Width . . . . . . . . . . . . . . . . . . . 185

6.4 Advanced Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 190

References and Further Reading 192

Index 194

Chapter 1

Beginnings, Axioms, andViewpoints

Since you are now studying geometry and trigonometry, I will giveyou a problem. A ship sails the ocean. It left Boston with a cargo ofwool. It grosses 200 tons. It is bound for Le Havre. The mainmastis broken, the cabin boy is on deck, there are 12 passengers aboard,the wind is blowing East-North-East, the clock points to a quarterpast three in the afternoon. It is the month of May. How old is thecaptain?

—Gustave Flaubert

1.1 Euclid and Beyond

1.1.1 The Most Successful Textbook Ever Written

Question Think of all the books that were ever written. What are some ofthe most influential of these?

?The Elements by the Greek mathematician Euclid of Alexandria should be

high on this list. Euclid lived in Alexandria, Egypt, around 300 BC. His book,The Elements is an attempt to compile and write down everything that wasknown about geometry. This book is perhaps the most successful textbook everwritten, having been used in nearly all universities up until the 20th century.Even today its heritage can be seen in scientific thought and writing. Here arethree reasons this book is so important:

(1) The Elements is of practical use.

(2) The Elements contains powerful ideas.

1

1.1. EUCLID AND BEYOND

(3) The Elements provides a playground for the development of logicalthought.

We’ll address each of these in turn.

The Elements is of practical use. Any time something large is built, somegeometry must be used. The roads we drive on every day, the buildings we livein, the malls we shop at, the stadiums our favorite sports teams compete in; inshort, if it is bigger than a shack, then geometry must have been used at somepoint. Moreover, geometry is crucial to modern transportation—in fact, anylarge scale transportation. An airplane could never make it to its destinationwithout geometry, nor could any ship at sea. People of the past were faced withthe difficulties of geometry on a continuing basis. Euclid’s The Elements wastheir handbook to solve everyday problems.

The Elements contains powerful ideas. Around 200 BC, the head librar-ian at the Great Library of Alexandria was a man by the name of Eratosthenes.Not only was Eratosthenes a great athlete, he was a scholar of astronomy, ethics,music, philosophy, poetry, theater, and important to this discussion, mathemat-ics. His nickname was Beta, the second letter in the Greek alphabet. This isbecause with so many interests and accomplishments, he seemed to be second

best at everything in the world.

It came to Eratosthenes that every year, on the longest day of the year, atnoon, sunlight would shine down to the bottom of a deep well located in presentday city of Aswan, Egypt. Eratosthenes reasoned that this meant that the sunwas directly overhead Aswan at this time. However, Eratosthenes knew thatthe sun was not directly overhead in Alexandria. He realized that the situationmust be something like this:

Aswan

Alexandria

Sun’s rays

center of the Earth

Using ideas found in The Elements, Eratosthenes realized that if he drew imagi-nary lines from Alexandria and Aswan down to the center of the Earth, and if hecould compute the angle between these lines, then to compute the circumference

2

CHAPTER 1. BEGINNINGS, AXIOMS, AND VIEWPOINTS

of the Earth he would need only to solve the equation:

total degrees in a circle

angle between the cities=

circumference of the Earth

distance between the cities

Thus Eratosthenes hired a man to pace the distance from Alexandria to Aswan.It was found to be about 5000 stadia. A stadia is an ancient unit of measurementwhich is the the length of a stadium. To measure the angle, he measured theangle of the shadow of a perpendicular stick in Alexandria, and found:

Sun’s rays

ground

stick

Where the lower left angle is about 83◦ and the upper right angle is about 7◦.Thus we have

360

7=

x

5000⇒ 360

7· 5000 = x.

So we see that x, the circumference of the Earth is about

250000 stadia.

Unfortunately, as you may realize, the length of a stadium can vary. If the lengthof the stadium is defined to be 157 meters, the length of an ancient Egyptianstadium, then it can be calculated that the circumference of the Earth is about39250 kilometers. Considering the true amount is 40075 kilometers, Eratos-thenes made a truly remarkable measurement. However, the most importantpart is not that his measurement was close to being exactly right, but that hislogic was correct.

Question What did Eratosthenes assume when he made his measurement ofthe circumference of the Earth?

?The Elements provides a playground for the development of logical

thought. By answering the question raised above, we see it is necessary tounderstand what one assumes when doing science. When Euclid wrote The

Elements, he started by stating his assumptions. By stating his assumptions,he gave rigor to his arguments. By focusing on the logical reasoning that goesinto problem solving, Euclid put the method of solving a problem, and notmerely the solution, into the spotlight.

Euclid’s assumptions were stated as five axioms1.

1Actually, in Euclid’s time the word axiom was reserved for something obvious, a commonnotion, while postulate meant something to be assumed. However, in present day languagewe use the word axiom to mean something that is assumed. Henceforth we will always usethe modern terminology.

3


Definition An axiom is a statement that is accepted without proof.

Euclid’s five axioms can be paraphrased as:

(1) A line can be drawn from a point to any other point.

(2) A finite line can be extended indefinitely.

(3) A circle can be drawn, given a center and a radius.

(4) All right angles are ninety degrees.

(5) If a line intersects two other lines such that the sum of the interior angleson one side of the intersecting line is less than the sum of two right angles,then the lines meet on that side and not on the other side.

The first four axioms are easy to understand, but the fifth is more complex. Weshould draw a picture describing the situation. Here is an example of how todraw pictures describing mathematical statements:

Example Here is a picture describing the fifth axiom above:

δ

γβ

α

the lines meet yonderthe lines don’t meet

on this side

The fifth axiom says that if α+ β is less than 180 degrees, the sum of two right

angles, then the lines will meet on that side. Likewise the axiom says that if

δ+ γ is less than than 180 degrees, then the angles will meet on that side. The

latter looks to be the case in the diagram above.

One may wonder, what if we just ignore the Euclid’s 5th Axiom? By remov-ing or changing the fifth axiom (or any independent axiom) a different geometryis created. The sort of geometry that Euclid wrote about takes place on a plane.We call this sort of geometry Euclidean Geometry in honor of Euclid. By chang-ing Euclid’s 5th Axiom, we stop doing geometry on the plane and start doingit on other types of surfaces, say spheres or other beasts.

While The Elements may be the most successful textbook ever written, withover one thousand editions and over two thousand years of usage, there is stillroom for improvement. In the early 20th century, mathematicians pointed outthat there are some logical flaws in the proofs that Euclid gives. David Hilbert,one of the great mathematicians of the 20th century, required around 20 axiomsto prove all the theorems in The Elements. Nevertheless most of the theoremsin The Elements are proved more-or-less correctly, and the text continues tohave influence to this day.

4


1.1.2 The Parallel Postulate

Euclidean Geometry seems to be a wonderful description of the universe inwhich we live. Is it really? How useful is it if you want to travel all the wayaround the world? What if you are standing in the middle of a city with largebuildings obstructing your path? In each of these situations, a different sort ofgeometry is needed. Let’s look at new geometries that are different from butclosely related to Euclidean Geometry.

The first four of Euclid’s axioms discussed in the previous section were alwayswidely accepted. The fifth attracted more attention. For convenience’s sake,here is the fifth axiom again:

(5) If a line intersects two other lines such that the sum of the interior angleson one side of the line is less than the sum of two right angles, then thelines meet on that side and not on the other side.

Here are some other statements closely related to Euclid’s fifth axiom:

(5A) Exactly one line can be drawn through any point not on a given lineparallel to the given line.

(5B) The sum of the angles in every triangle is equal to 180◦.

(5C) If two lines ℓ1 and ℓ2 are both perpendicular to some third line, then ℓ1and ℓ2 do not meet.

Question Can you draw pictures depicting these statements? Can you ex-plain why Euclid’s fifth axiom is sometimes called the parallel postulate?

?

Let’s replace Euclid’s fifth axiom by the following:

(⋆) Given a point and a line, there does not exist a line through that pointparallel to the given line.

There is a very natural geometry where this new axiom holds and theessences of the first four also still hold. Instead of working with a plane, we nowwork on a sphere. We call this sort of geometry Spherical Geometry. Points,circles, angles, and distances are exactly what we would expect them to be. Butwhat do we mean by lines on a sphere? Lines are supposed to be extendedindefinitely. In Spherical Geometry, the lines are the great circles.

Definition A great circle is a circle on the sphere with the same center asthe sphere.

5


Here is a picture to help you out. On the left, we have great circles drawn.On the right, we have regular old circles drawn.

A great circle cuts the sphere into two equal hemispheres. Great circles of theplanet Earth include the equator and the lines of longitude. A great circlethrough a point P also goes through the point directly opposite to P on thesphere. This point is the called the antipodal point for P . For example, anygreat circle through the North Pole also goes through the South Pole.

Question Why should we choose great circles to be the lines in SphericalGeometry?

I’ll take this one. It is a theorem of Euclidean Geometry that the shortestpath between any two points on a plane is given by a line segment. We have asimilar theorem in Spherical Geometry.

Theorem 1 The shortest path between any two points on a sphere is given

by an arc of a great circle.

This theorem is really handy! For one thing, it explains why an airplaneflies over Alaska when it is flying from Chicago to Tokyo.

Many of the results from Euclid’s Elements still hold once we make suitablechanges. For example, Euclid’s second axiom says that a finite line segment

6


can be extended. This idea still holds: Given a line segment (an arc of a greatcircle) we can extend it to a line (a great circle). However this line is no longerinfinite in length. It will loop around and meet itself after traveling around thecircumference of the sphere.

However, not all of our results will still hold.

Question Are statements (5A), (5B), and (5C) true in Spherical Geometry?

?Let’s take a closer look at (5B).

Question What is a triangle in Spherical Geometry? What is a polygon?

?

The picture above shows a triangle in Spherical Geometry. Here a regionbounded by three line segments that meet at their endpoints. The sum ofthe angles in this triangle is clearly greater than 180◦, contradicting (5B). InSpherical Geometry, the sum of the angles in a triangle can be any numberbetween 180◦ and 900◦.

Question How is it that in Spherical Geometry the angles of a triangle cansum to any number between 180◦ and 900◦?

?You can go further and develop a whole theory of spherical trigonometry.

This proved to be very important in cartography and navigation with hugerewards (more than $5,000,000 in today’s money) being offered to anyone whocould devise a practical, accurate way of determining a ship’s location when itis in the middle of the ocean, a problem that was not deemed fully solved until1828. Unless your journey is very short, the fact that the Earth is not flat makesa big difference.

7


Question Suppose you replaced Euclid’s fifth axiom by the statement: Givena line and a point not on that line, there exists multiple lines through that pointparallel to the given line. What kind of geometry would this lead to?

?

8


Problems for Section 1.1

(1) Briefly explain what Eratosthenes assumed when he computed the cir-cumference of the Earth.

(2) Doug drove from Columbus, Ohio to Urbana, Illinois in 5 hours. Thedrive is almost exactly 300 miles. Deena says, “Doug, it looks like youwere speeding.”

Doug replies, “No, I was driving 60 miles per hour.”

(a) How did Doug come to his conclusion?

(b) How did Deena come to her conclusion?

(c) What assumptions were made?

(d) Whose statement is correct? Explain your answer.

(3) Consider the following proposition of Euclid:

Given a line segment, one can construct an equilateral trianglewith the line segment as its side.

Draw a picture depicting this statement and give a short explanation ofhow your picture depicts the above statement.


If two lines intersect, then the opposite angles at the intersectionpoint are equal.



In any triangle, the sum of the lengths of any two sides is greaterthan the length of the third.


(6) Euclid’s fourth axiom states: “All right angles are ninety degrees.” This isnot quite what Euclid said. Euclid said that a right angle is formed whentwo lines intersect and adjacent angles on either side of one of the linesare equal. In particular, Euclid asserted that the angles in every such casewill be equal. Draw a picture depicting this statement and give a shortexplanation of how your picture depicts the above statement.

(7) Consider the following axiom of Hilbert:

9


Let AB and BC be two segments of a line ℓ that have no pointsin common aside from the point B, and, furthermore, let A′B′

and B′C ′ be two segments of another line ℓ′ having, likewise,no point other than B′ in common. If AB is the same length asA′B′ and BC is the same length as B′C ′, then AC is the samelength as A′C ′.


(8) Consider the following axiom of Hilbert:

Let A, B, and C be three points not lying on the same line, andlet ℓ be a straight line lying in the plane ABC and not passingthrough any of the points A, B, or C. Then, if the line ℓ passesthrough a point of the segment AB, it will also pass througheither a point of the segment BC or a point of the segment AC.


(9) Consider the following proposition:

If two lines ℓ1 and ℓ2 are both perpendicular to some third line,then ℓ1 and ℓ2 do not meet.


(10) State the definition of a great circle and compare/contrast it to a line inEuclidean Geometry.

(11) In Spherical Geometry, what is the difference between a great circle anda regular Spherical Geometry circle?

(12) One way of writing Euclid’s first axiom is “Any two distinct points deter-mine a unique line.” Explain how you would alter this so that it holds inSpherical Geometry.

(13) One way of writing Euclid’s second axiom is “A finite line segment can beextended to an infinite line.” Explain how you would alter this so that itholds in Spherical Geometry.

(14) One way of writing Euclid’s third axiom is “Given any point and anyradius, a circle can be drawn with this center and this radius.” Explainhow you would alter this so that it holds in Spherical Geometry.

(15) Explain why the following proposition from Euclidean Geometry does nothold in Spherical Geometry: If two lines ℓ1 and ℓ2 are both perpendicularto some third line, then ℓ1 and ℓ2 do not meet.

10


(16) Explain why the following proposition from Euclidean Geometry does nothold in Spherical Geometry: A triangle has at most one right angle. (Canyou find a triangle in Spherical Geometry with three right angles?)

(17) Explain why the following result from Euclidean Geometry does not holdin Spherical Geometry: When the radius of a circle increases, its circum-ference also increases.

(18) Define distinct lines to be parallel if they do not intersect. Can you haveparallel lines in Spherical Geometry? Explain why or why not.

(19) Come up with a definition of a circle that will be true in both Euclideanand Spherical Geometry.

(20) Come up with a definition of a polygon that will be true in both Euclideanand Spherical Geometry.

(21) True or False. Explain your conclusions.

(a) Any two distinct lines in Spherical Geometry have at most one pointof intersection.

(b) All polygons in Spherical Geometry have at least three sides.

(c) In Spherical Geometry there exist points arbitrarily far apart.

(d) In Spherical Geometry all triangles have finite area.

(e) In Spherical Geometry any two points can be connected by more thanone line.

(22) A mathematician goes camping. She leaves her tent, walks one mile duesouth, then one mile due east. She then sees a bear before walking onemile north back to her tent. What color was the bear?

(23) The great German mathematician Gauss measured the angles of the trian-gle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken.What reasons might one have for doing this?

11

1.2. POINTS OF VIEW

1.2 Points of View

By studying geometry from different viewpoints we gain insight. Consider some-thing very simple: U-shaped curves. U-shaped curves appear all the time innature. Pick something up and toss it into the air. The object should follow aU-shaped path, we hope an upside down U-shape!

Question Where else do U-shaped curves appear in nature?

?U-shaped curves often appear in mathematics. There are many different U-

shaped curves, including catenary curves, hyperbolas, and most famous of all,parabolas.

Question What is a parabola?

We will answer the above question multiple times in the discussion thatfollows. Here is our first definition of a parabola:

Definition Given a point and a line, a parabola is the set of points such thateach of these points is the same distance from the given point as it is from thegiven line.

Question Why study parabolas?

Well, the mirror in your makeup kit or reflecting telescope has paraboliccross-sections. The cables in suspension bridges approximate parabolas. Butprobably the most important application of parabolas is how they describe pro-jectile motion:

Of course, if we are actually interested in projectile motion, we are probablymost interested in two specific questions:

(1) At a given time, how fast is the object moving?

(2) At a given time, what direction is the object moving in?

12


Question Why are we interested in the above questions?

?The two questions above are directly related to the idea of a tangent line.

So if we are interested in the two above questions, then we are interested intangent lines.

Question What is a tangent line?

?So we want to know about parabolas and tangent lines of parabolas. We

will now look at these ideas in different ways using:

(1) Synthetic Geometry.

(2) Algebraic Geometry.

(3) Analytic Geometry.

1.2.1 Synthetic Geometry

When you study geometry without the use of a coordinate system (that is,an (x, y)-plane) you are studying synthetic geometry. Good examples of thisare when you study properties of triangles, circles, compass and straightedgeconstructions, or any other idea that goes back to classical Greek geometry.

Definition When studying synthetic geometry, the classical way to define aparabola is as a special slice of a cone:

Hence people often refer to a parabola as a conic-section.

13

1.2. POINTS OF VIEW

Question What curves do you get if you cut the cone some other way?

?Now how do we study lines that are tangent to some slice of a cone? I don’tknow! But here is another way to think about a parabola using synthetic ge-ometry that makes the job easy. Check this out:

How do we know that the above picture is a parabola? Well, we would need toprove this, but we will not do that here. So if you accept that the above pictureis a parabola, then you can just see those tangent lines.

But this interpretation really doesn’t help us solve problems. We need amore sophisticated approach.

1.2.2 Algebraic Geometry

In the 1600s, Rene Descartes revolutionized geometry. Descartes was a philoso-pher and a mathematician. Outside of mathematics, he is most famous for hisphrase:

Je pense, donc je suis.

Which is often translated as:

I think, therefore I am.

With that statement, Descartes was laying the foundations for his futurearguments on the nature of the universe around him, with his first argumentbeing that he, the arguer, actually exists. This rigor that Descartes employs isno doubt inherited from Euclid and other Greek mathematicians.

However, Descartes’ connection to geometry does not stop there. Descartesis best known in mathematics as the inventor of the (x, y)-plane, also called theCartesian plane in his honor. The (x, y)-plane was a brilliant breakthrough as itallowed geometry to be combined with algebra in ways that were not previouslyimagined.

Question What is a parabola?

14


Definition Algebraically, a parabola is the graph of:

y = ax2 + bx+ c


To answer this question, we must ask ourselves, “how do lines go throughparabolas?” Look at this:

Of the lines that intersect a parabola, most go through two points—ignorevertical lines. Take one of the good lines, one that intersects the parabola attwo points. We can slide this line around, without changing the slope, until itintersects the parabola at only one point. This line is tangent to the parabola.

Question A tangent line will go through a parabola at exactly one point. Isthis true for tangent lines of all curves?

?Question If line(x) is a line that goes through the parabola, how many rootsdoes the equation

parabola(x) = line(x)

have? How many roots does

parabola(x)− line(x) = 0

have?

?

15

1.2. POINTS OF VIEW

Example So suppose you wish to find the line tangent to the parabola y = x2

at x = 2. To do this, write

x2 − line(x) = (x− 2)(x− 2),

since x2 − line(x) must have a double-root at x = 2. Now we see that

x2 − line(x) = x2 − 4x+ 4

line(x) = 4x− 4.

So, the line tangent to the parabola y = x2 at x = 2 is line(x) = 4x− 4.

What if you want to find the line tangent to a higher degree polynomial? Inthat case, life gets a bit harder, but not impossible. Again we need to ask thequestion:


I’ll take this one. Above we see that a tangent line to a parabola is a linethat passes through the parabola in such a way that

parabola(x) = line(x)

has a double-root. So we’ll make the following definition:

Definition Algebraically speaking, a tangent line is a line that passes througha curve such that

curve(x) = line(x)

has a double-root.

Example Suppose you wish to find the line tangent to the curve y = x3 at

x = 2. To do this write

x3 − line(x) = (x− 2)(x− 2)(x− c),

where c is some unknown constant. We know that x3 − line(x) factors with two

(x− 2) since 2 must be a double-root. Now write

x3 − line(x) = (x2 − 4x+ 4)(x− c)

x3 − line(x) = x3 − 4x2 + 4x− cx2 + 4cx− 4c.

Since there are no terms involving x2 on the left-hand side of the equals sign,

we see that

−4x2 − cx2 = 0

−4x2 = cx2

−4 = c.

16


Plugging this back in, we find

x3 − line(x) = x3 − 4x2 + 4x− (−4)x2 + 4(−4)x− 4(−4)

x3 − line(x) = x3 − 4x2 + 4x+ 4x2 − 16x+ 16

x3 − line(x) = x3 − 12x+ 16

line(x) = 12x− 16.

So we see that the line tangent to the curve y = x3 at x = 2 is line(x) = 12x−16.

Question What are the limitations of this method?

?

1.2.3 Analytic Geometry

Let me tell you a story: A young man graduates from college at the age of 23.He graduated without honors and without distinction. He then traveled to thecountry to meditate on, among other things, the following question:

Question If we know the position of an object at every instant of time,shouldn’t we know its velocity?

This question helped lead this man to develop Calculus. The year was 1665and the man was Isaac Newton. OK, but what does this have to do with whatwe’ve been talking about? It turns out that if you have a line that representsthe position of an object, then the slope of that line is the velocity of the object.So now what we want to do is look at the “slope” of a curve. How do we dothis? We look at the slope of the line tangent to the curve. Fine—but howdo you do that? Here’s the idea: Suppose you want to find the slope of thefollowing curve at x = a. Look at:

a a+s

That s up there stands for a small (near zero) number. So if we look at theslope of that line we find

slopef(x)(a, s) =f(a+ s)− f(a)

s.

17

1.2. POINTS OF VIEW

If we want to find the slope of the tangent line at x = a, all we do is plug invalues for s that get closer and closer to zero.

Example Suppose you need to find the slope of the line tangent to the curve

f(x) = x2 at the point x = 2. So you write

slopef(x)(2, s) =(2 + s)2 − 22

s.

Now you plug in values for s that approach zero. Look at this:

s = 0.1 ⇒ slopef(x)(2, 0.1) = 4.1

s = 0.01 ⇒ slopef(x)(2, 0.01) = 4.01

s = 0.001 ⇒ slopef(x)(2, 0.001) = 4.001

Ah! It looks like as s gets really close to zero that slopef(x)(2, s) = 4. Now we

should check the slope when s is a small negative number.

Question What is slopef(x)(2, s) when s is a small negative number?

?Let’s see another example:

Example Suppose you need to find the slope of the line tangent to the curve

g(x) = x3 at the point x = 2. So you write

slopeg(x)(2, s) =(2 + s)3 − 23

s.

Now you plug in values for s that approach zero. Look at this:

s = 0.1 ⇒ slopeg(x)(2, 0.1) = 12.61

s = 0.01 ⇒ slopeg(x)(2, 0.01) = 12.0601

s = 0.001 ⇒ slopeg(x)(2, 0.001) = 12.006001

Ah! It looks like as s gets really close to zero that slopeg(x)(2, s) = 12. Now we

should check the slope when s is a small negative number.

Question What is slopeg(x)(2, s) when s is a small negative number?

?If you think that the above method is a bit sloppy and imprecise, then you

are correct. How do you clean up this sloppiness? You must learn the martialart known as Calculus!

Question Can you come up with a function f(x) (a sketch will suffice) where

slopef(x)(2, 0.1), slopef(x)(2, 0.01), slopef(x)(2, 0.001),

do not approach the slope of f(x) at x = 2?

?

18


Old Enemies

In a previous math course, you may have come upon the mysterious function:

f(x) = ex

What’s the deal with this? Some common answers are:

• e is easy to work with.

• e appears naturally in real world problems.

I don’t know about you, but I was never satisfied by answers like those above.Here’s the real deal: ex is a function such that

slopeex(a, s) ≈ ea

as s gets smaller and smaller. In fact, you can get slopeex(a, s) to be as close toea as you want!

Question Suppose that for some number a, ea = 0. What would you con-clude about ex then?

?Question What does the graph of ex look like?

?Question Can you explain ex in terms of:

• Driving.

• Speed-limit signs.

• Mile-marker signs.

Hint: What would happen if you got the mile-marker signs confused with thespeed-limit signs?

?

19

1.2. POINTS OF VIEW


(1) Explain the differences between the synthetic, algebraic, and analytic ap-proaches to geometry.

(2) Explain how to define a parabola knowing a point and a line.

(3) What is a tangent line?

(4) Explain how to define a parabola using conic-sections.

(5) Draw a parabola given two lines using tangent lines.

(6) Give an algebraic definition of a tangent line.

(7) Given:

3x7−x5+x4−16x3+27 = a7x7+a6x

6+a5x5+a4x

4+a3x3+a2x

2+a1x1+a0

Find a0, a1, a2, a3, a4, a5, a6, a7.

(8) Given:

6x5 + a4x4 − x2 + a0 = a5x

5 − 24x4 + a3x3 + a2x

2 − 5

Find a0, a1, a2, a3, a4, a5.

(9) Algebraically find the line tangent to y = x2 at the point x = 2. Explainyour work.

(10) Algebraically find the line tangent to y = x2 − 3x+ 1 at the point x = 3.Explain your work.

(11) Algebraically find the line tangent to y = x2 +12x− 4 at the point x = 0.Explain your work.

(12) Algebraically find the line tangent to y = −x2+4x−2 at the point x = 0.Explain your work.

(13) Algebraically find the line tangent to y = x2 at the point x = P , in termsof P . Explain your work.


(15) Algebraically find the line tangent to y = x3 − 3x2 + 4x − 1 at the pointx = 0. Explain your work.

(16) Algebraically find the line tangent to y = x3 +5x2 +2 at the point x = 1.Explain your work.

(17) Algebraically find the line tangent to y = x20 − 23x + 4 at the pointx = 0. Explain your work. Hint: If you have trouble with this one, doProblems (7) and (8) above.

20


(18) Algebraically find the line tangent to y = x4+3x3− 5x2+12 at the pointx = 0. Explain your work.


(20) Explain why

slopef(x)(a, s) =f(a+ s)− f(a)

s

gives you the slope of the tangent line that passes through the point(a, f(a)), when s is near zero.

(21) For a given function f(x), write out the formula for slopef(x)(a, 0).

(22) Approximate the slope of the line tangent to the function f(x) = x2 atx = 2 to 2 decimal places. Explain your work.


(24) Approximate the slope of the line tangent to the function f(x) = x2+2x+1at x = 1 to 2 decimal places. Explain your work.


(26) Approximate the slope of the line tangent to the function f(x) = x3 +5x2 + 2 at x = 1 to 2 decimal places. Explain your work.

(27) True or False: Explain your conclusions.

(a) A tangent line can intersect a curve at more than 1 point.

(b) Any line which intersects a curve at exactly one point is a tangentline.

(c) Some points on the graph of a function might not have a tangentline.

(d) Any quadratic equation will always have 2 distinct roots.

(e) If x10 − x + 1 − line(x) = x2g(x) where g(x) is a polynomial, thenline(x) = −x+ 1.

(28) Give an example of a curve C and a line ℓ where ℓ is not a tangent line ofC at any point and only intersects C at a single point. Clearly label yoursketch.

(29) Give an example of a curve C and a line ℓ where ℓ is a tangent line of Cat some point, but ℓ also intersects C in exactly 4 points. Clearly labelyour sketch.

21

1.2. POINTS OF VIEW

(30) Can you come up with a function f(x) (a sketch will suffice) where

slopef(x)(2, 0.1) = 0 and slopef(x)(2, 0.01) = 1?

Explain your answer.


slopef(x)(2, 0.1) = 0, slopef(x)(2, 0.01) = 1, slopef(x)(2, 0.001) = 0?

Explain your answer.


slopef(x)(2, 0.1) = 0, slopef(x)(2, 0.01) = 0, slopef(x)(2, 0.001) = 0,

but the slope of f(x) at x = 2 is 1? Explain your answer.

(33) Approximate the slope of the line tangent to the function f(x) = ex atx = 1 to 2 decimal places. Recall that e = 2.718281828459045 . . . . Explainyour work.



(36) Suppose that for some number a, ea = 0. In light of the previous threequestions, what would you conclude about ex then? Explain your answer.

(37) What does the graph of ex look like?

(38) Explain ex in terms of a combination of the following:

• Driving.

• Speed-limit signs.

• Mile-marker signs.

22


1.3 City Geometry

One day I was walking through the city—that’s right, New York City. I had themost terrible feeling that I was lost. I had just passed a Starbucks Coffee on myleft and a Sbarro Pizza on my right, when what did I see? Another Starbucks

Coffee and Sbarro Pizza! Three options occurred to me:

(1) I was walking in circles.

(2) I was at the nexus of the universe.

(3) New York City had way too many Starbucks and Sbarro Pizzas !

Regardless, I was lost. My buddy Joe came to my rescue. He pointed out thatthe city is organized like a grid.

“Ah! City Geometry!” I exclaimed. At this point all Joe could say was“Huh?”

Question What the heck was I talking about?

Most cities can be viewed as a grid of city blocks:

In City Geometry we have points and lines, just like in Euclidean Geometry.However, since we can only travel on city blocks, the distance between pointsis computed in a bit of a strange way. We don’t measure distance as the crowflies. Instead we use the Taxicab distance:

Definition Given two points A = (ax, ay) and B = (bx, by), we define theTaxicab distance as:

dT (A,B) = |ax − bx|+ |ay − by|

The approach taken in this section was adapted from [15].

23

1.3. CITY GEOMETRY

Example Consider the following points:

Let A = (0, 0). Now we see that B = (7, 4). Hence

dT (A,B) = |0− 7|+ |0− 4|= 7 + 4

= 11.

Of course in real life, you would want to add in the appropriate units to your

final answer.

Question How do you compute the distance between A and B as the crowflies?

?Here’s the scoop: When we consider our points and lines to be like those

in Euclidean Geometry, but when we use the Taxicab distance, we are workingwith City Geometry.

Question Compare and contrast the notion of a line in Euclidean Geometryand in City Geometry. In either geometry is a line the unique shortest pathbetween any two points?

?If you are interested in real-world types of problems, then maybe City Ge-

ometry is the geometry for you. The concepts that arise in City Geometry aredirectly applicable to everyday life.

Question Will just bought himself a brand new gorilla suit. He wants toshow it off at three parties this Saturday night. The parties are being heldat his friends’ houses: the Antidisestablishment (A), Hausdorff (H), and the

24


Wookie Loveshack (W ). If he travels from party A to party H to party W , howfar does he travel this Saturday night?

Solution We need to compute

dT (A,H) + dT (H,W )

Let’s start by fixing a coordinate system and making A the origin. Then H is(2,−5) and W is (−10,−2). Then

dT (A,H) = |0− 2|+ |0− (−5)|= 2 + 5

= 7

and

dT (H,W ) = |2− (−10)|+ | − 5− (−2)|= 12 + 3

= 15.

Will must trudge 7 + 15 = 22 blocks in his gorilla suit. �

1.3.1 Getting Work Done

Okay, that’s enough monkey business. Time to get some work done.

Question Brad and Melissa are going to downtown Champaign, Illinois.Brad wants to go to Jupiter’s for pizza (J) while Melissa goes to Boardman’s

Art Theater (B) to watch a movie. Where should they park to minimize the

25

1.3. CITY GEOMETRY

total distance walked by both?

Solution Again, let’s set up a coordinate system so that we can say whatpoints we are talking about. If J is (0, 0), then B is (−5, 4).

No matter where they park, Brad and Melissa’s two paths joined together mustmake a path from B to J . This combined path has to be at least 9 blocks longsince dT (B, J) = 9. They should look for a parking spot in the rectangle formedby the points (0, 0), (0, 4), (−5, 0), and (−5, 4).

Suppose they park within this rectangle and call this point C. Melissa nowwalks 4 blocks from C to B and Brad walks 5 blocks from C to J . The twopaths joined together form a path from B to J of length 9.

If they park outside the rectangle described above, for example at point D,then the corresponding path from B to J will be longer than 9 blocks. Anypath from B to J going through D goes a block too far west and then has tobacktrack a block to the east making it longer than 9 blocks. �

Question If we consider the same question in Euclidean Geometry, what isthe answer?

?Question Tom is looking for an apartment that is close to Altgeld Hall (H)but is also close to his favorite restaurant, Crane Alley (C). Where should Tom

26


live?

Solution If we fix a coordinate system with its origin at Altgeld Hall, H,then C is at (8, 2). We see that dT (H,C) = 10. If Tom wants to live as closeas possible to both of these, he should look for an apartment, A, such thatdT (A,H) = dT (A,C) = 5. He would then be living halfway along one of theshortest paths from Altgeld to the restaurant. Mark all the points 5 blocks awayfrom H. Now mark all the points 5 blocks away from C.

We now see that Tom should check out the apartments near (5, 0), (4, 1), and(3, 2). �

Question Johann is starting up a new business, Cafe Battle Royale. Heknows mathematicians drink a lot of coffee so he wants it to be near AltgeldHall. Balancing this against how expensive rent is near campus, he decides thecafe should be 3 blocks from Altgeld Hall. Where should his cafe be located?

Solution What are the possibilities? The cafe could be 3 blocks due northor due south of Altgeld Hall, which is labeled A in the figure below. It couldbe also be 2 blocks north and 1 block west or 1 block north and 2 blocks west.

27

1.3. CITY GEOMETRY

Continuing in this fashion, we obtain the following figure:

Johann can have his coffee shop on any of the point above surrounding AltgeldHall. �

1.3.2 (Un)Common Structures

How different is life in City Geometry from life in Euclidean Geometry? In thissection we’ll try to find out!

Triangles

If we think back to Euclidean Geometry, we may recall some lengthy discussionson triangles. Yet so far, we have not really discussed triangles in City Geometry.

Question What does a triangle look like in City Geometry and how do youmeasure its angles?

I’ll take this one. Triangles look the same in City Geometry as they do in Eu-clidean Geometry. Also, you measure angles in exactly the same way. However,there is one minor hiccup. Consider these two triangles in City Geometry:

28


Question What are the lengths of the sides of each of these triangles? Whyis this odd?

?

Hence we see that triangles are a bit funny in City Geometry.

Circles

Circles are also discussed in many geometry courses and this course is no dif-ferent. However, in City Geometry the circles are a little less round. The firstquestion we must answer is the following:

Question What is a circle?

Well, a circle is the collection of all points equidistant from a given point.So in City Geometry, we must conclude that a circle of radius 2 would look like:

Question How many points are there at the intersection of two circles inEuclidean Geometry? How many points are there at the intersection of twocircles in City Geometry?

?

Midsets

Definition Given two points A and B, their midset is the set of points thatare an equal distance away from both A and B.

Question How do we find the midset of two points in Euclidean Geometry?How do we find the midset of two points in City Geometry?

29

1.3. CITY GEOMETRY

In Euclidean Geometry, we just take the the following line:

If we had no idea what the midset should look like in Euclidean Geometry, wecould start as follows:

• Draw circles of radius r1 centered at both A and B. If these circles inter-sect, then their points of intersection will be in our midset. (Why?)

• Draw circles of radius r2 centered at both A and B. If these circles inter-sect, then their points of intersection will be in our midset.

• We continue in this fashion until we have a clear idea of what the midsetlooks like. It is now easy to check that the line in our picture is indeedthe midset.

How do we do it in City Geometry? We do it basically the same way.

Example Suppose you wished to find the midset of two points in City Geom-

etry.

We start by fixing coordinate axes. Considering the diagram below, if A =(0, 0), then B = (5, 3). We now use the same idea as in Euclidean Geometry.

Drawing circles of radius 3 centered at A and B respectively, we see that there

are no points 3 points away from both A and B. Since dT (A,B) = 8, this is

to be expected. We will need to draw larger Taxicab circles before we will find

points in the midset. Drawing Taxicab circles of radius 5, we see that the points

30


(1, 4) and (4,−1) are both in our midset.

Now it is time to sing along. You draw circles of radius 6, to get two more

points (1, 5) and (4,−2). Drawing circles with larger radii yields more and

more points “due north” of (1, 5) and “due south” of (4,−2). However, if we

draw circles of radius 4 centered at A and B respectively, their intersection is

the line segment between (1, 3) and (4, 0). Unlike Euclidean circles, distinct

City Geometry circles can intersect in more than two points and City Geometry

midsets can be more complicated than their Euclidean counterparts.

Question How do you draw the City Geometry midset of A and B? Whatcould the midsets look like?

?Parabolas

Recall that a parabola is a set of points such that each of those points is thesame distance from a given point, A, as it is from a given line, L.

31

1.3. CITY GEOMETRY

This definition still makes sense when we work with Taxicab distance insteadof Euclidean distance.

Draw a line parallel to L at Taxicab distance r away from L. Now draw aCity circle of radius r centered at A. The points of intersection of this line andthis circle will be r away from L and r away from A and so will be points onour City parabola. Repeat this process for different values of r.

Unlike the Euclidean case, the City parabola need not grow broader andbroader as the distance from the line increases. In the picture above, as we gofrom B to C on the parabola, both the Taxicab and Euclidean distances to theline L increase by 1. The Taxicab distance from the point A also increases by 1as we go from B to C but the Euclidean distance increases by less than 1. Forthe Euclidean distance from A to the parabola to keep increasing at the samerate as the distance to the line L, the Euclidean parabola has to keep spreadingto the sides.

Question How do you draw City Geometry parabolas? What do differentparabolas look like?

?

A Paradox

To be completely clear on what a paradox is, here is the definition we will beusing:

Definition A paradox is a statement that seems to be contradictory. Thismeans it seems both true and false at the same time.

There are many paradoxes in mathematics. By studying them we gaininsight—and also practice tying our brain into knots! Here is the first para-dox we will study in this course:

32


Paradox√2 = 2.

False-Proof Consider the following sequence of diagrams:

On the far right-hand side, we see a right-triangle. Suppose that the lengths ofthe legs of the right-triangle are one. Now by the Pythagorean Theorem, thelength of the hypotenuse is

√12 + 12 =

√2.

However, we see that the triangles coming from the left converge to thetriangle on the right. In every case on the left, the stair-step side has length2. Hence when our sequence of stair-step triangles converges, we see that thehypotenuse of the right-triangle will have length 2. Thus

√2 = 2. �

Question What is wrong with the proof above?

?

33

1.3. CITY GEOMETRY


(1) Given two points A and B in City Geometry, does dT (A,B) = dT (B,A)?Explain your reasoning.

(2) Explain how City Geometry shows that Euclid’s five axioms are not enoughto determine all of the familiar properties of the plane.

(3) Do Euclid’s axioms hold in City Geometry? How would you change theseaxioms so that they do hold in City Geometry?

(4) Brad and Melissa are going to downtown Champaign. Brad wants to goto Jupiter’s for pizza while Melissa goes to Boardman’s to watch a movie.Where should they park to minimize the total distance walked by bothand Brad insists that Melissa should not have to walk a longer distancethan him?

(5) Brad and Melissa are going to downtown Champaign. Brad wants to goto Jupiter’s for pizza while Melissa goes to Boardman’s to watch a movie.Where should they park to minimize the total distance walked by bothand Melissa insists that they should both walk the same distance?

(6) Lisa just bought a 3-wheeled zebra-striped electric car. It has a top speedof 40 mph and a maximum range of 40 miles. Suppose that there are 4

34


blocks to a mile and she wishes to drive 4 miles from her house. Whatpoints can she reach?

(7) A group of hooligans think it would be hilarious to place a bucket on theAlma Mater’s2 head, point A. Moreover, these hooligans are currentlyat point S and wish to celebrate their accomplishment at Murphy’s Pub,point M . If there are campus police at points P and Q, what path shouldthe hooligans take from S to A to M to best avoid detainment for theirhijinks?

(8) Scott wants to live within 4 blocks of a cafe, C, within 5 blocks of a bar, B,and within 10 blocks of Altgeld Hall, A. Where should he go apartmenthunting?

(9) The university is installing emergency phones across campus. Whereshould they place them so that their students are never more than a blockaway from an emergency phone?

(10) Suppose that you have two triangles△ABC and△DEF in City Geometrysuch that

(a) dT (A,B) = dT (D,E).

2The Alma Mater is a statue of a “Loving Mother” at the University of Illinois.

35

1.3. CITY GEOMETRY

(b) dT (B,C) = dT (E,F ).

(c) dT (C,A) = dT (F,D).

Is it necessarily true that △ABC ≡ △DEF? Explain your reasoning.

(11) In City Geometry, if all the angles of △ABC are 60◦, is △ABC necessarilyan equilateral triangle? Explain your reasoning.

(12) In City Geometry, if two right triangles have legs of the same length,is it true that their hypotenuses will be the same length? Explain yourreasoning.

(13) Considering that π is the ratio of the circumference of a circle to its diam-eter, what is the value of π in City Geometry? Explain your reasoning.

(14) Considering that the area of a circle of radius r is given by πr2, what isthe value of π in City Geometry. Explain your reasoning.

(15) How many points are there at the intersection of two circles in EuclideanGeometry? How many points are there at the intersection of two circlesin City Geometry?

(16) What would the City Geometry equivalent of a compass be?

(17) Cafe Battle Royale, Inc. is expanding. Johann wants his potential cus-tomers to always be within 4 blocks of one of his cafes. Where should hiscafes be located?

(18) When is the Euclidean midset of two points equal to their City Geometrymidset?

(19) Find the City Geometry midset of (−2, 2) and (3, 2).

(20) Find the City Geometry midset of (−2, 2) and (4,−1).

(21) Find the City Geometry midset of (−2, 2) and (2, 2).

(22) Draw the City Geometry parabola determined by the point (2, 0) and theline y = 0.

(23) Draw the City Geometry parabola determined by the point (2, 0) and theline y = x.

(24) Find the distance in City Geometry from the point (3, 4) to the line y =−1/3x. Explain your reasoning.

(25) Draw the City Geometry parabola determined by the point (3, 0) and theline y = −2x + 6. Explain your reasoning. This problem was suggestedby Marissa Colatosti.

36


(26) There are hospitals located at A,B, and C. Ambulances should be sentto medical emergencies from whichever hospital is closest. Divide the cityinto regions in a way that will help the dispatcher decide which ambulanceto send.

(27) Find all points P such that dT (P,A) + dT (P,B) = 8. Explain your work.(In Euclidean Geometry, this condition determines an ellipse. The solutionto this problem could be called the City Geometry ellipse.)

(28) True/False: Three noncollinear points lie on a unique Euclidean circle.Explain your reasoning.

(29) True/False: Three noncollinear points lie on a unique Taxicab circle. Ex-plain your reasoning.

(30) Explain why no Euclidean circle can contain three collinear points. Cana Taxicab circle contain three collinear points? Explain your conclusion.

(31) Can you find a false-proof showing that π = 2?

37

Chapter 2

Proof by Picture

A picture is worth a thousand words.

—Unknown

2.1 Basic Set Theory

The word set has more definitions in the dictionary than any other word. Inour case we’ll use the following definition:

Definition A set is any collection of elements for which we can always tellwhether an element is in the set or not.

Question What are some examples of sets? What are some examples ofthings that are not sets?

?If we have a set X and the element x is inside of X, we write:

x ∈ X

This notation is said “x in X.” Pictorially we can imagine this as:

38

CHAPTER 2. PROOF BY PICTURE

Definition A subset Y of a set X is a set Y such that every element of Y isalso an element of X. We denote this by:

Y ⊆ X

If Y is contained in X, we will sometimes loosely say that X is bigger thanY .

Question Can you think of a set X and a subset Y where saying X is biggerthan Y is a bit misleading?

?

Question How is the meaning of the symbol ∈ different from the meaning ofthe symbol ⊆?

?

2.1.1 Union

Definition Given two sets X and Y , X union Y is the set of all the elementsin X and all the elements in Y . We denote this by X ∪ Y .

Pictorially, we can imagine this as:

2.1.2 Intersection

Definition Given two sets X and Y , X intersect Y is the set of all theelements that are simultaneously in X and in Y . We denote this by X ∩ Y .

39

2.1. BASIC SET THEORY


Question Consider the sets X and Y below:

What is X ∩ Y ?

I’ll take this one: Nothing! We have a special notation for the set with noelements, it is called the empty set. We denote the empty set by the symbol∅.

2.1.3 Complement

Definition Given two sets X and Y , X complement Y is the set of all theelements that are in X and are not in Y . We denote this by X − Y .


40


Question Check out the two sets below:

What is X − Y ? What is Y −X?

?

2.1.4 Putting Things Together

OK, let’s try something more complex:

Question Prove that:

X ∪ (Y ∩ Z) = (X ∪ Y ) ∩ (X ∪ Z)

Proof Look at the left-hand side of the equation first:

41


And so we see:

Now look at the right-hand side of the equation:

And:

42


So we see that:

Comparing the diagrams representing the left-hand and right-hand sides of theequation, we see that we are done. �

43



(1) Given two sets X and Y , explain what is meant by X ∪ Y .

(2) Given two sets X and Y , explain what is meant by X ∩ Y .

(3) Given two sets X and Y , explain what is meant by X − Y .

(4) Explain the difference between the symbols ∈ and ⊆.

(5) Prove that:X = (X ∩ Y ) ∪ (X − Y )

(6) Prove that:X − (X − Y ) = (X ∩ Y )

(7) Prove that:X ∪ (Y −X) = (X ∪ Y )

(8) Prove that:X ∩ (Y −X) = ∅

(9) Prove that:

(X − Y ) ∪ (Y −X) = (X ∪ Y )− (X ∩ Y )

(10) Prove that:X ∪ (Y ∩ Z) = (X ∪ Y ) ∩ (X ∪ Z)

(11) Prove that:X ∩ (Y ∪ Z) = (X ∩ Y ) ∪ (X ∩ Z)

(12) Prove that:X − (Y ∩ Z) = (X − Y ) ∪ (X − Z)

(13) Prove that:X − (Y ∪ Z) = (X − Y ) ∩ (X − Z)

(14) If X ∪ Y = X, what can we say about the relationship between the setsX and Y ? Explain your reasoning.

(15) If X ∪ Y = Y , what can we say about the relationship between the setsX and Y ? Explain your reasoning.

(16) If X ∩ Y = X, what can we say about the relationship between the setsX and Y ? Explain your reasoning.

(17) If X ∩ Y = Y , what can we say about the relationship between the setsX and Y ? Explain your reasoning.

(18) If X − Y = ∅, what can we say about the relationship between the setsX and Y ? Explain your reasoning.

(19) If Y −X = ∅, what can we say about the relationship between the setsX and Y ? Explain your reasoning.

44


2.2 Logic

Logic is a great tool to have around. It turns out that we can solve lots of logicalproblems using simple tables. Moreover, one can often look at logic using theideas of Set Theory that we learned in the previous section.

When working with logic, there are certain buzz words you need to be onthe watch for:

not—, —and—, —or—, if—, then—, —if and only if—.

Any time you see the above buzz words you need to stop and think. We willaddress each of these words in turn.

The first buzz word above is not. Suppose you have a statement:

P = I love math!

We use symbol ¬ to mean not. To negate the above statement, you just putthe ¬ in front of the P :

¬P = It is not the case that I love math.

When is ¬P true? Well, only when P is false. We can display this with atruth-table:

P ¬PT FF T

When you apply not to a statement, it simply swaps true for false in the truth-tables:

Now consider the statement:

I’m strong︸︷︷︸

P

and︸︷︷︸

∧

I’m cool.︸︷︷︸

Q

and let P = I’m strong, while Q = I’m cool. When is the above statement true?Well it is only true when both P and Q are true. We can display this with atruth-table. Note that we use the symbol ∧ to mean and:

P Q P ∧QT T TT F FF T FF F F

Now what if we want to look at the statement:

I eat ice cream︸︷︷︸

P

or︸︷︷︸

∨

I eat cookies.︸︷︷︸

Q

When is the above statement true? It is true when either P or Q are true.In fact it is true even when they are both true. We can display this with atruth-table. Note that we use the symbol ∨ to mean or:

45

2.2. LOGIC

P Q P ∨QT T TT F TF T TF F F

If you look at the truth-tables for both and and or you see a sort of symmetry.This can best explained by the use of not.

WARNING Applying not changes an and to an or and vice versa.

So if we have the statement:

I’m strong︸︷︷︸

P

and︸︷︷︸

∧

I’m cool.︸︷︷︸

Q

Then

¬(P ∧Q) = (¬P ) ∨ (¬Q)

= I’m not strong︸︷︷︸

¬P

or︸︷︷︸

∨

I’m not cool.︸︷︷︸

¬Q

We can see this best using a truth-table:

P Q P ∧Q ¬(P ∧Q) ¬P ¬Q (¬P ) ∨ (¬Q)T T T F F F FT F F T F T TF T F T T F TF F F T T T T

Question What does the truth-table for ¬(P ∨Q) and (¬P )∧(¬Q) look like?

?While and, or, and not really aren’t all that bad, if-then is much trickier.

Allow me to demonstrate how tricky if-then can be. I will do this with theWason selection task : Suppose I had a set of cards each with a number on oneside and a letter on the other side, and I laid four of them on a table in front ofyou:

A 7 B 6

Consider the statement:

If one side of the card shows an even number, then the other side ofthe card shows a vowel.

46


Question Exactly which card(s) above do you need to flip over to see whethermy statement is true?

?Now suppose you are a police officer at a local bar that has four tables. At

the first table nobody is drinking alcohol, at the second table every customerlooks quite old, at the third table there are many pitchers of beer, and at thefourth table everybody looks quite young:

youngbeer

oldnopeoplepeoplealcohol

Consider the law:

If you are under 21, then you cannot drink alcohol.

Question Exactly which table(s) do you need to check to see if the law isbeing upheld?

?Question Are the two questions above different?

?Question Of the two questions above, which one was easier?

?I think that the two situations above involving if-then statements show that

we need to be careful when dealing with them, especially when the situation issomewhat abstract. Let’s look at the truth-tables for if-then. Note that we usethe symbol ⇒ to mean if-then:

P Q P ⇒ QT T TT F FF T TF F T

To make P ⇒ Q easier to read it is sometimes helpful to read it P implies Q.A curious fact is that often the easiest way to negate an if-then statement

is to rewrite it in terms of or and not:

47

2.2. LOGIC

P Q ¬P ¬P ∨Q P ⇒ QT T F T TT F F F FF T T T TF F T T T

From the truth-table we see that

P ⇒ Q = ¬P ∨Q.

Now we can negate this easily:

¬(P ⇒ Q) = ¬(¬P ∨Q) = P ∧ (¬Q).

Finally if you see if-and-only-if, denoted by the symbol ⇔, this is nothingmore than:

P ⇔ Q = (P ⇒ Q) ∧ (Q ⇒ P ).

Question Can you connect the ideas in this section to ideas in Set Theory?Specifically, let a statement be a set. The “points” that make it true are whatare inside the set. What do

not—, —and—, —or—, if—, then—, —if and only if—,

look like?

?

48



(1) Knowing that P ⇔ Q = (P ⇒ Q) ∧ (Q ⇒ P ), write a truth-table forP ⇔ Q.

(2) Use a truth-table to show that ¬(P ∨Q) = (¬P ) ∧ (¬Q).

(3) Use a truth-table to show that (P ⇒ Q) 6= (Q ⇒ P ).

(4) Explain why P ⇒ Q is not the same as Q ⇒ P by giving a real-worldsentence for P and a real world sentence for Q and analyzing what P ⇒ Qand Q ⇒ P mean.

(5) Use a truth-table to show that P ⇒ Q = (¬Q) ⇒ (¬P ).

(6) Explain why P ⇒ Q is the same as (¬Q) ⇒ (¬P ) by giving a real-worldsentence for P and a real world sentence for Q and analyzing what P ⇒ Qand (¬Q) ⇒ (¬P ) mean.

(7) Go out and find some friends. Set up the card example as explained above.See how many of them can get it right. Then set up the example of thetables at a bar as explained above. How many get it right now?

(8) Suppose I give you the statement:

If your name is Agatha, then you like to eat tomatoes.

Which of the following don’t contradict the above statement:

(a) Your name is Agatha and you like to eat tomatoes.

(b) Your name is Agatha and you don’t like to eat tomatoes.

(c) Your name is Joe and you like to eat tomatoes.

(d) Your name is Joe and you don’t like to eat tomatoes.

(9) Suppose I give you the statement:

If your name is Jen, then you have a cat named Hypie.

Which of the following don’t contradict the above statement:

(a) Your have a cat named Hypie and your name is Jen.

(b) Your have a cat named Hypie and your name is Joe.

(c) Your have no cats and your name is Jen.

(d) Your have no cats and your name is Joe.

(10) Give an if-then statement involving traffic laws and use it in an exampleto explain why (false ⇒ true) is a true statement. Explain your answer.

(11) Give an if-then statement involving traffic laws and use it in an exampleto explain why (false ⇒ false) is a true statement. Explain your answer.

49

2.2. LOGIC

(12) Let P and Q be true statements, and let X and Y be false statements.Determine the truth value of the following statements:

(a) P ∧ Y .

(b) X ∨Q.

(c) P ⇒ Q.

(d) X ⇒ Y .

(e) Y ⇒ Q.

(f) P ⇒ Y .

(g) P ⇒ (X ∨ Y ).

(h) ¬P ⇒ P .

(i) (X ∧Q) ⇒ Y .

(j) (P ∨ Y ) ⇒ Q.

(k) ¬(P ∧ (¬P )).

(l) ¬(X ∨ (¬X)).

(13) Here is the truth-table for neither-nor, denoted by the symbol ×:

P Q P ×QT T FT F FF T FF F T

(a) Make a truth-table for P × P .

(b) Make a truth-table for (P ×Q)× (P ×Q).

(c) Make a truth-table for (P × P )× (Q×Q).

(d) Make a truth-table for ((P × P )×Q)× ((P × P )×Q).

Use your work above to express not, and, or, and if-then purely in termsof neither-nor.

(14) Draw pictures showing the connection between intersection and and, andunion and or. What does not look like? What does if-then look like?What does if-and-only-if look like?

50


2.3 Tessellations

Go to the internet and look up M.C. Escher. He was an artist. Look at some ofhis work. When you do your search be sure to include the word “tessellation”OK? Back already? Very good. With some of Escher’s work he started with atessellation. What’s a tessellation? I’m glad you asked:

Definition A tessellation is a pattern of polygons fitted together to cover theentire plane without overlapping. A tessellation is called a regular tessellationif the polygons are regular and they have common vertexes.

Example Here are some examples of regular tessellations:

Johannes Kepler was one of the first people to study tessellations. He cer-tainly knew the next theorem:

Theorem 2 There are only 3 regular tessellations.

Since one can prove that there are only three regular tessellations, and wehave shown three above, then that is all of them. On the other hand there arelots of nonregular tessellations. Here are two different ways to tessellate theplane with a triangle:

Here is a way that you can tessellate the plane with any old quadrilateral:

51

2.3. TESSELLATIONS

2.3.1 Tessellations and Art

How does one make art with tessellations? To start, a little decoration goes along way. Check this out: Decorate two squares as such:

Tessellate them randomly in the plane to get this lightning-like picture:

Question What sort of picture do you get if you tessellate these decoratedsquares randomly in a plane?

?

Another way to go is to start with your favorite tessellation:

52


Then you modify it a bunch to get something different:

Question What kind of art can you make with tessellations?

?

53

2.3. TESSELLATIONS


(1) Show two different ways of tessellating the plane with a given scalenetriangle. Label your picture as necessary.

(2) Show how to tessellate the plane with a given quadrilateral. Label yourpicture.

(3) Show how to tessellate the plane with a nonregular hexagon. Label yourpicture.

(4) Give an example of a polygon with 9 sides that tessellates the plane.

(5) Give examples of polygons that tessellate and polygons that do not tes-sellate.

(6) Give an example of a triangle that tessellates the plane where both 4 and8 angles fit around each vertex.


(a) There are exactly 5 regular tessellations.

(b) Any quadrilateral tessellates the plane.

(c) Any triangle will tessellate the plane.

(d) If a triangle is used to tessellate the plane, then it is always the casethat exactly 6 angles will fit around each vertex.

(e) If a polygon has more than 6 sides, then it cannot tessellate the plane.

(8) Fill in the following table:

Regular Does it Measure If it tessellates, hown-gon tessellate? of angles many surround each vertex?

3-gon4-gon5-gon6-gon7-gon8-gon9-gon10-gon

Hint: A regular n-gon has interior angles of 180(n− 2)/n degrees.

(a) What do the shapes that tessellate have in common?

(b) Make a graph with the number of angles on the horizontal axis andthe measure of the angles on the vertical axis.

54


(c) What regular polygons could a bee use for building hives? Give somereasons that bees seem to use hexagons.

(9) Given a regular tessellation, what is the sum of the angles around a givenvertex?

(10) Given that the regular octagon has 135 degree angles, explain why youcannot give a regular tessellation of the plane with a regular octagon.

(11) Considering that the regular n-gon has interior angles of 180(n − 2)/ndegrees, and Problem (8) above, prove that there are only 3 regular tes-sellations of the plane.

55

2.4. PROOF BY PICTURE

2.4 Proof by Picture

Pictures generally do not constitute a proof on their own. However, a goodpicture can show insight and communicate concepts better than words alone.In this section we will show you pictures giving the idea of a proof and then askyou to supply the words to finish off the argument.

2.4.1 Proofs Involving Right Triangles

Let’s start with something easy:

Question Explain how the following picture “proves” that the area of a righttriangle is half the base times the height.

?That wasn’t so bad was it? Now for a game of whose-who:

Question What is the most famous theorem in mathematics?

Probably the Pythagorean Theorem comes to mind. Let’s recall the state-ment of the Pythagorean Theorem:

Theorem 3 (Pythagorean Theorem) Given a right triangle, the sum of the

squares of the lengths of the two legs equals the square of the length of the

hypotenuse. Symbolically, if a and b represent the lengths of the legs and c is

the length of the hypotenuse,

ac

b

then

a2 + b2 = c2.

Nearly all of the pictures from this section are adapted from the wonderful source books:[18] and [19].

56


Question What is the converse to the Pythagorean Theorem? Is it true?How do you prove it?

?

While everyone may know the Pythagorean Theorem, not as many knowhow to prove it. Euclid’s proof goes kind of like this:

Consider the following picture:

c2

b2

a2

Now, cut up the squares a2 and b2 in such a way that they fit into c2 perfectly.When you give a proof that involves cutting up the shapes and putting themback together, it is called a dissection proof. The trick to ensure that thisis actually a proof is in making sure that your dissection will work no matterwhat right triangle you are given. Does it sound complicated? Well it can be.

57


Is there an easier proof? Sure, look at:

Question How does the picture above “prove” the Pythagorean Theorem?

Solution Both of the large squares above are the same size. Moreover boththe unshaded regions above must have the same area. The large white squareon the left has an area of c2 and the two white squares on the right have acombined area of a2 + b2. Thus we see that:

c2 = a2 + b2

�

Let’s give another proof! This one looks at a tessellation involving 2 squares.

��

��

Question How does the picture above “prove” the Pythagorean Theorem?

Solution The striped triangle is our right triangle. The area of the overlaidsquare is c2, the area of the small squares is a2, and the area of the medium

58


square is b2. Now label all the “parts” of the large overlaid square:

��

��

2

3

4

1

5

From the picture we see that

a2 = {3 and 4}b2 = {1, 2, and 5}c2 = {1, 2, 3, 4, and 5}

Hence

c2 = a2 + b2

Since we can always put two squares together in this pattern, this proof willwork for any right triangle. �

Question Can you use the above tessellation to give a dissection proof of thePythagorean Theorem?

?

Now a paradox:

59


Paradox What is wrong with this picture?

Question How does this happen1?

?

2.4.2 Proofs Involving Boxy Things

Consider the problem of Doubling the Cube. If a mathematician asks us todouble a cube, he or she is asking us to double the volume of a given cube.One may be tempted to merely double each side, but this doesn’t double thevolume!

Question Why doesn’t doubling each side of the cube double the volume ofthe cube?

?1See [10] Chapter 8, for a wonderful discussion of puzzling pictures like this one.

60


Well, let’s answer an easier question first. How do you double the area of asquare? Does taking each side and doubling it work?

No! You now have four times the area. So you cannot double the area of asquare merely by doubling each side. What about for the cube? Can you doublethe volume of a cube merely by doubling the length of every side? Check thisout:

Ah, so the answer is again no. If you double each side of a cube you have 8times the volume.

The Arithmetic-Geometric Mean Inequality

The arithmetic mean of two numbers is just the average of those two numbers.However, the geometric mean is a bit more mysterious. Essentially with thegeometric mean, you are “squaring” a rectangle. What do we mean by this?We mean that you are finding a square whose area is the same as the originalrectangle.

Question Suppose you have have a rectangle with sides a and b. What is theside length of the square whose area is the same as that rectangle?

?Theorem 4 (Arithmetic-Geometric Mean Inequality) If a and b are positivenumbers then: √

a · b 6 a+ b

2

and the inequality above is an equality when a = b.

61


Question Can you state the above theorem in English? Can you give someexamples of how it is true? Can you show me a graph?

Now look at this picture:

b

a

a b

Question How does the picture above “prove” the Arithmetic-GeometricMean Inequality?

Solution Consider the area of the large square. This is

(a+ b)2.

On the other hand, the area of the four smaller rectangles is 4ab. Since wecan see from the picture that this area is less than the area of the large square(unless of course a = b), we have

4ab 6 (a+ b)2

ab 6(a+ b)2

4√a · b 6 a+ b

2,

which is what we wanted to show. �

OK, but in mathematics we really want to know that we are correct. So todo this, we will often give as many proofs of the same theorem as possible. Lookat this picture:

Question How does the picture above “prove” the Arithmetic-GeometricMean Inequality?

Solution As usual, in the above right-triangles, let the short leg be of lengtha, and the longer leg—that is not the hypotenuse, be of length b. Now the areaof the big square is

(a+ b)2.

62


But the area of the all the triangles is 4ab. Hence

4ab 6 (a+ b)2

ab 6(a+ b)2

4√a · b 6 a+ b

2,

which is what we wanted to show. Note that if a = b, then the triangles wouldfill the square and we would have equality. �

2.4.3 Proofs Involving Sums

Finite Sums

According to legend, when Gauss was in elementary school, his teacher gave theproblem of summing all the integers from 1 to 100. Supposedly Gauss gave theanswer in seconds, infuriating the teacher. How did he do it? Well he probablydid something like this:

Write the numbers in a funky array:

0 1 2 3 4 5 · · · 95 96 97 98 99 100+ + + + + + · · · + + + + + +100 99 98 97 96 95 · · · 5 4 3 2 1 0‖ ‖ ‖ ‖ ‖ ‖ · · · ‖ ‖ ‖ ‖ ‖ ‖100 100 100 100 100 100 · · · 100 100 100 100 100 100︸︷︷︸

101 terms

So if we sum up all the integers from 1 to 100 twice we get

100 · 101.

Thus, the sum of all the integers from 1 to 100 is

1 + 2 + 3 + 4 + 5 + · · ·+ 95 + 96 + 97 + 98 + 99 + 100 =100 · 101

2= 5050.

That’s nice. But it isn’t a very useful fact to know. I mean, suppose you nowwant to know what is:

1 + 2 + 3 + · · ·+ 9 + 10

Or:1 + 2 + · · ·+ 452 + 453

What we really want is a formula that somehow encapsulates what Gauss didabove.

Question What is the formula for the following sum of integers?

1 + 2 + 3 + 4 + 5 + · · ·+ n

63


?Consider this picture:

Question Explain how the picture above “proves” that:

1 + 2 + 3 + 4 + 5 + · · ·+ n =n(n+ 1)

2

Solution Since the light circles make up half of the rectangle above, and therectangle has n(n+ 1) circles in it, then the number of light circles is

n(n+ 1)

2.

However, from the picture, we can see that there are

1 + 2 + 3 + 4 + · · ·+ n

light colored circles. Therefore

1 + 2 + 3 + 4 + 5 + · · ·+ n =n(n+ 1)

2.

�

Now you may object to this proof because the specific picture shown, n is7 and not any old value. However, to this objection one could retort, that thepattern is clear, and that one need only continue the pattern to the desiredvalue of n.

Infinite Sums

As is our style, we will start off with a question:

Question Can you add up an infinite number of terms and still get a finitenumber?

64


Consider 1/3. Actually, consider the decimal notation for 1/3:

1

3= .333333333333333333333333333333 . . .

But this is merely the sum:

.3 + .03 + .003 + .0003 + .00003 + .000003 + · · ·

It stays less than 1 because the terms get so small so quickly. Are there otherinfinite sums of this sort? You bet! In fact:

1

2+

(1

2

)2

+

(1

2

)3

+

(1

2

)4

+

(1

2

)5

+ · · · = 1

Wow! How can we visualize this? Consider this picture:

1/2

1

(1/2)2

1


1

2+

(1

2

)2

+

(1

2

)3

+

(1

2

)4

+

(1

2

)5

+ · · · = 1

Solution So we take a unit square and divide it in half. This half piece hasan area of 1/2. Now look at the other half of the square, divide it in half. Thispiece has an area of (1/2)2. Look at the next half and so on. From the pictureabove we see that we will eventually fill the entire square. Therefore, summingthe areas we see:

1

2+

(1

2

)2

+

(1

2

)3

+

(1

2

)4

+

(1

2

)5

+ · · · = 1

�

65


Now look at:


1

4+

(1

4

)2

+

(1

4

)3

+

(1

4

)4

+

(1

4

)5

+ · · · = 1

3

Solution Let’s take it in steps. If the big triangle has area 1, the area of theshaded region below is 1/4.

We also see that the area of the shaded region below

is:1

4+

(1

4

)2

Continuing on in this fashion we see that the area of all the shaded regions is:

1

4+

(1

4

)2

+

(1

4

)3

+

(1

4

)4

+

(1

4

)5

+ · · ·

But look, the unshaded triangles have twice as much area as the shaded triangle.Thus the shaded triangles must have an area of 1/3. �

66


2.4.4 Proofs Involving Sequences

Get your calculator out and play along at home:

1

3=

1

31 + 3

5 + 7=?

1 + 3 + 5

7 + 9 + 11=?

1 + 3 + 5 + 7

9 + 11 + 13 + 15=?

1 + 3 + 5 + 7 + 9

11 + 13 + 15 + 17 + 19=?

Question What is happening here? Does it always happen?

?

One of the first people to study this phenomenon was the physicist GalileoGalilei. To help us understand it, look at this picture:


1 + 3 + · · ·+ (2n− 1)

(2n+ 1) + (2n+ 3) + · · ·+ (4n− 1)=

1

3

Solution Looking at the rows of the pyramid, the numerator of the fractionis represented by the top part and the denominator is represented by the bottompart. While it is not completely general since it stops at 5 circles, it is obvioushow to extend the picture to work for any number. Since we can see that thetop part of the pyramid is 1/3 of the bottom part,

1 + 3 + · · ·+ (2n− 1)

(2n+ 1) + (2n+ 3) + · · ·+ (4n− 1)=

1

3.

�

67


Now look at this:

1 · 1 = 1

11 · 11 = 121

111 · 111 = 12321

1111 · 1111 = 1234321

Question What is happening here? Does it always happen? Hint: You don’tneed a picture to figure this one out.

?

2.4.5 Thinking Outside the Box

A calisson is a French candy that sort of looks like two equilateral trianglesstuck together. They usually come in a hexagon-shaped box.

Question How do the calissons fit into their hexagon-shaped box?

If you start to put the calissons into a box, you quickly see that they can beplaced in there with exactly three different orientations:

Theorem 5 In any packing, the number of calissons with a given orientation

is exactly one-third the total number of calissons in the box.

Look at this picture:

Question How does the picture above “prove” Theorem 5? Hint: Thinkoutside the box!

?

68



(1) Explain how the following picture “proves” that the area of a right triangleis half the base times the height.

(2) Building on Problem (1), explain how the following picture “proves” thatthe area of any triangle is half the base times the height.

(3) You may have noticed Geometry Giorgio in your class. In an attempt toprove the formula for the area of a triangle, Geometry Giorgio draws thefollowing picture:

What is he doing wrong? How could he fix his “proof”?

(4) Again building on Problem (1), explain how the following picture “proves”that the area of any triangle is half the base times the height. Note, thisway of thinking is the basis for Cavalieri’s Principle.

��

��

��

��

69


(5) Explain how the following picture “proves” that the area of any parallel-ogram is base times height. Note, this way of thinking is the basis forCavalieri’s Principle.

��

��

��

��

(6) Explain how the following picture “proves” the Pythagorean Theorem.



70


Note: This proof is due to Leonardo da Vinci.


��

(10) Use the following tessellation to give a dissection proof of the PythagoreanTheorem.

��

71



ab

a

bc

b2

×a

ac ac

b

c

a2

c2bc

×b×c

(12) Recall that a trapezoid is a quadrilateral with two parallel sides. Considerthe following picture:

How does the above picture prove that the area of a trapezoid is

area =h(b1 + b2)

2,

where h is the height of the trapezoid and b1, b2, are the lengths of theparallel sides?


Note: This proof is due to James A. Garfield, the 20th President of theUnited States.

(14) Look at Problem (12). Can you use a similar picture to prove that the

72


area of a parallelogram

is the length of the base times the height?

(15) Explain how the following picture “proves” that the area of a parallelogramis base times height.

(16) You probably have noticed Geometry Giorgio in your class. In an attemptto prove the formula for the area of a parallelogram, Geometry Giorgio

draws the following picture:

What is he doing wrong? How could he fix his “proof”?

(17) Which of the above “proofs” for the formula for the area of a parallelogramis your favorite? Explain why.

(18) Explain how the following picture “proves” that if a quadrilateral has twoopposite angles that are equal, then the bisectors of the other two angles

73


are parallel or on top of each other.

(19) Explain how the following picture “proves” that the area of a quadrilateralis equal to half of the area of the parallelogram whose sides are parallel toand equal in length to the diagonals of the original quadrilateral.

74


(20) Why might someone find the following picture disturbing? How would youassure them that actually everything is good and well in the geometricalworld?

75


(21) Why might someone find the following picture disturbing? How would youassure them that actually everything is good and well in the geometricalworld?

(22) How could you explain to someone that doubling the lengths of each sideof a cube does not double the volume of the cube?

(23) Explain how the following picture “proves” that if a and b are positivenumbers then:

√a · b 6 a+ b

2


b

a

a b

(24) Explain how the following picture “proves” that the sum of a number x

76


and its inverse 1/x is at least 2.

1

x

x

1

x

x

11

1

1

(25) Explain how the following picture “proves” that if a and b are positivenumbers then: √

a · b 6 a+ b

2


(26) Explain how the following picture “proves” that:

1 + 2 + 3 + 4 + 5 + · · ·+ n =n(n+ 1)

2


1 + 2 + 3 + 4 + 5 + · · ·+ n =1

2(n2 + n) =

n(n+ 1)

2

77



1 + 2 + 3 + 4 + 5 + · · ·+ n =n2

2+

n

2=

n(n+ 1)

2


1 + 3 + 5 + · · ·+ (2n− 1) = n2

78



1 + 2 + 3 + · · ·+ (n− 1) + n+ (n− 1) + · · ·+ 3 + 2 + 1 = n2


1

2+

(1

2

)2

+

(1

2

)3

+

(1

2

)4

+

(1

2

)5

+ · · · = 1

1/2

1

(1/2)2

1

(32) Explain how the following picture “proves” that if 0 < r < 1:

r + r(1− r) + r(1− r)2 + r(1− r)3 + · · · = 1

79


1

1

r

r(1− r)

r(1− r)2

r(1− r)3

r(1− r)

r

r


1

4+

(1

4

)2

+

(1

4

)3

+

(1

4

)4

+

(1

4

)5

+ · · · = 1

3


1

2+

(1

2

)2

+

(1

2

)3

+

(1

2

)4

+ · · · = 1

80


Hint: Add up the area of the shaded regions and the area of the unshadedregions.


1 + 3 + · · ·+ (2n− 1)

(2n+ 1) + (2n+ 3) + · · ·+ (4n− 1)=

1

3


1 + 3 + · · ·+ (2n− 1)

(2n+ 1) + (2n+ 3) + · · ·+ (4n− 1)=

1

3

3

5

2n− 1

1

2n

(37) Look at:

1 · 1 = 1

11 · 11 = 121

111 · 111 = 12321

1111 · 1111 = 1234321

81


Does the pattern continue? Explain your answer.

(38) Explain how the following picture “proves” that in any packing, the num-ber of calissons with a given orientation is exactly one-third the totalnumber of calissons in the box.

82

Chapter 3

Topics in Plane Geometry

Geometry is the science of correct reasoning on incorrect figures.

—George Polya

3.1 Triangles

3.1.1 Centers in Triangles

The idea of a center for an equilateral triangle makes sense. However, for anarbitrary triangle, there can be several different ideas for what the center is.

Question How could one define the “center” of an arbitrary triangle?

?

The Circumcenter

Theorem 6 The perpendicular bisector of the sides of a triangle meet at a

point. This point is called the circumcenter.

Here is a picture illustrating the theorem above:

83

3.1. TRIANGLES

Paper-Folding Construction

To construct the circumcenter using paper-folding, perform the following steps:

(1) Fold the leg of the triangle over top of itself so that its endpoints meet.

(2) Repeat Step 1 for each of the 3 legs of the triangle.

(3) The creases made in Steps 1 and 2 above should meet at the circumcenterof the triangle.

What is it really?

The circumcenter is the center of the circle that circumscribes the triangle:

This circle is sometimes called the circumcircle.

The Incenter

Theorem 7 The interior bisectors of the angles of a triangle meet at a point.

This point is called the incenter.


To construct the incenter using paper-folding, perform the following steps:

(1) Choose a vertex of the triangle and fold the triangle over top of itself sothat legs adjacent to the chosen vertex line up.

(2) Repeat Step 1 for each of the 3 vertexes of the triangle.

84

CHAPTER 3. TOPICS IN PLANE GEOMETRY

(3) The creases made in Steps 1 and 2 above should meet at the incenter ofthe triangle.

What is it really?

If we draw a circle inside the triangle such that the circle touches each of thesides of the triangle, then the incenter is the center of this circle.

This circle is sometimes called the incircle.

The Orthocenter

Recall the following definition:

Definition An altitude of a triangle is a line segment originating at a vertexof the triangle that meets the line containing the opposite side at a right angle.

Since the perpendicular bisectors of the large triangle form the altitudes ofthe smaller triangle, we now have the following theorem:

Theorem 8 The lines containing the altitudes of a triangle meet at a point.

This point is called the orthocenter.


To construct the orthocenter using paper-folding, perform the following steps:

(1) Fold the leg of the triangle over top of itself so that the vertex oppositethe leg is on the crease.

(2) Repeat Step 1 for each of the 3 legs of the triangle.

(3) The creases made in Steps 1 and 2 above should meet at the orthocenterof the triangle.

85

3.1. TRIANGLES

The Centroid

Theorem 9 The lines of a triangle that connect the vertexes to the midpoints

of the opposite sides of a triangle meet at a point. This point is called the

centroid.

The line segments used in finding the centroid have a special name:

Definition A median of a triangle is a line segment that connects a vertexto the midpoint of the opposite side.


To construct the centroid using paper-folding, perform the following steps:

(1) Fold the leg of the triangle over top of itself so that its endpoints meet.

(2) Take the vertex opposite the leg above, and make a crease that starts atthe vertex and extends to where the crease made in Step 1 meets the legopposite the vertex.

(3) Repeat Steps 1 and 2 for each of the 3 legs of the triangle.

(4) The creases made in the paper in Steps 2 and 3 above should meet at thecentroid of the triangle.

What is it really?

The centroid is the center of mass of the triangle. The center of mass isthe balancing point of an object. That is, if we had a triangle made of saycardboard, then you could balance the cardboard triangle on its centroid.

Putting it all Together

The next turn we should make on our path is to understand the relationshipbetween the centers above.

Question Is it always/ever the case that the circumcenter, the orthocenter,the incenter, and the centroid are all actually the same point?

?

86


Question Can the circumcenter ever be outside the triangle?

?Question Can the incenter ever be outside the triangle?

?Question Can the orthocenter ever be outside the triangle?

?Question Can the centroid ever be outside the triangle?

?Keeping all these ideas straight can be tough, and unfortunately, I don’t

know of an easy way to learn them. However, a first step is knowing the namesof each of the centers described above. Here is simple mnemonic that may help:

ce

C O I N

i r n tr t c rc h e ou o n im c t dc e ee n rn tt ee rr

Note that the centers that correspond to the IN part of COIN are alwaysinside the triangle!

3.1.2 Theorems about Triangles

Now consider some triangle and look at the orthocenter, the centroid, and thecircumcenter. In the illustration below, O is the orthocenter, N is the centroid,and C is the circumcenter:

N CO

87

3.1. TRIANGLES

Do you notice anything?

Theorem 10 (Euler) The circumcenter, the centroid, and the orthocenter are

on a line. The centroid lies a third of the distance from the circumcenter to the

orthocenter. This line is called the Euler line.

What about all the other points on the above triangle? Well there are manymany theorems about all kinds of points. Here is one that relates nine differentpoints to each other:

Theorem 11 Given any triangle, three sets of three points all lie on a circle.

Those three sets are:

(1) The midpoints of the sides of the triangle.

(2) Where the altitudes meet the lines containing the sides of the triangle.

(3) Midpoints of the segments joining the orthocenter and the vertexes.

This circle is called the Nine-Point Circle.

Question How would one go about drawing this?

?Theorem 12 The center of the Nine-Point Circle bisects the segment joining

the orthocenter, point O, and the circumcenter, point C.

CO

While the previous theorems concerned themselves with points, the nexttheorem involves arcs.

88


Theorem 13 (Miquel) Consider three points, one on each side of a triangle,

none of which is on a vertex of the triangle. Then the three circles determined

by a vertex and the two points on adjacent sides meet at a point. This point is

called a Miquel point.

Question How would one go about drawing this?

?Question Why do we want the points in Miquel’s Theorem not to be on thevertexes of the triangle?

?A final theorem about triangles:

Theorem 14 (Morley) If you trisect the angles of any triangle with lines, then

those lines form a new equilateral triangle inside the original triangle.


To construct the triangle described in Morley’s Theorem using paper-folding,perform the following steps:

(1) We must trisect an angle of the triangle. Choose a vertex of the triangleand fold the paper so that the crease leads up to the vertex, with the edgeof the flap being folded over bisecting the new angle of the crease and theedge that was not moved.

89

3.1. TRIANGLES

(2) Now fold the edge that was not moved on top of the flap that was justmade. It should fit perfectly near the vertex. If done correctly, Steps 1and 2 should trisect the chosen vertex.

(3) Repeat Steps 1 and 2 for each vertex of the triangle.

(4) Now mark the adjacent intersections of creases coming from different ver-texes.

(5) Connect the marks made in Step 4 above. This should form an equilateraltriangle.

A Paradox

Here is another devilish paradox!

Paradox All triangles are isosceles.

False-Proof Consider

A

B CY

X Z

The central point is the point where the bisector of ∠XAZ meets the perpen-dicular bisector of line BC, we’ll call this point P . We want to show thatAB ≡ AC. To see this first note that

△AXP ≡ △AZP

since they have equal angles and share a side. Next note that

△BY P ≡ △Y CP.

Hence △BXP ≡ △ZCP as they are right triangles with two equal sides. ThusAB ≡ AC. �

This is a paradox as not all triangle are isosceles and yet we seem to haveproved that all triangles are isosceles! There must be something wrong withthe proof.

Question What is wrong with the proof above? Hint: Use paper-folding toconstruct the figure in the above false-proof.

?

90



(1) Use paper-folding to construct the circumcenter of a triangle.

(2) Use paper-folding to construct the orthocenter of a triangle.

(3) Use paper-folding to construct the incenter of a triangle.

(4) Use paper-folding to construct the centroid of a triangle.

(5) Explain how a perpendicular bisector is different from an altitude. Drawan example to illustrate the difference.

(6) Explain how a median different from an angle bisector. Draw an exampleto illustrate the difference.

(7) What is the name of the point that is the same distance from all threesides of a triangle? Explain your answer.

(8) What is the name of the point that is the same distance from all threevertexes of a triangle? Explain your answer.

(9) Could the circumcenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(10) Could the orthocenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(11) Could the incenter be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(12) Could the centroid be outside the triangle? If so, draw a picture andexplain. If not, explain why not using pictures as necessary.

(13) Are there shapes that do not contain their centroid? If so, draw a pictureand explain. If not, explain why not using pictures as necessary.

(14) Draw a triangle. Now draw the lines containing the altitudes of this trian-gle. How many orthocenters do you have as intersections of lines in yourdrawing? Hints:

(a) More than one.

(b) How many triangles are in the picture you drew?

(15) Where is the circumcenter of a right triangle?

(16) Where is the orthocenter of a right triangle?

(17) Can you draw a triangle where the circumcenter, orthocenter, incenter,and centroid are all the same point? If so, draw a picture and explain. Ifnot, explain why not using pictures as necessary.

91

3.1. TRIANGLES


(a) An altitude of a triangle is always perpendicular to a line containingsome side of the triangle.

(b) An altitude of a triangle always bisects some side of the triangle.

(c) The incenter is always inside the triangle.

(d) The circumcenter, the centroid, and the orthocenter always lie in aline.

(e) The circumcenter can be outside the triangle.

(f) The orthocenter is always inside the triangle.

(g) The orthocenter is always the center of the Nine-Point Circle.

(19) How many Euler-Lines does a given triangle have? Explain your reasoning.

(20) How many Nine-Point Circles does a given triangle have? Explain yourreasoning.

(21) How many Miquel points does a given triangle have? Explain your rea-soning.

(22) Sketch and label a Nine-Point Circle.

(23) Sketch and label an Euler-Line.

(24) True or False: Explain your conclusion.

(a) The centroid of a triangle always lies on the Nine-Point Circle.

(b) If a given altitude of a triangle is extended to an infinite line, thenthe Nine-Point Circle will always intersect that line in two distinctpoints.

(c) Given a triangle, the Nine-Point Circle for that triangle is the cir-cumscribed circle for a triangle whose vertexes are the midpoints ofthe sides of original triangle.

(d) A triangle does not ever share 9 points with its Nine-Point Circle.

(e) There is a triangle that only shares 3 points with its Nine-PointCircle.

(25) Does a Miquel point always lie inside the triangle? Explain your answer.

(26) Use paper-folding to illustrate Morley’s Theorem.

(27) Illustrate the statement of Morley’s Theorem using triangles with the fol-lowing angles:

(a) 60◦, 60◦, and 60◦.

(b) 30◦, 60◦, and 90◦.

(c) 15◦, 45◦, and 120◦.

Use a protractor as necessary. Give a short explanation of your illustra-tions.

92


3.2 Numbers

Numbers are synonymous with mathematics. In this section we will discussseveral ways that numbers come up in geometry.

3.2.1 Areas

Heron’s Formula

The standard formula for the area of a triangle is:

area = (1/2)base · heightCould we compute the area of a triangle just knowing the lengths of the sides?Well after a moments thought, and some sketches, one should decide that itis indeed possible, but the many different configurations of possible trianglesmight make it difficult. Enter Heron’s Formula:

Theorem 15 (Heron’s Formula) Given a triangle whose sides have length a,b, and c, then the area of the triangle is given by

area =

√(P

2− a

)(P

2− b

)(P

2− c

)(P

2

)

where P = a+ b+ c, the perimeter of the triangle.

Question Why does it make sense that a formula for the area of a trianglecan be given knowing only the sides? Could a similar statement be made for allquadrilaterals?

?Giving away part of this question, we only have similar formulas for quadri-

laterals in certain cases. Our next lemma will help us out. What is a lemma,you ask? A lemma is nothing but a little theorem that we have to help us solveanother problem. Note that a lemma should not be confused with the more sourlemon, as that is something different and unrelated to what we are discussing.

Lemma 16 A quadrilateral can be circumscribed in a circle if and only if

opposite angles sum to 180 degrees.

If a quadrilateral can be circumscribed by a circle then we have anotherversion of Heron’s Formula:

Theorem 17 (Heron’s Formula) Given a quadrilateral that can be circum-

scribed by a circle, whose sides have length a, b, c, and d, then the area of the

quadrilateral is given by

area =

√(P

2− a

)(P

2− b

)(P

2− c

)(P

2− d

)

where P = a+ b+ c+ d, the perimeter of the quadrilateral.

93

3.2. NUMBERS

It should be noted that we can see that Heron’s Formula for the area of aquadrilateral reduces to Heron’s Formula for the triangle when d = 0.

Lattice points

Lattice points are just points that are spaced 1 unit apart, horizontally andvertically, in the plane. The following picture shows some lattice points:

If a polygon has a vertexes on lattice points, then we can easily compute itsarea using Pick’s Theorem:

Theorem 18 (Pick) If a polygon has its vertexes on a lattice, then

area =b

2+ n− 1

where b is the number of lattice points on the border of the polygon and n is

the number of lattice points inside our polygonal region.

Example Suppose you want to find the area of the following polygon:

Since it has its vertexes on lattice points, we can use Pick’s Theorem:

area =8

2+ 2− 1 = 5 square units.

3.2.2 Ratios

Turning Tricks into Techniques

We will show you three separate tricks, which are all quite similar. By consid-ering these tricks, we will develop techniques for solving problems.

First Trick How is 0.999 . . . related to 1? I claim we have the followingparadox: I intend to show that

0.999 . . . = 1.

94


To see this, set:x = 0.999 . . .

Now we have:10x = 9.999 . . .

So we see that

10x− x = (9.999 . . .)− (0.999 . . .)

9x = 9,

and we are forced to conclude that x = 1, but we started off with the assumptionthat x = 0.999 . . . , hence 1 = 0.999 . . . .

This paradox challenges a common implicit (and false) notion that everynumber has exactly one decimal representation. We are forced to conclude that0.999 . . . and 1 are representations for the same number! To be completelyexplicit, using a similar method as described above we see that

4.999 . . . = 5,

7.3999 . . . = 7.4,

23.745999 . . . = 23.746,

and so on. Hence numbers can have multiple decimal representations.

Second Trick Keeping the first trick used above in the back of your mind,consider this next similar trick. What is:

x =

√

1 +

√

1 +

√

1 +√1 + · · ·

Now we have:x =

√1 + x

Squaring both sides we get:x2 = 1 + x.

Now putting everything on the left-hand side:

x2 − x− 1 = 0.

By the Quadratic Formula,

x =1±

√5

2,

and since we can see that x > 1, we must conclude that

x =1 +

√5

2.

Well this seems like a strange number. Oh well, let’s keep on going.

95

3.2. NUMBERS

Third Trick Now what about:

x = 1 +1

1 +1

1 +1

1 +1

1 + · · ·

Again using a similar trick as above,

x = 1 +1

x.

Multiplying both sides by x we get:

x2 = x+ 1.

Now putting everything on the left-hand side:

x2 − x− 1 = 0.

By the Quadratic Formula,

x =1±

√5

2,

and since we can see that x > 1 we must conclude that

x =1 +

√5

2.

Wait a minute, this says that:

√

1 +

√

1 +

√

1 +√1 + · · · = 1 +

1

1 +1

1 +1

1 +1

1 + · · ·

Wow! Who would have ever thought that? These two crazy looking formulasare equal, and despite the fact that the only number in them is a one, they areboth equal to the messy number:

1 +√5

2= 1.6180339887 . . .

Continued Fractions

Can we use similar techniques (tricks) to study other numbers that have a nastyform? You bet! Before we do that, we’ll need a definition:

96


Definition A fraction of the form

a0 +a1

b1 +a2

b2 +a3

b3 +a4

b4 + · · ·

is called a continued fraction. If a1, a2, a3, . . . are all 1, we will call this asimple continued fraction.

So what about the continued fraction:

x = 1 +1

2 +1

2 +1

2 +1

2 + · · ·

Using the technique as above,

x = 1 +1

1 + x.

Multiplying both sides by 1 + x we get:

x+ x2 = (1 + x) + 1.

Now putting all the x’s on the left-hand side and all the numbers on the right:

x2 = 2.

Ah! So x =√2. Using your calculator, you can see that:

√2 = 1.4142135623 . . .

This means that:

1.4142135623 . . . = 1 +1

2 +1

2 +1

2 +1

2 + · · ·

Note that on the left-hand side you don’t see much of a pattern. However, onthe right-hand side a clear pattern is formed. This is part of the beauty ofcontinued fractions. Now it turns out that you can do this with other numbersand and get lots of other cool patterns!

97

3.2. NUMBERS

Check out e = 2.718281828459045 . . . . It turns out that

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 + · · ·Also check out π = 3.14159265358 . . . . It turns out that we can get a nice

continued fraction for π:

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 +112

6 + · · ·

Going the Other Way Given a number, can you find the continued fractionof the number? It turns out that if the number is sufficiently nice, then this isnot so hard. Before we start with an example, let’s get some definitions and alemma out of the way:

Definition Thewhole-number part of a number is the largest whole numberwhich is less than or equal to the given number.

Example The whole-number part of 2 is 2, while the whole-number part of

5.32 is 5.

Definition The fractional part of a number is the number minus its whole-number part.

Example The fractional part of 2 is 0, while the fractional part of 5.32 is 0.32.

Question Why don’t we just describe the fractional part of a number as thepart that is to the right of the decimal point? Hint: Think about 0.99999 . . . .

?Given any number, we can write it as a simple continued fraction. Consider

13/5. To start note that

3 >13

5> 2.

98


So this means that13

5= 2 +

3

5.

Here 2 is the whole-number part and 3/5 is the fractional part of 13/5. But inthe simple continued fraction, our numerator is 1, not 3. How do we deal withthis? Well,

13

5= 2 +

3

5= 2 +

153

.

This is an improvement but we only want whole numbers in our simple continuedfractions and not 5/3. So we write

5

3= 1 +

2

3

which gives us13

5= 2 +

1

1 +2

3

.

Again, we want our numerator to be 1, not 2 so we will repeat the steps aboveto get

13

5= 2 +

1

1 +2

3

= 2 +1

1 +132

= 2 +1

1 +1

1 +1

2

and this last expression is the simple continued fraction for 13/5. We could alsolist our steps as:

13

5= 2+

3

5135

=5

3= 1+

2

3

123

=3

2= 1+

1

2

112

= 2+ 0

These boldface numbers tell us our continued fraction expansion.We can also find the simple continued fraction of numbers which are not

already fractions (otherwise this would all be a bit silly). Consider√2. To start

note that 2 >√2 > 1. So this means that

√2 = 1 + (

√2− 1).

Where 1 is the whole-number part and (√2 − 1) is the fractional part of

√2.

Alright, now look at 1/(√2− 1). Again we want to separate the whole-number

part and the fractional part. With a little algebra we see that

1√2− 1

=

√2 + 1

2− 1=

√2 + 1 = 2 + (

√2 + 1− 2) = 2 + (

√2− 1).

99

3.2. NUMBERS

Now don’t you get bogged down in the steps. Here it is in fast forward:√2 = 1+ (

√2− 1)

1

(√2− 1)

= 2+ (√2− 1)

1

(√2− 1)

= 2+ (√2− 1)

1

(√2− 1)

= 2+ (√2− 1),

...

At each step we want:

number = whole-number part + fractional part

Now from the bold-faced numbers above we will make our continued fraction:√2 = 1+

1

2+1

2+1

2+1

2+ · · ·Question Can you explain why this works?

?The Golden Ratio

It turns out that the number

φ =1 +

√5

2= 1 +

1

1 +1

1 +1

1 +1

1 + · · ·is a special number that we call the golden ratio. We denote the golden ratioby the symbol φ.

Question What’s so special about the golden ratio?

Given any rectangle you can divide it into a square and another smallerrectangle:

100


Suppose that you want the new smaller rectangle to have the same proportionsas the original rectangle. This will only happen if the ratio of the sides of therectangle are φ to 1.

We can show this to be true algebraically. Start by making a sketch of arectangle:

1

x

x− 1

So we want1

x=

x− 1

1and so:

1 = x2 − x

Thusx2 − x− 1 = 0.

But we have already solved this equation several times, its solution is:

x =1 +

√5

2= φ.

Definition A rectangle with the proportions of φ for one of its sides and 1 forthe other is called a golden rectangle.

Given a golden rectangle, we can put a spiral in side, by making a quarterof a circle in every square:

Moreover, we can make what is called a golden triangle, an isosceles tri-angle with two long sides being related to the shorter side by a ratio of φ to 1.We can place a similar spiral in this shape as before:

101

3.2. NUMBERS

Question How would you draw the above figures?

?

3.2.3 Combining Areas and Ratios—Probability

Definition The probability of an event occurring is a number between 0 and1 giving a linear scale of the likelihood of the event occurring, with 0 meaningimpossible, and 1 meaning certain.

Question Could we build a machine that would take random numbers andfrom them produce a closer and closer approximation of π?

?To get at the above question, let’s talk about probability.

Question What is the probability that a point chosen at random in the squarebelow will land in the shaded region?

?Question What is the probability that a point chosen at random in the squarebelow will land in the shaded region?

?Question What is the probability that a point chosen at random in the squarebelow will land in the shaded region?

102


?OK! Now a tricky question:

Question What is the probability that a point chosen at random in the squarebelow will land inside the shaded area of the arc?

I’ll help you solve this one. What if we expand our view a bit? Say thesquare above has area 1. Now consider

In this case the area of the big square is 4 and the area of the shaded region isπ, as it is a circle of radius 1. Thus if a point is chosen at random in the bigsquare, the probability that it lands in the shaded circle is π/4. However, thelittle square and the little quarter-circle above are only 1/4 of this each. So wehave a probability of

π/4

4/4= π/4

of landing in the shaded region in the little square above.So the above example gives us a hint how we could take random numbers

and turn them into an approximation for π. Here is an algorithm that shoulddo the trick:

(1) Take a random set of numbers all between 0 and 1.

(2) Take pairs of these numbers (a, b). Let n be the total number of pairs thatwe have.

(3) To see if a pair (a, b) lands inside the circle use the Pythagorean Theorem,that is if

a2 + b2 6 1

then the point is inside the circle, otherwise the point is outside the circle.

103

3.2. NUMBERS

(4) Count how many pairs land inside the circle and divide by the total numberof pairs.

(5) This ratio approximates π/4. To get an approximation for π, multiplyyour answer by 4.

The Monty Hall Problem

There used to be a TV show called Let’s Make a Deal. It was hosted by MontyHall. At the end of the show something like this was presented to the leadingcontestant.

31 2

These are three doors. Behind two of the doors is something you don’twant—say a goat. But behind one of the doors is something you might like—say a brand new car! Here is how the game works: You point to a door andthen Monty Hall opens another door revealing a goat. He then offers to let youswitch or stay. If you stay, then you open your door to reveal either the secondgoat or the fabulous car. If you decide to switch, you open the other remainingdoor to reveal either the second goat or the fabulous car.

Question Considering the Monty Hall problem, is it better to switch or is itbetter to stay? Hint: This problem is very tricky and has fooled many a goodmathematician!

?

Bertrand’s Paradox

Here is an innocent looking question:

Question Given a circle, find the probability that a chord chosen at randomis longer than the side of an inscribed equilateral triangle.

Let’s use our new found skills in probability to hack this one to pieces.

Solution 1 How do we pick a random chord? Well imagine two spinnersmounted at the center of the circle. When they get done spinning, just connectthe points that they point at to get your chord. Let the one spinner finishanywhere. Now the second spinner will tell us whether the chord is longer than

104


the edge of the triangle.

Since there are 60 degrees in a triangle, and there are 180 degrees in a straightline, the probability that the random chord will be longer than the edge of thetriangle is

60

180= 1/3.

�

Here is another solution:

Solution 2 Let’s consider another way of picking a random chord. By draw-ing some pictures, one can see that a chord can be determined completely byits midpoint, unless the midpoint is the center of the circle—but what is theprobability of a random point landing in the exact center of a circle?

Thus a chord is longer than the edge of the triangle if and only if its midpointlands inside the circle inscribed inside of the triangle. However, it can be shownthat the radius of the circle inside the triangle is half of the radius of the largecircle and hence the ratio of the areas of the circles is:

π(1/2)2

π12= 1/4,

and this must be the probability that the random chord will be longer than theedge of the triangle. �

D’oh! 1/3 6= 1/4!

Question What is wrong in the above discussion?

?

105

3.2. NUMBERS

Wacky Dice

Consider the game Rock-Paper-Scissors. In this game two players make handgestures. Each player’s hand gesture represents one of the following: a rock, apiece of paper, or scissors. According to the rules of the game:

• Rock beats scissors, by breaking them of course.

• Scissors beats paper, by cutting it of course.

• Paper beats rock, by covering it of course1.

Now consider these dice:

3

3

3

3

33 5 1 5

1

5

4

4

4

2

2

2

2

6 6

1 0

04

(3) (2) (1) (0)

Here is a new game. Considering the dice above, let your opponent pick a die.Then you pick one of the remaining three dice. Each player throws their dieand the highest number wins a point. Play until someone reaches 10 points.

You’ll have a much better chance of winning the game if you know thesefacts: Die (3) has a probability of 2/3 beating Die (2), Die (2) has a probabilityof 2/3 beating Die (1), Die (1) has a probability of 2/3 beating Die (0), andDie (0) has a probability of 2/3 beating Die (3). Just as in Rock-Paper-Scissorswe have made a full circle. These crazy dice were invented by Bradley Efron.

Die (3) has a 2/3 probability of beating Die (2) since no matter what Die (3)rolls, there are 4 losing squares out of a total of 6 squares on Die (2):

2

2

2

6 6

(2)

2

Before we can compute the next probability we need two lemmas.

1OK I admit it, the method through which paper actually beats rock has always been amystery to me.

106


Lemma 19 If two events E1 and E2 are disjoint, meaning that they cannot

both occur together, then the probability that E1 or E2 will happen is the

sum of the probability that E1 happens with the probability that E2 happens.

Symbolically, if P (Ei) is the probability that event Ei happens, then

P (E1 ∨ E2) = P (E1) + P (E2).

Lemma 20 If two events E1 and E2 are independent, meaning that the occur-

rence of one has no effect on the occurrence of the other, then the probability

that E1 and E2 will happen is the product of the probability that E1 happens

with the probability that E2 happens. Symbolically, if P (Ei) is the probability

that event Ei happens, then

P (E1 ∧ E2) = P (E1) · P (E2).

Die (2) has a 2/3 probability of beating Die (1) since you roll a 6 with Die (2)and win

2

2

2

2

6 6

(2)

with a 2/6 = 1/3 probability or you roll as follows:

5 1 5

1

5

1

(1)

2

2

2

2

6 6

(2)

and

Since there is an and there, we must multiply the probabilities to get a proba-bility of

4

6· 12=

1

3.

Since there was an or above, we must add

1

3+

1

3=

2

3.

Thus Die (2) has a 2/3 probability of beating Die (1).

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3.2. NUMBERS

Die (1) has a 2/3 probability of beating Die (0) since you roll a 5 with Die (1)and win

5 1 5

1

5

1

(1)

with a 3/6 = 1/2 probability or you roll as follows:

5 1 5

1

5

1

(1)

4

4

4

0

04

(0)

and

Since there is an and there, we must multiply the probabilities to get a proba-bility of

1

2· 26=

1

6.

Since there was an or above, we must add

1

2+

1

6=

2

3.

Thus Die (1) has a 2/3 probability of beating Die (0).Finally Die (0) has a 2/3 probability of beating Die (3) since you roll a 4

with Die (0) and win

4

4

4

0

04

(0)

with a 4/6 = 2/3 probability.

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(1) Find the area of the triangle whose sides are of length 3, 4, and 5. Explainyour work.

(2) Find the area of the triangle whose sides are of length 10, 6, and 10.Explain your work.

(3) Find the area of the triangle whose sides are of length 5, 5, and 5. Explainyour work.

(4) Find the area of the triangle whose sides are of length 4, 6, and 10. Explainyour work. Does this answer make sense? Why or why not?

(5) Find the area of the quadrilateral that can be inscribed in a circle whosesides are of lengths 9, 3, 5, and 3. Explain your work.



(8) Find the area of the quadrilateral that can be inscribed in a circle whosesides are of lengths 2, 10, 5, and 3. Explain your work. Does your answermake sense? Why or why not?

(9) Compute the area of:

Explain your work and include units with your answer.




109

3.2. NUMBERS


(12) Explain what φ is in terms of squares and rectangles.

(13) Explain what φ is in terms of a continued square-root.

(14) Explain what φ is in terms of a continued fraction.

(15) Courtney Gibbons is someone who has a rather unusual tattoo. She waskind enough to let an unusual person like me take a picture of it. Whatdoes her tattoo represent? Explain your reasoning.

(16) Find the exact value for x when:

x =

√

2 +

√

2 +

√

2 +√2 + · · ·

Explain your work.


x =

√

6 +

√

6 +

√

6 +√6 + · · ·

Explain your work.


x =

√

12 +

√

12 +

√

12 +√12 + · · ·

Explain your work.

110



x =

√

20 +

√

20 +

√

20 +√20 + · · ·

Explain your work.


x =

√

30 +

√

30 +

√

30 +√30 + · · ·

Explain your work.


x = 2 +1

4 +1

4 +1

4 +1

4 + · · ·Explain your work.

(22) Find x when:

x = 4 +1

6 +1

6 +1

6 +1



x = 4 +1

8 +1

8 +1

8 +1



x = 3 +1

10 +1

10 +1

10 +1


111

3.2. NUMBERS

(25) Explain what the whole-number part and what the fractional part ofa number are. Give examples.

(26) Find the simple continued fraction expansion of 5/3. Explain your work.



(29) Using a calculator, find the first five terms in the simple continued fractionexpansion of π. What number do you get by only considering the firstterm? The first four?

(30) Find the simple continued fraction expansion of√5. Explain your work.





(35) Find the simple continued fraction expansion of 11. Explain your work.

(36) What is it about the numbers 2, 5, 10, 17, 26 that makes it easy to computethe continued fraction expansion of the square-roots of these numbers?Explain your answer.

(37) What is the definition of probability?

(38) Aloof old Professor Rufus came into his Calculus class one day and said,“I have chosen a real number randomly and will give anyone an ‘A’ on thenext exam who can guess it!” Aloof old Professor Rufus knew that theprobability of someone finding a single real number with a finite number ofguesses is 0, and so was quite confident that none could guess his number.It should be no surprise to you that aloof old Professor Rufus turned quitewhite when after 37 guesses, Smart Sally guessed his number which was 13.How is it that the students were able to guess aloof old Professor Rufus’number? How was aloof old Professor Rufus wrong about the probability?

(39) What is the probability that a point chosen at random in the larger rect-angle below will land in the shaded region? Explain your answer.

112




(42) What is the probability that a point chosen at random in the larger regionbelow will land in the smaller shaded region? Explain your answer.



113

3.2. NUMBERS

(45) Explain how someone could start to approximate π using a dart-boardand darts.

(46) Here are two dice. Which die has a better probability of rolling higher,and what is that probability? Explain your answer.

3

4

3

3

4

2

2

2

2

6 6

(a) (b)

4


3

3

3 2

2

(a) (b)

4 4 4

5

5

5

2


3

3

3 2

6

(a) (b)

4 4 4

2

6

2

6

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Chapter 4

Compass and StraightedgeConstructions

Mephistopheles: I must say there is an obstacleThat prevents my leaving:It’s the pentagram on your threshold.

Faust: The pentagram impedes you?Tell me then, you son of hell,If this stops you, how did you come in?

Mephistopheles: Observe! The lines are poorly drawn;That one, the outer angle,Is open, the lines don’t meet.

—Gothe, Faust act I, scene III

4.1 Constructions

About a century before the time of Euclid, Plato—a student of Socrates—declared that the compass and straightedge should be the only tools of thegeometer. Why would he do such a thing? For one thing, both the the compassand straightedge are fairly simple instruments. One draws circles, the otherdraws lines—what else could possibly be needed to study geometry? Moreover,rulers and protractors are far more complex in comparison and people back thencouldn’t just walk to the campus bookstore and buy whatever they wanted.However, there are other reasons:

(1) Compass and straightedge constructions are independent of units.

(2) Compass and straightedge constructions are theoretically correct.

(3) Combined, the compass and straightedge seem like powerful tools.

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4.1. CONSTRUCTIONS

Compass and straightedge constructions are independent of units.

Whether you are working in centimeters or miles, compass and straightedgeconstructions work just as well. By not being locked to set of units, the con-structions given by a compass and straightedge have certain generality that isappreciated even today.

Compass and straightedge constructions are theoretically correct. Inmathematics, a correct method to solve a problem is more valuable than a cor-rect solution. In this sense, the compass and straightedge are ideal tools for themathematician. Easy enough to use that the rough drawings that they producecan be somewhat relied upon, yet simple enough that the tools themselves canbe described theoretically. Hence it is usually not too difficult to connect a givenconstruction to a formal proof showing that the construction is correct.

Combined, the compass and straightedge seem like powerful tools.

No tool is useful unless it can solve a lot of problems. Without a doubt, thecompass and straightedge combined form a powerful tool. Using a compass andstraightedge, we are able to solve many problems exactly. Of the problems thatwe cannot solve exactly, we can always produce an approximate solution.

We’ll start by giving the rules of compass and straightedge constructions:

Rules for Compass and Straightedge Constructions

(1) You may only use a compass and straightedge.

(2) You must have two points to draw a line.

(3) You must have a point and a line segment to draw a circle. The point isthe center and the line segment gives the radius.

(4) Points can only be placed in two ways:

(a) As the intersection of lines and/or circles.

(b) As a free point, meaning the location of the point is not importantfor the final outcome of the construction.

Our first construction is also Euclid’s first construction:

Construction (Equilateral Triangle) We wish to construct an equilateral tri-

angle given the length of one side.

(1) Open your compass to the width of the line segment.

(2) Draw two circles, one with the center being each end point of the line

segment.

(3) The two circles intersect at two points. Choose one and connect it to both

of the line segment’s endpoints.

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CHAPTER 4. COMPASS AND STRAIGHTEDGE CONSTRUCTIONS

Euclid’s second construction will also be our second construction:

Construction (Transferring a Segment) Given a segment, we wish to move

it so that it starts on a given point.

(1) Draw a line through the point in question.

(2) Open your compass to the length of the line segment and draw a circle

with the given point as its center.

(3) The line segment consisting of the given point and the intersection of the

circle and the line is the transferred segment.

If you read The Elements, you’ll see that Euclid’s construction is much morecomplicated than ours. Apparently, Euclid felt the need to justify the ability tomove a distance. Many sources say that Euclid used what is called a collapsing

compass, that is a compass that collapsed when it was picked up. However, I donot believe that such an invention ever existed. Rather this is something thatlives in the conservative geometer’s head.

Regardless of whether the difficulty of transferring distances was theoreticalor physical, we need not worry when we do it. In fact, Euclid’s proof of theabove theorem proves that our modern way of using the compass to transferdistances is equivalent to using the so-called collapsing compass.

Question Exactly how would one prove that the modern compass is equiva-lent to the collapsing compass? Hint: See Euclid’s proof.

?Construction (Bisecting a Segment) Given a segment, we wish to cut it in

half.

(1) Open your compass to the width of the segment.

(2) Draw two circles, one with the center being at each end point of the line

segment.

(3) The circles intersect at two points. Draw a line through these two points.

117

4.1. CONSTRUCTIONS

(4) The new line bisects the original line segment.

Construction (Perpendicular through a Point) Given a point and a line, we

wish to construct a line perpendicular to the original line that passes through

the given point.

(1) Draw a circle centered at the point large enough to intersect the line in

two distinct points.

(2) Bisect the line segment. The line used to do this will be the desired line.

Construction (Bisecting an Angle) We wish to divide an angle in half.

(1) Draw a circle with its center being the vertex of the angle.

(2) Draw a line segment where the circle intersects the lines.

(3) Bisect the new line segment. The bisector will bisect the angle.

118


We now come to a very important construction:

Construction (Copying an Angle) Given a point on a line and some angle,

we wish to copy the given angle so that the new angle has the point as its vertex

and the line as one of its edges.

(1) Open the compass to a fixed width and make a circle centered at the

vertex of the angle.

(2) Make a circle of the same radius on the line with the point.

(3) Open the compass so that one end touches the 1st circle where it hits an

edge of the original angle, with the other end of the compass extended to

where the 1st circle hits the other edge of the original angle.

(4) Draw a circle with the radius found above with its center where the second

circle hits the line.

(5) Connect the point to where the circles meet. This is the other leg of the

angle we are constructing.

119

4.1. CONSTRUCTIONS

Construction (Parallel through a Point) Given a line and a point, we wish to

construct another line parallel to the first that passes through the given point.

(1) Draw a circle around the given point that passes through the given line

at two points.

(2) We now have an isosceles triangle, duplicate this triangle.

(3) Connect the top vertexes of the triangles and we get a parallel line.

Question Can you give another different construction?

?

Construction (Tangent to a Circle) Given a circle and a point, we wish to

construct a line tangent to the circle that goes through the point.

(1) Draw a line segment connecting the point to the center of the circle.

(2) Bisect the above segment.

(3) Draw a circle centered at the bisector whose radius is half the length of

the above segment.

(4) Draw lines connecting the given point to where the two circles intersect.

120


Question What if the point is inside the circle? What if the point is on thecircle?

?

121

4.1. CONSTRUCTIONS


(1) What is a collapsing compass? Why don’t we use them or worry aboutthem any more?

(2) Prove that the collapsing compass is equivalent to the modern compass.

(3) Given a line segment, construct an equilateral triangle whose edge has thelength of the given segment. Explain the steps in your construction.

(4) Use a compass and straightedge to bisect a given line segment. Explainthe steps in your construction.

(5) Given a line segment with a point on it, construct a line perpendicularto the segment that passes through the given point. Explain the steps inyour construction.

(6) Use a compass and straightedge to bisect a given angle. Explain the stepsin your construction.

(7) Given an angle and some point, use a compass and straightedge to copythe angle so that the new angle has as its vertex the given point. Explainthe steps in your construction.

(8) Given a point and line, construct a line perpendicular to the given line thatpasses through the given point. Explain the steps in your construction.

(9) Given a point and line, construct a line parallel to the given line thatpasses through the given point. Explain the steps in your construction.

(10) Given a circle and a point, construct a line tangent to the given circle thatpasses through the given point.

(11) Given a triangle, construct the circumcenter. Explain the steps in yourconstruction.

(12) Given a triangle, construct the orthocenter. Explain the steps in yourconstruction.

(13) Given a triangle, construct the incenter. Explain the steps in your con-struction.

(14) Given a triangle, construct the centroid. Explain the steps in your con-struction.

(15) Given a triangle, construct the incircle. Explain the steps in your con-struction.

(16) Given a triangle, construct the circumcircle. Explain the steps in yourconstruction.

122


(17) Given a triangle, construct the Euler line. Explain the steps in yourconstruction.

(18) Given 3 distinct points not all in a line, construct a circle that passesthrough all three points. Explain the steps in your construction.

(19) Give 3 different constructions of the Nine-Point Circle. Explain the stepsin your constructions.

(20) Given a triangle with a point on each side, construct a Miquel point.Explain the steps in your construction.

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4.2. TRICKIER CONSTRUCTIONS

4.2 Trickier Constructions

Question How do you construct regular polygons? In particular, how doyou construct regular: 3-gons, 4-gons, 5-gons, 6-gons, 7-gons, 8-gons, 10-gons,12-gons, 17-gons, 24-gons, and 144-gons?

?Well the equilateral triangle is easy. It was the first construction that we

did. What about squares? What about regular hexagons? It turns out thatthey aren’t too difficult. What about pentagons? Or say n-gons? We’ll have tothink about that. Let’s leave the difficult land of n-gons and go back to thinkingabout nice, three-sided triangles.

Construction (SAS Triangle) Given two sides with an angle between them,

we wish to construct the triangle with that angle and two adjacent sides.

(1) Transfer the one side so that it starts at the vertex of the angle.

(2) Transfer the other side so that it starts at the vertex.

(3) Connect the end points of all moved line segments.

The “SAS” in this construction’s name spawns from the fact that it requirestwo sides with an angle between them. The SAS Theorem states that we canobtain a unique triangle given two sides and the angle between them.

Construction (SSS Triangle) Given three line segments we wish to construct

the triangle that has those three sides if it exists.

(1) Choose a side and select one of its endpoints.

(2) Draw a circle of radius equal to the length of the second side around the

chosen endpoint.

(3) Draw a circle of radius equal to the length of the third side around the

other endpoint.

(4) Connect the end points of the first side and the intersection of the circles.

This is the desired triangle.

Question Can this construction fail to produce a triangle? If so, show how.If not, why not?

?Question Remember earlier when we asked about the converse to the PythagoreanTheorem? Can you use the construction above to prove the converse of thePythagorean Theorem?

124


?Question Can you state the SSS Theorem?

?Construction (SAA Triangle) Given a side and two angles, where the given

sided does not touch one of the angles, we wish to construct the triangle that

has this side and these angles if it exists.

(1) Start with the given side and place the adjacent angle at one of its end-

points.

(2) Move the second angle so that it shares a leg with the leg of the first

angle–not the leg with the side.

(3) Extend the side past the first angle, forming a new angle with the leg of

the second angle.

(4) Move this new angle to the other endpoint of the side, extending the legs

of this angle and the first angle will produce the desired triangle.

Question Can this construction fail to produce a triangle? If so, show how.If not, why not?

?Question Can you state the SAA Theorem?

?

4.2.1 Challenge Constructions

Question How can you construct a triangle given the length of one side,the length of the the median to that side, and the length of the altitude ofthe opposite angle? Hint: Recall that the median connects the vertex to themidpoint of the opposite side.

Solution

(1) Start with the given side.

(2) Since the median hits our side at the center, bisect the given side.

(3) Make a circle of radius equal to the length of the median centered at thebisector of the given side.

(4) Construct a line parallel to our given line of distance equal to the lengthof the given altitude away.

125


(5) Where the line and the circle intersect is the third point of our triangle.Connect the endpoints of the given side and the new point to get thetriangle we want.

�

Question How can you construct a triangle given one angle, the length of anadjacent side and the altitude to that side?

Solution

(1) Start with a line containing the side.

(2) Put the angle at the end of the side.

(3) Draw a parallel line to the side of the length of the altitude away.

(4) Connect the angle to the parallel side. This is the third vertex. Connectthe endpoints of the given side and the new point to get the triangle wewant.

126


�

Question How can you construct a circle with a given radius tangent to twoother circles?

Solution

(1) Let r be the given radius, and let r1 and r2 be the radii of the given circles.

(2) Draw a circle of radius r1 + r around the center of the circle of radius r1.

(3) Draw a circle of radius r2 + r around the center of the circle of radius r2.

(4) Where the two circles drawn above intersect is the center of the desiredcircle.

�

127


Question Place two tacks in a wall. Insert a sheet of paper so that the edgeshit the tacks and the corner passes through the imaginary line between thetacks. Mark where the corner of the piece of paper touches the wall. Repeatthis process, sliding the paper around. What curve do you end up drawing?

?Question How can you construct a triangle given an angle and the length ofthe opposite side?

Solution We really can’t solve this problem completely because the infor-mation given doesn’t uniquely determine a triangle. However, we can still saysomething. Here is what we can do:

(1) Put the known angle at one end of the line segment. Note in the picturebelow, it is at the left end of the line segment and it is opening downwards.

(2) Bisect the segment.

(3) See where the bisector in Step 2 intersects the perpendicular of the otherleg of the angle drawn from the vertex of the angle.

(4) Draw the circle centered at the point found in Step 3 that touches theendpoints of the original segment.

All points on the circle could be the vertex. �

Question Why does the above method work?

?

128


Question You are on a boat at night. You can see three lighthouses, andyou know their position on a map. Also you know the angles of the light raysbetween the lighthouses as measured from the boat. How do you figure outwhere you are?

?

4.2.2 Problem Solving Strategies

The harder constructions discussed in this section can be difficult to do. Thereis no rote method to solve these problems, hence you must rely on your brain.Here are some hints that you may find helpful:

Construct what you can. You should start by constructing anything youcan, even if you don’t see how it will help you with your final construction. Indoing so you are “chipping away” at the problem just as a rock-cutter chips awayat a large boulder. Here are some guidelines that may help when constructingtriangles:

(1) If a side is given, then you should draw it.

(2) If an angle is given and you know where to put it, draw it.

(3) If an altitude of length ℓ is given, then draw a line parallel to the side thatthe altitude is perpendicular to. This new line must be distance ℓ fromthe side.

(4) If a median is given, then bisect the segment it connects to and drawa circle centered around the bisector, whose radius is the length of themedian.

(5) If you are working on a figure, construct any “mini-figures” inside thefigure you are trying to construct. For example, many of the problemsbelow ask you to construct a triangle. Some of these constructions haveright-triangles inside of them, which are easier to construct than the finalfigure.

Sketch what you are trying to find. It is a good idea to try to sketchthe figure that you are trying to construct. Sketch it accurately and label allpertinent parts. If there are special features in the figure, say two segmentshave the same length or there is a right-angle, make a note of it on your sketch.Also mark what is unknown in your sketch. We hope that doing this will helporganize your thoughts and get your “brain juices” flowing.

Question Why are the above strategies good?

?

129



(1) Construct a square. Explain the steps in your construction.

(2) Construct a regular hexagon. Explain the steps in your construction.

(3) Your friend Margy is building a clock. She needs to know how to align thetwelve numbers on her clock so that they are equally spaced on a circle.Explain how to use a compass and straightedge construction to help herout. Illustrate your answer with a construction and explain the steps inyour construction.

(4) Construct a triangle given two sides of a triangle and the angle betweenthem. Explain the steps in your construction.

(5) Construct a triangle given three sides of a triangle. Explain the steps inyour construction.

(6) Construct a triangle given two adjacent sides of a triangle and a medianto one of the given sides. Explain the steps in your construction.

(7) Construct a figure showing that a triangle cannot always be uniquelydetermined when given an angle, a side adjacent to that angle, and theside opposite the angle. Explain the steps in your construction and explainhow your figure shows what is desired. Hint: Draw many pictures to helpyourself out.

(8) Give a construction showing that a triangle is uniquely determined if youare given a right-angle, a side touching that angle, and another side nottouching the angle. Explain the steps in your construction and explainhow your figure shows what is desired.

(9) Construct a triangle given a side, the median to the side, and the angleopposite to the side. Explain the steps in your construction.

(10) Construct a triangle given two sides and the altitude to the third side.Explain the steps in your construction.

(11) Construct a triangle given two altitudes and an angle touching one ofthem. Explain the steps in your construction.

(12) Construct a triangle given an altitude, and two angles not touching thealtitude. Explain the steps in your construction.

(13) Construct a triangle given the length of one side, the length of the themedian to that side, and the length of the altitude of the opposite angle.Explain the steps in your construction.

(14) Construct a triangle, given one angle, the length of an adjacent side andthe altitude to that side. Explain the steps in your construction.

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(15) Construct a circle with a given radius tangent to two other given circles.Explain the steps in your construction.

(16) Does a given angle and a given opposite side uniquely determine a triangle?Explain your answer.

(17) You are on the bank of a river. There is a tree directly in front of you onthe other side of the river. Directly left of you is a friend a known distanceaway. Your friend knows the angle starting with them, going to the tree,and ending with you. How wide is the river? Explain your work.

(18) You are on a boat at night. You can see three lighthouses, and you knowtheir position on a map. Also you know the angles of the light rays fromthe lighthouses. How do you figure out where you are? Explain your work.

(19) Construct a triangle given an angle, the length of the angle’s bisector tothe opposite side, and the length of a side adjacent to the given angle.Explain the steps in your construction.

(20) Construct a triangle given an angle, the length of the opposite side andthe length of the altitude of the given angle. Explain the steps in yourconstruction.

(21) Construct a triangle given one side, the altitude of the opposite angle, andthe radius of the circumcircle. Explain the steps in your construction.

(22) Construct a triangle given one side, the altitude of an adjacent angle, andthe radius of the circumcircle. Explain the steps in your construction.

(23) Construct a triangle given one side, the length of the median connectingthat side to the opposite angle, and the radius of the circumcircle. Explainthe steps in your construction. Hint: Recall that the median connects thevertex to the midpoint.

(24) Given a circle and a line, construct another circle of a given radius thatis tangent to both the original circle and line. Explain the steps in yourconstruction.

(25) Construct a circle with a given radius tangent to two given intersectinglines. Explain the steps in your construction.

(26) Construct a triangle given one angle and the altitudes to the two otherangles. Explain the steps in your construction.

(27) Construct a circle with three smaller circles of equal size inside such thateach smaller circle is tangent to the other two and the larger outside circle.Explain the steps in your construction.

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4.3. CONSTRUCTIBLE NUMBERS

4.3 Constructible Numbers

First of all, what do we mean by the words constructible numbers? Imagine aline with two points on it:

0 1

Label the left point 0 and the right point 1. If we think of this as a starting pointfor a number line, then a constructible number is nothing more than a pointwe can obtain on the above number line using only a compass and a straightedgestarting with the points 0 and 1. Call the set of constructible numbers C.Question Exactly which numbers are constructible?

?How do we attack this question? Well first let’s get a bit of notation. Recall

that we use the symbol “∈” to mean is in. So we know that 0 and 1 are in theset of constructible numbers. So we write

0 ∈ C and 1 ∈ C.

If we could use constructions to make the operations +, −, ·, and ÷, then wewould be able to say a lot more. In fact we will do just this. In the followingconstructions, the segments of length 1, a, and b are as given below:

a1 b

Construction (Addition) Adding is simple, use the compass to extend the

given line segment as necessary.

a b

a+b

Construction (Subtraction) Subtracting is easy too:

a−b

Question What does our number line look like at this point?

At this point we have all the whole numbers and their negatives. We have aspecial name for this set, we call it the integers and denote it by the letter Z:

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . . }.

But we still have some more operations:

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Construction (Multiplication) The idea of multiplication is based on the idea

of similar triangles. Start with given segments of length a, b, and 1:

(1) Make a small triangle with the segment of length 1 and segment of length

b.

(2) Now place the segment of length a on top of the unit segment with one

end at the vertex.

(3) Draw a line parallel to the segment connecting the unit to the segment of

length b starting at the other end of segment of length a.

(4) The length from the vertex to the point that the line containing b intersectsthe line drawn in Step 3 is of length a · b.

a

1

b

ab

Construction (Division) The idea of division is also based on the idea of

similar triangles. Again, you start with given segments of length a, b, and 1:

(1) Make a triangle with the segment of length a and the segment of length b.

(2) Put the unit along the segment of length a starting at the vertex where

the segment of length a and the segment of length b meet.

(3) Make a line parallel to the third side of the triangle containing the segment

of length a and the segment of length b starting at the end of the unit.

(4) The distance from where the line drawn in Step 3 meets the segment of

length b to the vertex is of length b/a.

a

1

b

b/a

Question What does our number line look like at this point?

Currently we have Z, the integers, and all of the fractions. In other words:

Q ={a

bsuch that a ∈ Z and b ∈ Z with b 6= 0

}

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Fancy folks will replace the words such that with a colon “:” to get:

Q ={a

b: a ∈ Z and b ∈ Z with b 6= 0

}

We call this set the rational numbers. The letter Q stands for the wordquotient, which should remind us of fractions.

In mathematics we study sets of numbers. In any field of science, the firststep to understanding something is to classify it. One sort of classification thatwe have is the notion of a field.

Definition A field is a set of numbers, which we will call F , that is closedunder two associative and commutative operations + and · such that:

(1) (a) There exists an additive identity 0 ∈ F such that for all x ∈ F ,

x+ 0 = x.

(b) For all x ∈ F , there is an additive inverse −x ∈ F such that

x+ (−x) = 0.

(2) (a) There exists a multiplicative identity 1 ∈ F such that for all x ∈ F ,

x · 1 = x.

(b) For all x ∈ F where x 6= 0, there is a multiplicative inverse x−1 suchthat

x · x−1 = 1.

(3) Multiplication distributes over addition. That is, for all x, y, z ∈ F

x · (y + z) = x · y + x · z.

Now, a word is in order about three tricky words I threw in above: closed,associative, and commutative:

Definition A set F is closed under an operation ⋆ if for all x, y ∈ F , x⋆y ∈ F .

Example The set of integers, Z, is closed under addition, but is not closed

under division.

Definition An operation ⋆ is associative if for all x, y, and z

x ⋆ (y ⋆ z) = (x ⋆ y) ⋆ z.

Definition An operation ⋆ is commutative if for all x, y

x ⋆ y = y ⋆ x.

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Question Is Z a field? Is Q a field? Can you think of other fields? Whatabout the set of constructible numbers C?

?From all the constructions above we see that the set of constructible numbers

C is a field. However, which field is it? In fact, the set of constructible numbersis bigger than Q!

Construction (Square-Roots) Start with given segments of length a and 1:

(1) Put the segment of length a immediately to the left of the unit segment

on a line.

(2) Bisect the segment of length a+ 1.

(3) Draw an arc centered at the bisector that starts at one end of the line

segment of length a+ 1 and ends at the other end.

(4) Construct the perpendicular at the point where the segment of length ameets the unit.

(5) The line segment connecting the meeting point of the segment of length aand the unit to the arc drawn in Step 3 is of length

√a.

a 1

This tells us that square-roots are constructible. In particular, the square-root of two is constructible. But the square-root of two is not rational! That is,there is no fraction

a

b=

√2 such that a, b ∈ Z.

How do we know the above fact? Well there are several ways to do it. Why notuse the Rational Roots Test?

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Theorem 21 (Rational Roots Test) If we have a polynomial with integer

coefficients,

anxn + an−1x

n−1 + · · ·+ a1x+ a0

then given any rational root r/s of the above polynomial, r must be a factor of

a0 and s must be a factor of an.

Now what does it mean to be the square-root of 2? It means that you are asolution to the following equation:

x2 − 2 = 0.

By the Rational Roots Test, if r/s is a rational root then r is a factor of −2 ands is a factor of 1. So here are our choices:

r = ±1,±2

s = ±1.

But no combination of the above numbers form fractions r/s such that

(r/s)2 = 2.

Thus, the square root of two is not rational.OK, so how do we talk about a field that contains both Q and

√2? Simple,

use this notation:

Q(√2) = {the smallest field containing both Q and

√2}

So the set of constructible numbers contains all of Q(√2). Does the set of

constructible numbers contain even more numbers? Yes! In fact the√3 is also

not rational, but is constructible. So here is our situation:

Z ⊆ Q ⊆ Q(√2) ⊆ Q(

√2,√3) ⊆ C

So all the numbers in Q(√2,√3) are also in C. But is this all of C? Hardly!

We could keep on going, adding more and more square-roots ’til the cows comehome, and we still will not have our hands on all of the constructible numbers.But all is not lost. We can still say something:

Theorem 22 The use of compass and straightedge alone on a field F can at

most produce numbers in a field F (√α) where α ∈ F .

The upshot of the above theorem is that the only numbers that are con-structible are expressible as a combination of rational numbers and the symbols:

+ − · ÷ √

So what are examples of numbers that are not constructible? Well to start3√2 is not constructible. Also π is not constructible. While both of these facts

can be carefully explained, we will spare you gentle reader—for now.

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Question Which of the following numbers are constructible?

3.1415926,16√5,

3√27,

6√27.

?

Question Is the golden ratio constructible?

Well

φ =1 +

√5

2,

so of course it is. But, how do you actually construct it? Here is an easy way:

Construction (Golden Ratio)

(1) Consider a unit on a line.

(2) Construct a perpendicular of unit length at the right end point of the unit.

(3) Bisect the original unit.

(4) Draw an arc, centered at the point found in Step 3 that goes through the

top of the perpendicular drawn in Step 2.

(5) The segment starting at the left end of the unit and ending at the point

found in Step 4 is of length φ.

Question How how do you construct a regular pentagon?

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One way is to use golden triangles:

1 1

1

φ

11

φ

What about other regular n-gons? Carl Friedrich Gauss, one of the greatestmathematicians of all time, solved this problem when he was 18. He did thisaround the year 1800, nearly 2000 years after the time of the Greeks. Howdid he do it? He thought of constructions algebraically as we have been doing.Using these methods, he discovered this theorem:

Theorem 23 (Gauss) One can construct a regular n-gon if n > 3 and

n = 2i · p1 · p2 · · · pj

where each subscripted p is a distinct prime number of the form

2(2k) + 1

where i, j, and k are nonnegative integers.

Around thirty years later, Wantzel proved that these were the only regularpolygons that could be constructed.

Question Find i, j, and k for a regular 3-gon, 4-gon, 5-gon, and 6-gon.

?

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(1) Explain what the set denoted by Z is.

(2) Explain what the set denoted by Q is.

(3) Explain what the set C of constructible numbers is.

(4) Given two line segments a and b, construct a + b. Explain the steps inyour construction.

(5) Given two line segments a and b, construct a − b. Explain the steps inyour construction.

(6) Given three line segments 1, a, and b, construct a · b. Explain the steps inyour construction.

(7) Given three line segments 1, a, and b, construct a/b. Explain the steps inyour construction.

(8) Given a unit, construct 4/3. Explain the steps in your construction.

(9) Given a unit, construct 3/4. Explain the steps in your construction.

(10) Given a unit, construct√2. Explain the steps in your construction.

(11) Use the construction for multiplication to explain why when multiplyingtwo numbers between 0 and 1, the product is always still between 0 and1.

(12) Use the construction for division to explain why when dividing a positivenumber by a number between 0 and 1, the quotient is always larger thanthe initial positive number.

(13) Fill in the following table:

+ − · ÷ ∧Commutative yesAssociative yesClosed in Z: yesClosed in Q: yesClosed in C: yes

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(14) Which of the following are constructible numbers? Explain your answers.

(a) 3.141

(b) 3√5

(c)√

3 +√17

(d) 8√5

(e) 10√37

(f) 16√37

(g) 3√28

(h)√

13 + 3√2 +

√11

(i) 3 + 5√4

(j)√

3 +√19 +

√10

(15) Suppose that you know that all the roots of

x4 − 10x3 + 35x2 − 50x+ 24

are rational. Find them and explain your work.


x4 − 5x2 + 4



x4 − 2x3 − 13x2 + 14x+ 24


(18) Is√7 a rational number? Is it a constructible number? Explain your

answers.


answers.


answers.

(21) Is 3√7 a rational number? Is it a constructible number? Explain your

answers.


answers.


answers.

140


(24) The Rational Roots Test, Theorem 21, is actually more general than youreally need to merely prove that a number is not rational. Can you statea more specialized theorem, based upon the Rational Roots Test, thatwould do everything we need to prove that a number is not rational?Give examples of your theorem in action.

(25) Construct the golden ratio. Explain the steps in your construction.

(26) Construct the golden ratio given a unit using only the basic constructionsfor addition, division, and square roots, and the fact that:

φ =1 +

√5

2

Explain the steps in your construction.

(27) Construct a golden rectangle. Explain the steps in your construction.

(28) Construct a golden triangle. Explain the steps in your construction.

(29) Construct a golden spiral associated to a golden rectangle. Explain thesteps in your construction.

(30) Construct a golden spiral associated to a golden triangle. Explain thesteps in your construction.

(31) Construct a regular pentagon. Explain the steps in your construction.

(32) Explain Gauss’ Theorem, Theorem 23 above.

(33) Which of the following polygons can be constructed using a compass andstraightedge? Explain your answers.

(a) A regular 3-gon.

(b) A regular 5-gon.

(c) A regular 7-gon.

(d) A regular 9-gon.

(e) A regular 11-gon.

(f) A regular 12-gon.

(g) A regular 13-gon.

(h) A regular 15-gon.

(i) A regular 17-gon.

(j) A regular 34-gon.

(k) A regular 2-gon.

(l) A regular 4-gon.

(m) A regular 10-gon.

(n) A regular 20-gon.

(o) A regular 70-gon.

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4.4. IMPOSSIBILITIES

4.4 Impossibilities

Oddly enough, the importance of compass and straightedge constructions is notso much what we can construct, but what we cannot construct. It turns out thatclassifying what we cannot construct is an interesting question. There are threeclassic problems which are impossible to solve with a compass and straightedgealone:

(1) Doubling the cube.

(2) Squaring the circle.

(3) Trisecting the angle.

4.4.1 Doubling the Cube

The goal of this problem is to double the volume of a given cube. This boilsdown to trying to construct roots to the equation:

x3 − 2 = 0

But we can see that the only root of the above equation is 3√2 and we already

know that this number is not constructible.

Question Why does doubling the cube boil down to constructing a solutionto the equation x3 − 2 = 0?

?

4.4.2 Squaring the Circle

Given a circle of radius r, we wish to construct a square that has the same area.Why would someone want to do such a thing? Well to answer this question youmust ask yourself:

Question What is area?

?So what is the deal with this problem? Well suppose you have a circle of

radius 1. Its area is now π square units. How long should the edge of a squarebe if it has the same area? Well the square should have sides of length

√π units.

In 1882, it was proved that π is not the root of any polynomial equation, andhence

√π is not constructible. Therefore, it is impossible to square the circle.

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4.4.3 Trisecting the Angle

This might sound like the easiest to understand, but it’s a bit subtle. Given anyangle, the goal is to trisect that angle. It can be shown that this cannot be doneusing a compass and straightedge. However, we are not saying that you cannottrisect some angles with compass and straightedge alone, in fact there are specialangles which can be trisected using a compass and straightedge. However themethods used to trisect those special angles will fail miserably in nearly all othercases.

Question Can you think of any angles that can be trisected using a compassand straightedge?

?Just because it is impossible to trisect an arbitrary angle with compass and

straightedge alone does not stop people from trying.

Question If you did not know that it was impossible to trisect an arbitraryangle with a compass and straightedge alone, how might you try to do it?

?One common way that people try to trisect angles is to take an angle, make

an isosceles triangle using the angle, and divide the line segment opposite theangle into three equal parts. While you can divide the opposite side into threeequal parts, it in fact never trisects the angle. When you do this procedureto acute angles, it seems to work, though it doesn’t really. You can see that itdoesn’t by looking at an obtuse angle:

Trisecting the line segment opposite the angle clearly leaves the middle anglemuch larger than the outer two angles. This happens regardless of the measureof the angle. This mistake is common among people who think that they cantrisect an angle with compass and straightedge alone.

How to Trisect the Angle with a Marked Straightedge

The ancient Greek mathematicians were a tenacious bunch. While they wantedvery badly to trisect the angle with a compass and straightedge alone, this didnot stop them from devising other methods of doing it. So they cheated and useda marked straightedge. Now to trisect an angle using a marked straightedge,we will use a method attributed to Archimedes:

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4.4. IMPOSSIBILITIES

Construction (Cheating) We will show how to trisect an arbitrary angle using

a marked straightedge.

(1) Take an angle and draw a circle centered at the vertex.

(2) Draw a line extending one leg of the angle.

(3) Draw a line from the intersection of the circle and the other leg that

intersects the line drawn in Step 2, such that the part of the line outside

the circle is as long as the radius of the circle.

(4) The angle formed by the lines drawn in Step 2 and Step 3 is one third of

the original angle.

WARNING The above method of trisecting an angle is not a legitimate com-

pass and straightedge construction. If you are attempting a problem in this

chapter and the problem says to “construct,” then you may not use the above

method. You may only use the above method if the question explicitly says

“use a marked straightedge and compass.”

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(1) Explain the three classic problems that cannot be solved with a compassand straightedge alone.

(2) Use a compass and straightedge construction to trisect an angle of 90◦.Explain the steps in your construction.



(5) Use a compass and straightedge construction to trisect an angle of 67.5◦.Explain the steps in your construction.

(6) Use a marked straightedge and compass to trisect a given angle. Explainthe steps in your construction.

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Chapter 5

Isometries

And since you know you cannot see yourself, so well as by reflection,I, your glass, will modestly discover to yourself, that of yourselfwhich you yet know not of.

—William Shakespeare

5.1 Matrices as Functions

We’re going to be discussing some basic functions in geometry. Specifically, weare talking about translations, reflections, and rotations. To start us off, weneed a little background on matrices.

Question What is a matrix?

You might think of a matrix as just a jumble of large brackets and numbers.However, we are going to think of matrices as functions. Just as we write f(x)for a function f acting on a number x, we’ll write:

Mp = q

to represent a matrix M mapping point p to point q.To make things work out nice, we need to write our points all straight and

narrow, with a little buddy at the end:

(x, y)

xy1

Throughout this chapter, we will abuse notation slightly, freely interchangingseveral notations for a point:

p! (x, y)!

xy1

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CHAPTER 5. ISOMETRIES

With this in mind, our work will be done via matrices and points that look likethis:

M =

a b cd e f0 0 1

and p =

xy1

Now recall the nitty gritty details of matrix multiplication:

Mp =

a b cd e f0 0 1

xy1

=

ax+ by + c · 1dx+ ey + f · 1

0 · x+ 0 · y + 1 · 1

=

ax+ by + cdx+ ey + f

1

Question Fine, but what does this have to do with geometry?

In this chapter we are going to study a special type of functions, calledisometries.

Definition An isometry is a function M that maps points in the plane toother points in the plane such that

d(p,q) = d(Mp,Mq),

where d is the distance function.

Question How do you compute the distance between two points again?

?

We’re going to see that several ideas in geometry, specifically translations, re-flections, and rotations which all seem very different, are actually all isometries.Hence, we will be thinking of these concepts as matrices.

5.1.1 Translations

Of all the isometries, translations are probably the easiest. With a translation,all we do is move our object in a straight line. Let’s see what happens to Louie

147

5.1. MATRICES AS FUNCTIONS

Llama:

Pretty simple eh? We can give a more “mathematical” definition of a trans-lation using our newly-found knowledge of matrices! Check it:

Definition A translation, denoted by T(u,v), is a function that moves everypoint a given distance u in the x-direction and a given distance v in the y-direction. We will use the following type of matrix to represent translations:

T(u,v) =

1 0 u0 1 v0 0 1

Example Suppose that you have a point p = (−3, 2) and you want to translate

it 5 units right and 4 units down.

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Here is how you do it:

T(5,−4)p =

1 0 50 1 −40 0 1

−321

=

−3 + 0 + 50 + 2− 40 + 0 + 1

=

2−21

Hence, we end up with the point (2,−2). But you knew that already, didn’t

you?

Question Can you demonstrate with algebra why translations are isometries?

?

Question We know how to translate individual points. How do we moveentire figures and other funky shapes?

?

5.1.2 Reflections

The act of reflection has fascinated humanity for millennia. It has a strong effecton our perception of beauty and has a defined place in the field of art. We allthink we know what reflections are, here is our definition:

Definition A reflection about a line ℓ, denoted by Fℓ, is function that movesevery point p to a point Fℓp such that:

(1) If p is on ℓ, then Fℓp = p.

(2) If p is not on ℓ, then ℓ is the perpendicular bisector of the segment con-necting p and Fℓp.

You might be saying, “Huh?” It’s not as hard as it looks. Check out this

149


picture of the situation, again Louie Llama will help us out:

A Collection of Reflections

We are going to look at a trio of reflections. We’ll start with a horizontal

reflection over the y-axis. Using our matrix notation, we write:

Fx=0 =

−1 0 00 1 00 0 1

The next reflection in our collection is a vertical reflection over the x-axis.Using our matrix notation, we write:

Fy=0 =

1 0 00 −1 00 0 1

The final reflection to add to our collection is a diagonal reflection over theline y = x. Using our matrix notation, we write:

Fy=x =

0 1 01 0 00 0 1

Example Consider the point p = (3,−1). What would p look like if we

150


reflected it across the line y = x?


Fy=xp =

0 1 01 0 00 0 1

3−11

=

0− 1 + 03 + 0 + 00 + 0 + 1

=

−131

Hence we end up with the point (−1, 3).

Question Let p be some point. It is not hard to see that Fy=xp lands inthe first quadrant of the (x, y)-plane. What reflection will place this point inQuadrant II? What about Quadrant IV? What about Quadrant III?

?Question Can you demonstrate with algebra why our collection of reflectionsabove are isometries?

?Question How do we deal with reflections that are not about the lines y = 0,x = 0, or y = x? How would you reflect points about the line y = 1?

?

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5.1.3 Rotations

Imagine that you are on a swing set, going higher and higher until you areactually able to make a full circle1. At the point where you are directly abovewhere you would be if the swing were at rest, where is your head, comparatively?Your feet? Your hands?

Rotations should bring circles to mind. This is not a coincidence. Check outour definition of a rotation:

Definition A rotation of n degrees about the origin, denoted by Rn, is afunction that moves every point p to a point Rnp such that:

(1) The points p and Rnp are equidistant from the origin.

(2) An angle of n degrees is formed by p, the origin, and Rnp.

Louie Llama, can you do the honors?

WARNING We always measure our angles in a counterclockwise fashion.

Looking back on your trigonometry, there were a few special angle measure-ments, namely 90◦, 60◦, and 45◦. We’ll focus on these degrees as well.

R90 =

0 −1 01 0 00 0 1

R60 =

12

−√3

2 0√32

12 0

0 0 1

R45 =

1√2

−1√2

01√2

1√2

0

0 0 1

Example Suppose that you have a point p = (4,−2) and you want to rotate

1Face it, I think we all dreamed of doing that when we were little—or in my case, lastweek.

152


it 60◦ about the origin.


R60p =

12

−√3

2 0√32

12 0

0 0 1

4−21

=

2 +√3 + 0

2√3− 1 + 0

0 + 0 + 1

=

2 +√3

2√3− 11

Hence, we end up with the point (2 +√3, 2

√3− 1).

Question Do the numbers in the matrices above look familiar? If so, why?

?Question How do you rotate a point 180 degrees?

?Question Can you demonstrate with algebra why our rotations above areisometries?

?

153



(1) How do you compute the distance between two points p and q in theplane?

(2) What is an isometry?

(3) What is a translation?

(4) What is a rotation?

(5) What is a reflection?

(6) In what direction does a positive rotation occur?

(7) Devise a way to explain translations using compass and straightedge con-structions.

(8) Devise a way to explain translations using origami constructions.

(9) Devise a way to explain reflections using origami constructions.

(10) Devise a way to explain rotations using compass and straightedge con-structions.

(11) Consider the following matrix:

M =

0 0 82 1 00 0 1

Is M an isometry? Explain your reasoning.


M =

2 3 22 3 20 0 1



I =

1 0 00 1 00 0 1

Is I an isometry? Explain your reasoning.


M =

0 2 0−3 0 00 0 1


154



M =

1 3 23 9 60 0 1



M =

0 2 01 −1 00 0 1


(17) Use a matrix to translate the point (−1, 6) three units right and two unitsup. Sketch this situation and explain your reasoning.

(18) The matrix T(−2,6) was used to translate the point p to (−1,−3). Whatis p? Sketch this situation and explain your reasoning.

(19) Given the point p = (0,−7), what would be the result of T(4,2)T(6,−5)p?Sketch this situation and explain your reasoning.

(20) Use a matrix to reflect the point (5, 2) across the x-axis. Sketch thissituation and explain your reasoning.

(21) Use a matrix to reflect the point (−3, 4) across the y-axis. Sketch thissituation and explain your reasoning.

(22) Use a matrix to reflect the point (−1, 1) across the line y = x. Sketch thissituation and explain your reasoning.

(23) Use a matrix to reflect the point (1, 1) across the line y = x. Sketch thissituation and explain your reasoning.

(24) The matrix Fy=0 was used to reflect the point p to (4, 3). What is p?Explain your reasoning.

(25) The matrix Fy=0 was used to reflect the point p to (0,−8). What is p?Explain your reasoning.

(26) The matrix Fx=0 was used to reflect the point p to (−5,−1). What is p?Explain your reasoning.

(27) The matrix Fy=x was used to reflect the point p to (9,−2). What is p?Explain your reasoning.

(28) The matrix Fy=x was used to reflect the point p to (−3,−3). What is p?Explain your reasoning.

155


(29) Considering the point (3, 2), use a matrix to rotate this point 60◦ aboutthe origin. Sketch this situation and explain your reasoning.

(30) Considering the point (√2,−

√2), use a matrix to rotate this point 45◦

about the origin. Sketch this situation and explain your reasoning.

(31) Considering the point (−7, 6), use a matrix to rotate this point 90◦ aboutthe origin. Sketch this situation and explain your reasoning.

(32) Considering the point (−1, 3), use a matrix to rotate this point 0◦ aboutthe origin. Sketch this situation and explain your reasoning.

(33) Considering the point (0, 0), use a matrix to rotate this point 120◦ aboutthe origin. Sketch this situation and explain your reasoning.

(34) Considering the point (1, 1), use a matrix to rotate this point −90◦ aboutthe origin. Sketch this situation and explain your reasoning.

(35) The matrix R90 was used to rotate the point p to (2,−5). What is p?Explain your reasoning.

(36) The matrix R60 was used to rotate the point p to (0, 2). What is p?Explain your reasoning.

(37) The matrix R45 was used to rotate the point p to (− 12 ,

52 ). What is p?

Explain your reasoning.

(38) The matrix R−90 was used to rotate the point p to (4, 3). What is p?Explain your reasoning.

(39) If someone wanted to plot y = 3x, they might start by filling in thefollowing table:

x 3x

01

−12

−23

−3

Reflect each point you obtain from the table above about the line y = x.Give a plot of this situation. What curve do you obtain? What is itsrelationship to y = 3x? Explain your reasoning.

(40) Some translation T was used to map point p to point q. Given p = (1, 2)and q = (3, 4), find T and explain your reasoning.

(41) Some translation T was used to map point p to point q. Given p = (−2, 3)and q = (2, 3), find T and explain your reasoning.

156


(42) Some reflection F was used to map point p to point q. Given p = (1, 4)and q = (1,−4), find F and explain your reasoning.

(43) Some reflection F was used to map point p to point q. Given p = (5, 0)and q = (0, 5), find F and explain your reasoning.

(44) Some rotation R was used to map point p to point q. Given p = (3, 0)and q = (0, 3), find R and explain your reasoning.

(45) Some rotation R was used to map point p to point q. Given p = (√2,√2)

and q = (0, 2), find R and explain your reasoning.

(46) Some isometry M maps

(0, 0) 7→ (3, 2),

(1, 0) 7→ (4, 2),

(0, 1) 7→ (3, 3).

Find M and explain your reasoning.


(0, 0) 7→ (−1, 1),

(1, 0) 7→ (0, 1),

(0, 1) 7→ (−1, 2).



(0, 0) 7→ (0, 0),

(1, 0) 7→ (1, 0),

(0, 1) 7→ (0,−1).



(0, 0) 7→ (0, 0),

(1, 0) 7→ (0, 1),

(0, 1) 7→ (1, 0).



(0, 0) 7→ (0, 0),

(1, 0) 7→ (0, 1),

(0, 1) 7→ (−1, 0).


157



(0, 0) 7→ (0, 0),

(1, 0) 7→(

1

2,

√3

2

)

,

(0, 1) 7→(

−√3

2,1

2

)

.


(52) Consider the line 3x+ 4y = 2.

(a) If p is a point on the line, and the x-coordinate of p is 7, what is they coordinate?

(b) If p is a point on the line, and the x-coordinate of p is −2, what isthe y coordinate?

(c) If p is a point on the line, what are the coordinates of p in terms ofx and y?

(d) Now consider the line ax + by = c. If p is a point on the line, whatare the coordinates of p in terms of x and y?

(53) Consider T(u,v) and a line ℓ : ax + by = c. Use algebra to explain whyT(u,v)ℓ is a new line and not some other curve. Hint: See Problem (52).

(54) Consider T(u,v) and a line ℓ : ax + by = c. Use algebra to explain whyT(u,v)ℓ is a new line that is parallel to the original line. Hint: See Prob-lem (52).

158


5.2 The Algebra of Matrices

5.2.1 Matrix Multiplication

We know how to multiply a matrix and a column. Multiplying two matrices isa similar procedure:

a b cd e fg h i

j k lm n op q r

=

aj + bm+ cp ak + bn+ cq al + bo+ crdj + em+ fp dk + en+ fq dl + eo+ frgj + hm+ ip gk + hn+ iq gl + ho+ ir

Variables are all good and well, but let’s do this with actual numbers. Con-sider the following two matrices:

M =

1 2 34 5 67 8 9

and I =

1 0 00 1 00 0 1

Let’s multiply them together and see what we get:

MI =

1 2 34 5 67 8 9

1 0 00 1 00 0 1

=

1 · 1 + 2 · 0 + 3 · 0 1 · 0 + 2 · 1 + 3 · 0 1 · 0 + 2 · 0 + 3 · 14 · 1 + 5 · 0 + 6 · 0 4 · 0 + 5 · 1 + 6 · 0 4 · 0 + 5 · 0 + 6 · 17 · 1 + 8 · 0 + 9 · 0 7 · 0 + 8 · 1 + 9 · 0 7 · 0 + 8 · 0 + 9 · 1

=

1 2 34 5 67 8 9

= M

Question What is IM equal to?

?

It turns out that we have a special name for I. We call it the identity

matrix.

WARNING Matrix multiplication is not generally commutative. Check it

out:

F =

1 0 00 −1 00 0 1

and R =

0 −1 01 0 00 0 1

159

5.2. THE ALGEBRA OF MATRICES

When we multiply these matrices, we get:

FR =

1 0 00 −1 00 0 1

0 −1 01 0 00 0 1

=

1 · 0 + 0 · 1 + 0 · 0 1 · (−1) + 0 · 0 + 0 · 0 1 · 0 + 0 · 0 + 0 · 10 · 0 + (−1) · 1 + 0 · 0 0 · (−1) + (−1) · 0 + 0 · 0 0 · 0 + (−1) · 0 + 0 · 10 · 0 + 0 · 1 + 1 · 0 0 · (−1) + 0 · 0 + 1 · 0 0 · 0 + 0 · 0 + 1 · 1

=

0 −1 0−1 0 00 0 1

On the other hand, we get:

RF =

0 1 01 0 00 0 1

Question Is it always the case that (LM)N = L(MN)?

?

5.2.2 Compositions of Matrices

It is often the case that we wish to apply several isometries successively to apoint. Consider the following:

M =

a b cd e f0 0 1

N =

g h ij k l0 0 1

and p =

xy1

Now let’s compute

M(Np) =

a b cd e f0 0 1

g h ij k l0 0 1

xy1

=

a b cd e f0 0 1

gx+ hy + ijx+ ky + l

1

=

agx+ ahy + ai+ bjx+ bky + bl + cdgx+ dhy + di+ ejx+ eky + el + f

1

Now you compute (MN)p and compare what you get to what we got above.

160


Compositions of Translations

A composition of translations occurs when two or more successive translationsare applied to the same point. Check it out:

T(5,−4)T(−3,2) =

1 0 50 1 −40 0 1

1 0 −30 1 20 0 1

=

1 0 20 1 −20 0 1

= T(5+(−3),(−4)+2)

= T(2,−2)

Theorem 24 The composition of two translations T(u,v) and T(s,t) is equiva-

lent to the translation T(u+s,v+t).

Question How do you prove the theorem above?

?Question Can you give a single translation that is equivalent to the followingcomposition?

T(−7,5)T(0,−6)T(2,8)T(5,−4)

?Question Are compositions of translations commutative? Are they associa-tive?

?

Compositions of Reflections

A composition of reflections occurs when two or more successive reflections areapplied to the same point. Check it out:

Fy=0Fy=x =

1 0 00 −1 00 0 1

0 1 01 0 00 0 1

=

0 1 0−1 0 00 0 1

Question Which line does the composition Fy=0Fy=x reflect points over?

161


?

Question Are compositions of reflections commutative? Are they associa-tive?

?

Compositions of Rotations

A composition of rotations occurs when two or more successive rotations areapplied to the same point. Check it out:

R60R60 =

12

−√3

2 0√32

12 0

0 0 1

12

−√3

2 0√32

12 0

0 0 1

=

−12

−√3

2 0√32

−12 0

0 0 1

Theorem 25 The product of two rotations Ra and Rb with the same center is

equivalent to the rotation Ra+b.

From this we see that:

R120 =

−12

−√3

2 0√32

−12 0

0 0 1

Question What is the rotation matrix for a 360◦ rotation? What about a405◦ rotation?

?

Question What makes a rotation different from a reflection?

?

Question Are compositions of rotations commutative? Are they associative?

?

162


5.2.3 Mixing and Matching

Life gets interesting when we start composing translations, reflections, and ro-tations together. First we’ll take a look at a reflection mixed with a rotation:

Fy=0R60 =

1 0 00 −1 00 0 1

12

−√3

2 0√32

12 0

0 0 1

=

12

−√3

2 0

−√32 − 1

2 00 0 1

Question Does this result look familiar?

?Now how about a rotation with a translation:

R90T(3,−4) =

0 −1 01 0 00 0 1

1 0 30 1 −40 0 1

=

0 −1 41 0 30 0 1

Question What do you think it would look like if instead we had T(3,−4)R90?Would the result be the same?

?Question Find a matrix that represents the reflection Fy=−x.

I’ll take this one. Note that

Fy=−x = R180Fy=x

= R90R90Fy=x

=

0 −1 01 0 00 0 1

0 −1 01 0 00 0 1

0 1 01 0 00 0 1

=

0 −1 0−1 0 00 0 1

OK looks good, but you, the reader, are going to have to check the abovecomputation yourself.

163


Question When we mix isometries, will we retain commutativity? Associa-tivity?

?

164



(1) Give a single translation that is equivalent to T(−3,2)T(5,−1). Explain yourreasoning.

(2) Consider the two translations T(−4,8) and T(4,−8). Do these commute?Explain your reasoning.

(3) Given the point p = (7, 4), use matrices to compute T( 1

2,6)T(2,−1)p. Sketch

this situation and explain your reasoning.

(4) Given the point p = (−√3,− 1

4 ), use matrices to compute T(2, 12)T(

√3,− 1

4)p.

Sketch this situation and explain your reasoning.

(5) Give a matrix representing Fy=−x. Explain your reasoning.

(6) Give a matrix representing R−45. Explain your reasoning.

(7) Give a matrix representing R−60. Explain your reasoning.

(8) Give a single reflection that is equivalent to Fx=0Fy=0. Explain your rea-soning.

(9) Given the point p = (−4, 2), use matrices to compute Fy=0Fy=xp. Sketchthis situation and explain your reasoning.

(10) Given the point p = (5, 0), use matrices to compute Fy=xFy=−xp. Sketchthis situation and explain your reasoning.

(11) Give a single rotation that is equivalent to R45R60. Explain your reasoning.

(12) Given the point p = (1, 3), use matrices to compute R45R90p. Sketch thissituation and explain your reasoning.

(13) Given the point p = (−7, 2), use matrices to compute R45R−45p. Sketchthis situation and explain your reasoning.

(14) Given the point p = (−2, 5), use matrices to compute R90R−90R360p.Sketch this situation and explain your reasoning.

(15) Given the point p = (5, 4), use matrices to compute Fy=0T(2,−4)p. Sketchthis situation and explain your reasoning.

(16) Given the point p = (−1, 6), use matrices to compute R45T(0,0)p. Sketchthis situation and explain your reasoning.

(17) Given the point p = (11, 13), use matrices to compute T(−6,−3)R135p.Sketch this situation and explain your reasoning.

(18) Given the point p = (−7,−5), use matrices to compute R540Fx=0p. Sketchthis situation and explain your reasoning.

165


(19) Given the point p = (8, 1), use matrices to compute R90Fy=xT(−2,3)p.Sketch this situation and explain your reasoning.

(20) Give a single isometry that is equivalent to Fy=−xT(−3,10). Also give asingle isometry that is equivalent to T(−3,10)Fy=−x. Explain your reason-ing.

(21) Give a single isometry that is equivalent to R30Fy=0. Also give a singleisometry that is equivalent to Fy=0R30. Explain your reasoning.

166


5.3 The Theory of Groups

One of the most fundamental notions in all of modern mathematics is that of agroup. Sadly, many students never see a group in their education.

Definition A group is a set of elements (in our case matrices) which wewill call G that is closed under an associative operation (in our case matrixmultiplication) such that:

(1) There exists an identity I ∈ G such that for all M ∈ G,

MI = M.

(2) For all M ∈ G there is an inverse M−1 such that

MM−1 = I.

5.3.1 Groups of Reflections

Let’s see a basic group:

Question Consider the following equilateral triangle.

How many different ways can we reflect this triangle back on to itself?

With a triangle there are only three reflections that preserve the location ofthe vertexes of the triangle. The easiest of these is the reflection over Fx=0.

Let’s use this as the basis for our first group table. We’ll start with just twoelements: I and Fx=0.

◦ I Fx=0

I I Fx=0

Fx=0 Fx=0 I

167

5.3. THE THEORY OF GROUPS

Notice what happens when we apply Fx=0 twice, we’re right back where westarted. Hence, Fx=0Fx=0 = I. Since matrix multiplication is associative, we seethat

{I,Fx=0}forms a group. Specifically this is a group of reflections of the triangle.

Question Above, we reflected across the line x = 0. What are the equationsof the other two lines of symmetry for the triangle?

?Question Would these same equations work with a square? If not, whatequations would they be?

?

5.3.2 Groups of Rotations

How many degrees does it take to rotate an equilateral triangle so that thevertexes are still at the same positions? Well, we have 3 angles and 360◦ ofrotation. That gives us 120◦ for each rotation. Remember the matrix for a 120◦

rotation?

R120 =

−12

−√3

2 0√32

−12 0

0 0 1

Question Consider the following equilateral triangle.

How many different ways can we rotate this triangle back on to itself?

Because R120R120 does not give us the identity, we need to include it in ourgroup table. We will write this as R2

120. However, if we apply the R120 rotationone more time, we do get back to the identity. This shows that R120 and R

2120

are inverses. Everything now has an inverse and we can complete our rotationgroup table:

◦ I R120 R2120

I I R120 R2120

R120 R120 R2120 I

R2120 R

2120 I R120

168


Question What kind of rotation matrices would we use when working witha square? A pentagon? A hexagon?

?

5.3.3 Symmetry Groups

What happens when we mix rotations and reflections? Take for instance if wego back to our triangle and do Fx=0R120:

What you may not immediately notice is that we obtain the same transformationby taking the original triangle and reflecting it over the line y = 1√

3x.

As it turns out, every possible arrangement of the vertexes of the equilat-eral triangle can be represented using compositions of reflections and rotations.Hence we’ll call such a composition a symmetry of the triangle. The collectionof all symmetries forms a group called the symmetry group of the equilateraltriangle. Let’s see the group table:

(D3, ◦) I R120 R2120 Fx=0 Fx=0R120 Fx=0R

2120

I I R120 R2120 Fx=0 Fx=0R120 Fx=0R

2120

R120 R120 R2120 I Fx=0R

2120 Fx=0 Fx=0R120

R2120 R

2120 I R120 Fx=0R120 Fx=0R

2120 Fx=0

Fx=0 Fx=0 Fx=0R120 Fx=0R2120 I R120 R

2120

Fx=0R120 Fx=0R120 Fx=0R2120 Fx=0 R

2120 I R120

Fx=0R2120 Fx=0R

2120 Fx=0 Fx=0R120 R120 R

2120 I

This table shows every symmetry of the triangle. By comparing the rows andcolumns of the group table, you can see that every element has an inverse andthe identity is included. This combined with the fact that matrix multiplicationis associative shows that the symmetries of the triangle form a group.

169


Question Can you express the symmetries of the square in terms of reflec-tions and rotations? What does the group table look like for the symmetrygroup of the square?

?

170



(1) How many lines of symmetry exist for a square? Provide a drawing tojustify your answer.

(2) How many lines of symmetry exist for a hexagon? Provide a drawing tojustify your answer.

(3) What are the equations for the lines of symmetry that exist for the square?Explain your answers.

(4) What are the equations for the lines of symmetry for a hexagon? Explainyour answers.

(5) Give the group table for the reflections of a square.

(6) Give the group table for the reflections of a hexagon.

(7) Give the group table for the rotations of a square.

(8) Give the group table for the rotations of a hexagon.

(9) Give the group table for the symmetries of a square.

(10) Give the group table for the symmetries of a hexagon.

(11) How many consecutive rotations are needed to return the vertexes of asquare to their original position? Provide a drawing to justify your answer,labeling the vertexes.

(12) How many degrees are in one rotation of the square? Explain your answer.

(13) How many rotations are needed to return to the identity in a hexagon?Provide a drawing to justify your answer, labeling the vertexes.

(14) How many consecutive rotations are needed to return the vertexes of ahexagon to their original position? Provide a drawing to justify youranswer, labeling the vertexes.

(15) In this section, we’ve focused on a 3-sided figure, a 4-sided figure, and a6-sided figure. Why do we not include the rotation group for the pentagonin this section? If we did, how many degrees would be in one rotation?

(16) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is R120. Explain and illustrate youranswer.

(17) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is Fx=0. Explain and illustrate youranswer.

171


(18) Find two symmetries of the equilateral triangle, neither of which is theidentity, such that their composition is R2

120. Explain and illustrate youranswer.

(19) Find two symmetries of the square, neither of which is the identity, suchthat their composition is R180. Explain and illustrate your answer.

(20) Find two symmetries of the square, neither of which is the identity, suchthat their composition is Fy=x. Explain and illustrate your answer.

(21) Find two symmetries of the square, neither of which is the identity, suchthat their composition is R270. Explain and illustrate your answer.

(22) Find two symmetries of the square, neither of which is the identity, suchthat their composition is Fx=0. Explain and illustrate your answer.

172

Chapter 6

Convex Sets

If people do not believe that mathematics is simple, it is only becausethey do not realize how complicated life is.

—John von Neumann

6.1 Basic Definitions

The following pictures are examples of convex sets :

The next set of pictures are not examples of convex sets:

Question What is the difference between a set which is convex and a setwhich is not convex?

Definition A set X is convex if for all points A, B in X the line segment ABis also in X.

Question Is a straight line a convex set?

?

173

6.1. BASIC DEFINITIONS

From two separate convex sets, we can build new ones:

Theorem 26 The intersection of two convex sets is a convex set.

Proof Let X1 and X2 be two convex sets. Let

Y = X1 ∩X2.

If A, B are points in X1, then the line segment AB is a subset of X1. Likewise,if A, B are points in X2, then the line segment AB is a subset of X2. Thus wesee that if A, B are points in Y , the line segment AB is a subset of Y . ThusY = X1 ∩X2 is convex. �

Question If X1, X2, and X3 are convex, is X1 ∩X2 ∩X3 convex?

?Question Can you generalize the above question? Is the intersection of 4convex sets convex? What about 5 convex sets? What about n convex sets?

?Imagine you are blind. How would you identify convex sets in real life? One

way to identify convex sets would be with your hands. A mathematical analogueof this is the idea of supporting lines. A supporting line is like a rigid stick thatone presses against the set in question. If you can “wedge” a point of the stickinto the set, then it is not convex. To be more explicit let’s look at the nextdefinition:

Definition If X is a set with interior points, a supporting line for X is aline that goes through at least one boundary point of X and cuts the plane inhalf such that X is only on one side.

Pictorially we have this as an example:

Theorem 27 A line is a supporting line for a convex set with interior points if

and only if it goes through at least 1 boundary point of the set and no interior

points.

Question In the above theorem, why does the set need to have interiorpoints? Why does the set need to be convex?

?

174

CHAPTER 6. CONVEX SETS

Since we are studying mathematics, we should ask ourselves, “How do weprove the above theorem?” Well, the proof is not hard, but you need to remem-ber the logic we did earlier.

Proof Let

P = {a line is a supporting line for a convex set}and

Q =

{a line goes through at least 1 boundary pointof a convex set and no interior points

}

Since the statement of the theorem above is:

P ⇔ Q

we must prove:(P ⇒ Q) ∧ (Q ⇒ P )

We will do this in two steps, proving the left-hand side first, and then provingthe right-hand side.

(⇒) Suppose that P ⇒ Q is false. In other words, suppose that

¬(P ⇒ Q) = P ∧ (¬Q)

is true. So we are supposing a line is a supporting line and it does not go throughat least 1 boundary point or it goes through some interior points. Since it isa supporting line, it must touch the boundary. So could it go through someinterior points? If so then there is a small zone around those points containedentirely inside the convex set. Thus we may pick points in our set on either sideof the line. Thus our line cannot be a supporting line, a contradiction.

(⇐) Suppose that Q ⇒ P is false. In other words, suppose that

¬(Q ⇒ P ) = Q ∧ (¬P )

is true. So we are supposing a line goes through at least 1 boundary point andno interior points and it is not a supporting line for a convex set. The points ofthe set must be on both sides of the line. Therefore, each line connecting thepoints on both sides of the line are in the set, as the set is convex. Hence, theline must touch interior points. �

Definition A supporting half-plane is the half-plane formed by a support-ing line which contains the set.

Here is a picture of a supporting half plane to help you understand thedefinition:

The supporting half-plane is shaded in the picture above.

175


Theorem 28 A convex set is the intersection of all its supporting half-planes.

Question How could you use the above theorem to define a triangular region?How could you use the above theorem to prove that a triangular region is convex?

?

6.1.1 An Application

Convex sets actually have uses in the real world. One use that we can discusseasily is that they can help us maximize or minimize certain functions. Checkout the next theorem:

Theorem 29 Consider a function f(x, y) = ax+ by+ c, and consider a convex

polygonal region P in the (x, y)-plane. Then:

(1) The maximum value for f(x, y) occurs at a vertex of P .

(2) The minimum value for f(x, y) occurs at a vertex of P .

Question How do you think about functions of two variables? Hint: Haveyou ever seen a topographical map?

?Example Suppose you’re taking a math exam, 60 problems total, 30 of which

are worth 7 points and 30 of which are worth 6 points. You are instructed to do

only 30 problems and you have 105 minutes to do it. It takes you 4 minutes to

do the 7 point problems and 3 minutes to do the 6 point problems. Assuming

you make no mistakes, how many of each problem should you do in order to

maximize your score?

To solve this problem first set

x = number of 7 point problems done.

y = number of 6 point problems done.

Now we have

score(x, y) = 7x+ 6y

We want to maximize score(x, y). What sort of constraints do we have? Well,

we can only do 30 problems so we have:

x+ y 6 30 ⇒ y 6 x+ 30

We only have 105 minutes to do our problems so:

4x+ 3y 6 105 ⇒ y 6 (−4/3)x+ 35

176


So what is the mystery point which maximizes the score? Well we have two

equations:

y = −x+ 30

y = (−4/3)x+ 35.

Setting them equal to each other we find:

(−4/3)x+ 35 = −x+ 30

Solving for x we get that x = 15 and by plugging 15 in for x in one of the above

equations we see that y = 15 too. Now look at our region:

5 10 15 20 25

10

20

30

5 10 15 20 25

10

20

30

Since the shaded region is a convex set, the theorem above says that the maxi-

mum must occur at a vertex. Plugging the appropriate points in we find:

score(0, 0) = 0,

score(0, 30) = 180,

score(26.25, 0) = 183.75,

score(15, 15) = 195.

So we see that solving 15 of the 7 point problems and 15 of the 6 point problems

is best.

177



(1) What is a convex set?

(2) State which of the following are convex sets:

(a) A straight line.

(b) Two lines intersecting at a point.

(c) Two parallel lines.

(d) A half-plane.

(e) The entire plane.

(f) A solid disk.

(g) An empty box.

(h) A solid 7 pointed star.

Explain your answers.

(3) Describe all possible 1-dimensional convex sets.

(4) Draw a series of diagrams illustrating the steps in the proof of Theorem 26.

(5) Draw a diagram showing how the conclusion of Theorem 26 is false ifeither X1 or X2 are not convex.

(6) Prove that the intersection of two convex sets is a convex set.

(7) Is the union of two convex sets ever convex? Is the union of two convexsets always convex?

(8) Draw a diagram showing how the conclusion of Theorem 27 is false if theset in question is not convex.

(9) Draw a series of diagrams illustrating the steps in the proof of Theorem 27.

(10) Use the fact that a half-plane is a convex set to prove that a triangularregion is a convex set.

(11) Use the fact that a half-plane is a convex set to prove that a square regionis a convex set.

(12) Draw a diagram showing how the conclusion of Theorem 28 is false if theset in question is not convex.

(13) Draw three nonconvex sets and identify one point on the boundary of eachthrough which no supporting line passes.

(14) Consider a bounded convex set. How many boundary points does a rayoriginating the from the interior of the convex set hit? Explain youranswer.

178


(15) Is the following picture a counterexample to Theorem 29? Why or whynot?

(16) Dieter is a farmer who owns 45 acres of land on each acre of which heplans to grow either wheat or corn. Here are the facts:

• Each acre of wheat yields $200 profit and requires 2 tons of fertilizer.

• Each acre of corn yields $300 profit and requires 4 tons of fertilizer.

Only 120 tons of fertilizer are available. How many acres of wheat and howmany acres of corn should Dieter plant to maximize his profit? Explainyour work.

(17) Jennifer is a carpenter who makes desks and chairs. Here are the facts:

• Each desk sells for a profit of $40 and requires 4 units of wood.

• Each chair sells for a profit of $25 and requires 2 units of wood.

If only 16 units of wood are available and she has to make at least twiceas many chairs as desks, how many desks and how many chairs shouldJennifer make to maximize her profit? Explain your work.

(18) Brubaki Breweries produces two type of beer: David’s Death-From-AboveStout and Han’s Honey-Delight Ale. Here are the facts:

• David’s Death-From-Above Stout makes a profit of $5 a barrel, andeach barrel requires 6 lbs of barley and 3 oz of hops.

• Han’s Honey-Delight Ale makes a profit of $2 a barrel, and eachbarrel requires 3 lbs of barley and 1 oz of hops.

If 60 lbs of barley and 25 oz of hops are available, how many barrels ofeach type of beer should be made to maximize profit? Explain your work.

(19) A clock-smith and her assistant produce two types of watches, a fancywatch and a utilitarian watch. Here are the facts:

• The fancy watch sells for a profit of $50, and requires 1 hour of workfrom the clock-smith and 2 hours of detailing by the assistant.

• The utilitarian watch sells for a profit of $40, and requires 1 hour ofwork from the clock-smith and 1/2 hour of detailing by the assistant.

If both people will only work 12 hours a day, how many of each type ofwatch should be manufactured each day for a maximum profit? Explainyour work.

179

6.2. CONVEX SETS IN THREE DIMENSIONS

6.2 Convex Sets in Three Dimensions

6.2.1 Analogies to Two Dimensions

Before we were doing things in two dimensions. A central idea in all of math-ematics is that of generalization. In this section we are going to generalize theideas we had before to three dimensions. Let’s start with a question:

Question Does the idea of a supporting line make sense in three dimensions?

?Definition A plane π is called a supporting plane for a three-dimensionalset S if π contains a boundary point of S and S lies entirely in one of thehalf-spaces determined by π.

WARNING Be careful, we currently have two very different ideas, the sup-

porting half-plane and supporting plane. Make sure you know the difference!

Since the supporting plane is the analogue of the supporting line, we have anew idea for the analogue of the supporting half-plane:

Definition A supporting half-space is the half-space determined by a sup-porting plane containing the set.

Now that we have our new definitions, we also have analogous theorems:

Theorem 30 (See Theorem 27) A plane is a supporting plane for a convex

set if and only if it goes through at least 1 boundary point of the set and no

interior points.

Theorem 31 (See Theorem 28) A three-dimensional convex set is the inter-

section of all its supporting half-spaces.

6.2.2 Platonic Solids

Between 500 BC and 300 BC, there was a group of mystics—today we mightcall it a cult—who called themselves the Pythagoreans. Being mystics, thePythagoreans had some strange ideas, but on the other hand they were anenlightened group of people, because they believed that they could better un-derstand the universe around them by studying mathematics. Being that thecurrent physical sciences have strong roots in mathematics, you must agree thatwe side with the Pythagoreans on this issue.

As part of the Pythagoreans’ numerological religion, they thought that someconvex solid bodies had special powers. The Pythagoreans associated regularconvex polyhedra to elements of nature as follows:

180


Fire: Tetrahedron Air: Octahedron

Earth: Cube Water: Icosahedron

However, the Pythagoreans knew of one more regular convex polyhedra, thedodecahedron:

Æther: Dodecahedron

Since the first four regular convex solids had elements associated to them, thePythagoreans reasoned that this fifth solid must also be associated to an el-ement. However, all the elements were accounted for. So the Pythagoreansassociated the dodecahedron to what we might call the æther, a mysteriousnon-earthly substance.

Question Look at the regular convex polyhedra above. What does the wordregular mean in the phrase regular convex polyhedra? In particular why is atriangular dipyramid, the shape which is two tetrahedrons joined at a face,not a regular polyhedra. How does your definition of the word regular abovediscount the above polyhedra from being regular?

?Question How could we use Theorem 31 to define the Platonic Solids?

?

181


Question Are there other regular convex polygons other than the ones shownabove?

We’ll give you the answer to this one:

Theorem 32 There are at most 5 regular convex polyhedra.

Proof To start, a corner of a three-dimensional object made of polygons musthave at least 3 faces. Now start with the simplest regular polygon, an equilateraltriangle. You can make a corner by placing:

(1) 3 triangles together.



Thus each of the above configuration of triangles could give rise to a regular con-vex polyhedra. However, you cannot make a corner out of 6 or more equilateraltriangles, as 6 triangles all connected lie flat on the plane.

Now we can make a corner with 3 square faces, but we cannot make a cornerwith 4 or more square faces as 4 or more squares lie flat on the plane.

Finally we can make a corner with 3 faces each shaped like a regular pen-tagon, but we cannot make a corner with 4 or more regular pentagonal faces asthey will be forced to overlap.

Could we make a corner with 3 faces each shaped line a regular hexagon? No,because any number of hexagons will lie flat on the plane. A similar argumentworks to show that we cannot make a corner with 3 faces shaped like a regularn-gon where n > 6, except now instead of the shapes lying flat on the plane,they overlap.

Thus we could at most have 5 regular convex solids. �

Question Above we prove that there are at most 5 regular convex polyhedra.How do we know that these actually exist?

?The five regular convex polyhedra came to be known as thePlatonic Solids,

after Plato discussed them in his work Timaeus. To help you on your way, hereis a table of facts about the Platonic Solids:

Solid Faces Edges Vertexes

tetrahedron 4 6 4octahedron 8 12 6cube 6 12 8icosahedron 20 30 12dodecahedron 12 30 20

182


These solids have haunted men for centuries. People who were trying tounderstand the universe wondered what was the reason that there were only5 regular convex polyhedra. Some even thought there was something special

about certain numbers.

Question In what ways does the number 5 come up in life that might lead aperson into believing it is a special number? Does this mean that the number5 is more special than any other number?

?

183



(1) Explain what a supporting half-plane and a supporting plane are, anddiscuss how they are different.

(2) State which of the following are convex sets:

(a) A hollow sphere.

(b) A half-space.

(c) The intersection of two spherical solids.

(d) A solid cube.

(e) A solid cone.

(f) The union of two spherical solids.

Explain your answers.

(3) Use words and pictures to describe the following objects:

(a) A tetrahedron.

(b) An octahedron.

(c) A cube.

(d) An icosahedron.

(e) A dodecahedron.

(f) A triangular dipyramid.

(4) How is a regular convex polyhedra different from any old convex polyhe-dra?

(5) True or False: If true, explain why. If false explain why.

(a) There are only 5 convex polyhedra.

(b) The icosahedron has exactly 12 faces.

(c) Every vertex of the tetrahedron touches exactly 3 faces.

(d) The octahedron has exactly 6 vertexes.

(e) Every vertex of the dodecahedron touches exactly 3 faces.

(6) While there are only 5 regular convex polyhedra, there are also nonconvexregular polyhedra. Draw some examples.

(7) Draw picture illustrating the steps of the proof of Theorem 32.

(8) Where does the proof of Theorem 32 use the fact that the solids areconvex?

(9) A dual polyhedron is the polyhedron obtained when one connects thecenters of all the pairs of adjacent faces of a given polyhedron. What arethe dual polyhedra of the Platonic Solids?

184


6.3 Ideas Related to Convexity

6.3.1 The Convex Hull

So far we have talked a lot about convex sets, but what about nonconvex sets?For this occasion, we have the idea of a convex hull, which is sometimes calledthe convex envelope or convex cover :

Definition The convex hull of a set S is the intersection of all the convexsets containing S.

Here are some examples of sets and their convex hull:

Theorem 33 A set is convex if and only if it is its own convex hull.

Proof (⇒) This is clear from the definition of convex hull.(⇐) If a set is its own convex hull, then it is the intersection of convex sets

and so it is convex. �

Question Can you define the Platonic Solids using convex hulls?

?

6.3.2 Sets of Constant Width

NASA

When the space shuttle launches into space there are three sets of rocket enginesused:

• The main shuttle engines.

• Two rocket boosters on either side of the shuttle.

Minutes after the shuttle is launched, the two rocket boosters separate from theshuttle and fall back to Earth, landing in the ocean. Later people retrieve therocket boosters and they are reused in future missions.

Question What happens when you heat up metal and then cool it downrapidly?

185

6.3. IDEAS RELATED TO CONVEXITY

?Since the rocket boosters are being heated and rapidly cooled, and also are

being stressed as by the enormous forces of the launch, they become slightlywarped. Being that the boosters are reused, they come apart, but if they arewarped, then they do not go back together as well as they should. If the boostersdo not go back together very well, then the boosters could start burning theirfuel in unexpected ways, and cause an explosion.

Question If you worked at NASA, how would you check to see if the boostersare still round?

?Before the Challenger accident, here is how NASA did it: They measured

the width of the booster every 60 degrees. However, this doesn’t work! Thereare many shapes with constant width besides the circle! To explain this, firstwe should say what we mean by the width of a set:

Definition If we have a set, with two parallel supporting lines, then a width

of the set is the distance between those supporting lines.

Most sets do not have constant width. However, there are sets with constantwidth. The easiest example is a circle. But another example is the Reuleaux

triangle:

Thus the method NASA was using to see how round their boosters were wasnot a sound one, as there are many noncircular sets with constant width.

To obtain more examples of sets of constant width, you can draw one your-self. We will give two methods of drawing sets of constant width.

Drawing a Set of Constant Width: Method 1

(1) Start with a regular polygon with a odd number of sides.

(2) Place the pointy end of a compass on a vertex of the polygon, and drawan arc spanning the edge opposite to the vertex you started with.

(3) Repeat the process for every vertex of the polygon.

Method 1 is how the Reuleaux triangle is drawn. Any set of constant widthdrawn using Method 1 is called a Reuleaux polygon. Here is another way ofdrawing sets of constant width:

186


Drawing a Set of Constant Width: Method 2

(1) Draw some straight lines that all intersect each other.

(2) Put the pointy end of your compass on an intersection of two of the lines.

(3) Draw an arc connecting the lines that intersect on one side of their inter-section.

(4) Moving clockwise around, extend the arc that was drawn before connectingthe next two lines with the pointy end of the compass at their intersection.You will need to pick up the compass and resize it to do this.

(5) Repeat step 5 until the curve closes.

Here is a picture which may help explain the above algorithm:

In the above picture, the first three arcs are drawn and the point which eacharc was drawn around is circled. If the reader continues around in the samefashion, the curve will close and the final set will be of constant width.

187

6.3. IDEAS RELATED TO CONVEXITY


(1) Draw the convex hull of the following:

(a) A circle.

(b) Two distinct points.

(c) Three distinct points, not all on the same line.

(d) Two parallel line segments.

(e) Two intersecting Line segments.

(f) Seven points in a line.

(g) Two circles that intersect.

(h) Two circles that do not intersect.

(i) The shape of the letter “D”.

(j) The shape of the letter “X”.

(2) True or False: If true, explain why. If false, give a counterexample.

(a) The convex hull of a set is always a convex set.

(b) The convex hull of a finite set of points is a finite set of points.

(c) A set of points could have two different convex hulls.

(d) The convex hull of a half-plane is the whole plane.

(e) The convex hull of a two-dimensional set is always two-dimensional.

(f) A line is its own convex hull.

(3) True or False: If true, explain why. If false give a counterexample.

(a) Every width of the Reuleaux triangle is the same.

(b) The intersection of a two Reuleaux triangles is always a Reuleauxtriangle.

(c) The intersection of a two Reuleaux triangles is never a Reuleauxtriangle.

(d) The Reuleaux triangle has 3 corner points.

(e) The Reuleaux triangle is a polygon.

(4) Draw a set of constant width using one of the methods described above.

(5) Find the perimeter of a Reuleaux triangle in terms of its constant width.Explain your work.

(6) Find the perimeter of a Reuleaux pentagon in terms of its constant width.Explain your work. Hint: Recall that if an angle of measure d degreeshas its vertex on a circle of circumference 1, then the arc spanned by theintersection of the edges of the angle with the circle is 2d/360.

188


(7) Find the perimeter of a Reuleaux 7-gon in terms of its constant width.Explain your work. Hint: Recall that if an angle of measure d degreeshas its vertex on a circle of circumference 1, then the arc spanned by theintersection of the edges of the angle with the circle is 2d/360.

(8) Considering the previous 3 exercises, make a conjecture about the perime-ter of a set of constant width.

189

6.4. ADVANCED THEOREMS

6.4 Advanced Theorems

In this section we simply state several advanced theorems related to the aboveconcepts and then ask questions about their statements.

Theorem 34 Let X1 and X2 be two convex sets. If X2 ⊆ X1, then the

perimeter of X1 is greater than or equal to the perimeter of X2.

Question What does this this theorem look like?

?Question If you leave off or change the assumptions of the theorem, is it stilltrue?

?Theorem 35 (Helly) Let X1, . . . , Xn be n convex sets lying in d-space with

n > d+1 such that any collection of d+1 of the sets has a nonempty intersection.

Then the intersection of all the sets is nonempty.



?Theorem 36 (Radon) Let S = {P1, . . . , Pn} be a set of points in d-space, ifn > d + 2, then we can partition the set into two disjoint sets whose convex

hulls intersect.



?Theorem 37 Let S = {P1, . . . , Pn} be a set of n points in the plane. Then

there is a point A such that every half-plane determined by a line through Acontains at least n/4 points of S.


190


?Question Will any point for A work? Or does A need to be chosen somewhatcarefully?

?Theorem 38 If we have a set S of n points in a plane such that each set of

3 points can be enclosed in a circle of radius 1, then every point in S can be

enclosed in a single circle of radius 1.



?Theorem 39 Given n parallel line segments in a plane, if there exists a line

that intersects all sets of 3 of them, then there is a line that intersects all of

them.



?

191

6.4. ADVANCED THEOREMS


(1) Draw a picture depicting the statement of Theorem 34 and give a shortexplanation of how your picture depicts the statement.

(2) Give an example showing that the conclusion of Theorem 34 does not holdfor nonconvex sets.

(3) Draw a picture depicting the statement of Helly’s Theorem and give ashort explanation of how your picture depicts the statement.

(4) Give an example where d = 2 showing that the conclusion of Helly’sTheorem does not hold for nonconvex sets.

(5) Draw a picture depicting the statement of Radon’s Theorem and give ashort explanation of how your picture depicts the statement.

(6) Give an example where n = 3 and d = 2 showing that the conclusion ofRadon’s Theorem does not hold in this case.


(8) Give an example showing that the conclusion Theorem 37 does not holdfor all choices of the point A.



(11) Give an example of four parallel line segments such that a line intersects3 of them, yet no line intersects all the segments simultaneously.

(12) Give an example of three parallel line segments such that some line in-tersects every pair of line segments, yet no line intersects all three of thesegments simultaneously.

192

References and FurtherReading

[1] N. Bourbaki. Elements of the History of Mathematics. Springer-Verlag,1984.

[2] C.B. Boyer and Uta C. Merzbach. A History of Mathematics. Wiley, 1991.

[3] R.G. Brown. Transformational Geometry. Ginn and Company, 1973.

[4] H. Dorrie. 100 Great Problems of Elementary Mathematics: Their History

and Solution. Dover, 1965.

[5] Euclid and T.L. Heath. The Thirteen Books of Euclid’s Elements: Volumes

1, 2, and 3. Dover, 1956.

[6] H.M. Enzensberger. The Number Devil: A Mathematical Adventure.Metropolitan Books, 1998.

[7] H. Eves. An Introduction to the History of Mathematics. Saunders CollegePublishing, 1990.

[8] R.P. Feynman. “What Do You Care What Other People Think?”. Norton,2001.

[9] M. Gardner. The Colossal Book of Mathematics. Norton, 2001.

[10] . Mathematics, Magic and Mystery. Dover, 1956.

[11] J. Gullberg. Mathematics: From the Birth of Numbers. Norton, 1997.

[12] S. Hawking. God Created the Integers. Running Press, 2005.

[13] J.L. Heilbron. Geometry Civilized. Oxford, 1998.

[14] M. Jeger. Transformation Geometry. John Wiley & Sons, 1966.

[15] E.F. Krause. Taxicab Geometry: An Adventure in Non-Euclidean Geome-

try. Dover, 1986.

193

REFERENCES AND FURTHER READING

[16] M. Livio. The Golden Ratio: The Story of Phi, the World’s Most Aston-

ishing Number. Broadway Books, 2002.

[17] C.H. MacGillavry. Fantasy & Symmetry: The Periodic Drawings of M.C.

Escher. Harry N. Abrams Inc, 1965.

[18] R.B. Nelsen. Proofs Without Words. Mathematical Association of America,1993.

[19] . Proofs Without Words II. Mathematical Association of America,2000.

[20] G. Polya. Mathematical Discovery: On Understanding, Learning, and

Teaching Problem Solving. Wiley, 1981.

[21] C. Sagan. Cosmos. Random House, 2002.

[22] T.S. Row. Geometric Exercises in Paper Folding. The Open Court Pub-lishing Company, 1901.

[23] J.R. Smart. Modern Geometries. Brooks/Cole, 1998.

[24] E.W. Weisstein. MathWorld—A Wolfram Web Resource.

http://mathworld.wolfram.com/

[25] D. Wells. The Penguin Dictionary of Curious and Interesting Geometry.

Penguin Books, 1991.

194

Index

æther, 181algebraic geometry, 14altitude, 85, 88analytic geometry, 17and, 45

and probability, 107antipodal point, 6Archimedes, 143area

Heron’s Formula, 93Pick’s Theorem, 94

arithmetic mean, 61Arithmetic-Geometric Mean Inequality,

61associative, 134axiom, 4

Battle Royale, 27beauty, see truthbeer, 47, 179bees, 55Bertrand’s paradox, 104boat

lost at night, 129brain juices, 129bucket

Alma Mater, 35

C, 132calisson, 68Cartesian plane, 14Cavalieri’s Principle, 69, 70center of mass, 86centroid, 86, 88, 122circle

circumcircle, 84City Geometry, 29

incircle, 85Spherical Geometry, 6

circumcenter, 83, 88, 122circumcircle, 84, 122circumscribe, 84, 93City Geometry, 23, 24

circle, 29midset, 29parabola, 32triangle, 28

closed, 134collapsing compass, 117commutative, 134compass

collapsing, 117compass and straightedge

addition, 132bisecting a segment, 117bisecting an angle, 118copying an angle, 119division, 133equilateral triangle, 116golden ratio, 137impossible problems, 142multiplication, 133parallel through a point, 120pentagon, 137perpendicular through a point, 118SAA triangle, 125SAS triangle, 124SSS triangle, 124subtraction, 132tangent to a circle, 120transferring a segment, 117

complement, 40conic-section, 13constructible numbers, 132

195

INDEX

constructions, see compass and straight-edge or paper-folding

continued fraction, 97convex, 173convex hull, 185convex polyhedra

regular, 181Crane Alley, 26

diagonal reflection, 150dissection proof, 57doubling the cube, 60, 142dual

polyhedron, 184

∈, 38, 132e, 98The Elements, 1empty set, 40equilateral triangle, 89Eratosthenes, 2Escher, M.C., 51Euclid, 1, 115Euclidean Geometry, 4Euler line, 88, 123

field, 134fractional part, 98free point, 116

Galileo Galilei, 67Gauss, Carl Friedrich, 63, 138geometric mean, 61geometry

algebraic, 14analytic, 17City, 24Spherical, 5synthetic, 13

geometry,City, 23goat, 104golden

ratio, 100, 137compass and straightedge, 137

rectangle, 101spirals, 101

triangle, 101, 138gorilla suit, 24great circle, 5group, 167

Helly’s Theorem, 190Heron’s Formula, 93

for quadrilaterals, 93for triangles, 93

horizontal reflection, 150

identity matrix, 159if-and-only-if, 48if-then, 46, 47iff, 48incenter, 84, 122incircle, 85, 122integers, 132intersection, 39isometry, 147

Kepler, Johannes, 51

lemma, 93lemon, see lemmaLet’s Make a Deal, 104logical symbols

∧, 45, 107⇔, 48⇒, 47¬, 45∨, 45, 107

Louie Llama, 148, 150, 152

matrix, 146multiplication, 147

median, 86midset

City Geometry, 29Miquel point, 89, 123Miquel’s Theorem, 89Monty Hall problem, 104Morley’s Theorem, 89

NASA, 185nexus of the universe, 23Nine-Point Circle, 88, 123

196

INDEX

nontransitive dice, 106not, 46

or, 45and probability, 107

orthocenter, 85, 88, 122

paper-foldingMorley’s Theorem, 89the centroid, 86the circumcenter, 84the incenter, 84the orthocenter, 85

parabolaalgebraic definition, 15City Geometry, 32conic-section definition, 13intrinsic definition, 12

paradox, 32√2 = 2, 33

1 = 0.999 . . . , 94all triangles are isosceles, 90Bertrand’s paradox, 104involving dice, 106the Monty Hall problem, 104triangle dissection, 59

parallel postulate, 5pentagon, 137φ, 100π, 98, 103Pick’s Theorem, 94Plato, 115Platonic Solids, 180, 182polyhedra

convex regular, 181prime numbers, 138probability, 102Pythagorean Theorem, 56, 124The Pythagoreans, 180

Q, 133Q(α), 136quadrilateral

tessellation of, 51

Radon’s Theorem, 190

rational numbers, 134rational roots test, 136reflection, 149

diagonal, 150horizontal, 150vertical, 150

regularconvex polyhedra, 180tessellation, 51

Reuleauxpolygon, 186triangle, 186

Rock-Paper-Scissors, 106rotation, 152

set, 38set theory symbols

−, 40∅, 40∈, 38, 132∩, 39⊆, 39∪, 39

sets of constant width, 185drawing, 186, 187

slopef(x)(x, s), 17Socrates, 115Spherical Geometry, 5

great circle, 5Spherical Geometry

circle, 5line, 5triangle, 7

spirals, 101√2, 33, 97, 99

squaring the circle, 142subset, 39supporting

half-plane, 175half-space, 180line, 174plane, 180

symmetry groupgroup, 169

synthetic geometry, 13

197

INDEX

tablestruth, 45

tangent line, 13, 15, 16Taxicab distance, 23tessellation, 51

any quadrilateral, 51regular, 51triangles, 51

three doors, 104translation, 148triangle

altitude, 85centroid, 86circumcenter, 83circumcircle, 84City Geometry, 28incenter, 84incircle, 85orthocenter, 85Spherical Geometry, 7

triangular dipyramid, 181trisecting the angle, 89, 143

by cheating, 144truth, see beautytruth-table, 45

∪, 39union, 39

vertical reflection, 150

Wantzel, Pierre, 138Wason selection task, 46whole-number part, 98width of a set, 186

Z, 132

198

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