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SME 430: HISTORY OF MATHEMATICSWEEK 8: NON-EUCLIDEAN GEOMETRIES

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ITINERARY

ANNOUNCEMENTS

BIOGRAPHIES

TIMELINES

REMAINING MATH HISTORY IN EDUCATIONPRESENTATIONS

ACTIVITIES (REVISITED)

DISCUSSION ON THE READINGS

FRACTAL VIDEO

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Items Expected in Portfolio

Everyone

Mathematician Facts from

weeks 3-6 (Aristotle, Aryabhata,

Fibonacci, Brahmagupta, Euler,

Pythagoras, Euclid, Thales,

Archimedes, Zeno, De Morgan,

& Poincar)

Completed In-Class Activities

(Multiplication, Egyptian

Fractions, Negatives, Pi,

Elements)

Arithmetic Journal Entry

Classroom Notes

If you have done presentations

Biography You Wrote

Abbagail H, Kayla V, Rachael B,Clare A, Laura M, Danielle D,Janae B, Nichole S, Taryn K,Lauren M, Emily K, Nicole K

Timeline you Presented

Rebecca M, Emily S, Kristina W,

Beth R, Danielle G, Siedah K,Alexandrea T, Krystal T, ElizabethW, Jaime S, Aimee M, Rachael K,

Jane M

ANNOUNCEMENTS

BINDERS HAVE BEENGRADED.

INSIDE YOU WILL SEEA PIECE OF PAPERWITH THEBREAKDOWN OFPOINTS

IF WE MISSEDSOMETHING, MUSTTAKE CARE OF ITBEFORE LEAVINGCLASS TODAY

Student Name: Adamus, Clare Elizabeth

1.Biography Facts of any 6 Mathematicians (1 points per Mathematician)Mathematician

Week 1 Aristotle Aryabhata Fibonacci

Week 2 Brahmagupta Euler Pythagoras

Week 3 Euclid Thales Archimedes

Week 4 Zeno DeMorgan Poincar

Circle first six found

Points Earned: ______/6

2.In-Class Activities (1.4 point per Activity)Activities

Multiplication Negatives Packet

Pi Euclid!s Elements

Egyptian Fractions (EitherActivity or Homework)

Cross-out those not found

Points Earned: ______/7

4.Arithmetic Journal Entry (5 points)Points Earned: ______/5

5.Notes from Class (2 points)Points Earned: ______/2

Additional Points

Biography Writeup Timeline Presentation

______/5 N/A

Total Points: ______/25

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BIOGRAPHIES - BENOT MANDELBROT, WACAW SIERPISKINIKOLAI LOBACHEVSKY

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TIMELINE

150A.D.-300A.D.

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150A.D.-300A.D.

Diophantus (~200-~284) - Father of Algebra, Arithmetica, 130 problems of (in)determinite equations

Porphyry (233-309) - Book brought together Aristotle and Plato, theory of logic

Sporus (~240-~300) - Wrote on squaring circle and doubling cube

150 A.D. 200A.D. 250 A.D. 300 A

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PRESENTATIONS

USING HISTORY IN MATHEMATICS EDUCATION5 DOWN - 8 TO GO

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ACTIVITY REVISITED

A36284480-A38556458 ORIGAMIA38654380-A42384979 SPHERE

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DISCUSSION QUESTIONS

NON-EUCLIDEAN GEOMETRY

HOW IS NON-EUCLIDEAN GEOMETRY DIFFERENT THANEUCLIDEAN GEOMETRY? GIVE AN EXAMPLE.

WHICH OF EUCLIDS 5 POSTULATES IS NOT TRUE INSPHERICAL GEOMETRY. EXPLAIN.

EXPLAIN WHY THE STATEMENT, EUCLIDEANGEOMETRY IS BETTER THAN NON-EUCLIDEANGEOMETRIES. IS AN UNFAIR STATEMENT TO MAKE.

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DISCUSSION QUESTIONS(CONTD)

PROJECTIVE GEOMETRY

WHAT MOTIVATED THE DEVELOPMENT OF PROJECTIVEGEOMETRY?

WHAT IS THE PRINCIPLE OF DUALITY? EXPLAIN WITHAN EXAMPLE.

PERSPECTIVE IN ART

HOW DID THE USE OF PERSPECTIVE CHANGE THEPERCEPTION AND PURPOSE OF ART IN SOCIETY.

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FRACTAL GEOMETRY VIDEO

FROM THE PBS SHOW NOVAHUNTING THE HIDDEN DIMENSION

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HOMEWORK

READ SKETCH 18 - HALF IS BETTER (SIN AND COSINE)P. 147-152

FINISH READING THE HISTORY OF MATHEMATICS IN A LARGENUTSHELL

P. 24-60

GEOMETRY JOURNAL ENTRY (SUBMIT TO DROPBOX ON ANGEL)

CONSIDER THE FOLLOWING TOPICS: GEOMETRY, MATHHISTORY, AND EDUCATION. WRITE A COUPLE OFPARAGRAPHS THAT DESCRIBES YOUR UNDERSTANDING OF(AND HOW YOUR UNDERSTANDING HAS IMPROVED IN)THESE THREE TOPICS AND HOW THEY INTERSECT WITHEACH OTHER.

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Non-Euclidean Geometry

How is non-Euclidean Geometry different than Euclidean Geometry? Give anexample.

Euclidean geometry is on the surface of a plane (non-Euclidean is not). An example was

Riemann which was on the surface of a sphere.

A sphere has no parallel lines, so the parallel postulate doesnt hold. Triangle angle

measures add to over 180 degrees. Pi is different

Which of Euclids 5 postulates is not true in spherical geometry. Explain.

Postulate 2 doesnt. Lines are finite (can be measured).

Postulate 5 doesnt. There are no parallel lines.

Explain why the statement, Euclidean Geometry is better than non-Euclideangeometries. is an unfair statement to make.

While we may know Euclidean and be more familiar with it, other subject areas rely on

and use different kinds of geometries. (For example, spherical geometry is good for

astronomy.)

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Projective Geometry

What motivated the development of projective geometry?

Based on a need from philosophers, artists, and scientists. Wanted to represent three

dimensional scenes on paper (in two dimensions). Wanted to look at the relationship

between distance on the paper versus distance in real life.

What is the principle of duality? Explain with an example.

Point-Line duality - For a statement that includes the words line and point,

interchanging the words still produces a true statement. Points are collinear if they all

lie on the same line and Lines are concurrent if they all intersect at the same point.

Perspective in Art

How did the use of perspective change the perception and purpose of art insociety.

Non-perspective art (such as Egyptian) was used to tell a story (or portray a message)for record keeping. Perspective allowed art to be more enjoyable and more abstract.

Perspective drawing allowed better blueprints for planning.

In art, perspective fed a realism movement. Going from cartoonish icons to more real

world descriptions. Perspective allowed showing depth in space.

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SME 430: History of Mathematics 02/23/10

Name:____________________________

Watch the video clip about changing maps from the episode of TheWest Wingtitled Somebody's Going to Emergency, Somebody's

Going to Jail (Episode 16, Season 2). Pay attention to the mapsthat are presented in this clip. Fill in the chart below with what youheard in the video clip.

Maps Advantages Disadvantages

MercatorMap

PetersProjection

Map

What problems occur when trying to represent the surface of the world on a flat map?

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What does the group in this clip claim that maps have to do with social equality?

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Adam Clements

Good for navigation to

cross an ocean (travel

east to west)

Distorts relative size

of countries

Distortion of distance

Distorts location

Relative size and where

countries are is better

represented

Unfamiliar

Confusing

Overlap occurs and distortion will occur at some point.

Size is thought of as power. North is on top of what is on bottom.

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SME 430: History of Mathematics 02/23/10

Mercator Map

Peters Projection Map

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SME 430: History of Mathematics 02/23/10

We will now attempt to go the other direction, and create a sphere from a flat map(instead of creating a flat map from a sphere).

In your groups, cut out the pieces for the Northern Hemisphere, then SouthernHemisphere.

Take the pieces from each hemisphere, and tape them together only at their

point. In each Lnrt Sphere box, you will find a sphere, several clear, half shells, anda doughnut to serve as a stabilizing base. Take these out of the box.

Fit each paper hemisphere inside one clear shell, making sure to arrange thepaper flaps as best as possible

Slowly slide the sphere into the shell, sandwiching the paper hemispherebetween the sphere and the shell. Readjust your flaps as needed. Do this forboth hemispheres.

Looking at your just created globe, how accurate is this creation? What parts, if any,are inaccurate? How could its accuracy be increased?

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More accurate than any of the flat maps, but could be more accurate

if you cut the strips smaller to preserve distance more.

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SME 430: History of Mathematics 02/23/10

Consider the following two screen captures of Earth from GoogleEarth.

What are the advantages/disadvantages of using satellite images for our maps.

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(-) You can preserve distance in the middle, but the further away

from the center you get, the less accurate and more distorted it becom

(+) Real image, better preserves size

(-) Can't see all of the world at once.

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