Chapter 18 Symmetry 18.1 Symmetry of Shapes in a Plane

Shapes and Measurement Chapter 18 page 73

Chapter 18 Symmetry

Symmetry is of interest in art—and design in general—and in the study of

molecules, for example. This chapter begins with a look at two types of

symmetry of 2-dimensional shapes, and then moves on to introduce

symmetry of polyhedra (and of 3-dimensional objects in general).

18.1 Symmetry of Shapes in a Plane

Symmetry of plane figures may appear as early

as Grade 1, where symmetry is restricted to

reflection symmetry, or line symmetry, for a

figure, as illustrated to the right. The reflection

line—the dashed line in the figure—cuts the

figure into two parts, each of which would fit

exactly onto the other part if the figure were

folded on the reflection line. Many flat shapes

in nature have reflection symmetry, and many

human-made designs incorporate reflection

symmetry into them.

You may have made symmetric designs

(snowflakes, Valentine's Day hearts, for

example) by first folding a piece of

paper, then cutting something from the

folded edge, and then unfolding. The

line of the folded edge is the reflection

line for the resulting figure.

thenunfold

A given shape may have more than one

reflection symmetry. For example, for a

square there are four lines, each of

which gives a reflection symmetry for

the square. Hence, a square has four

reflection symmetries.

line 1 line 2line 3

line 4

A second kind of symmetry for some shapes in the plane is rotational

symmetry. A shape has rotational symmetry if it can be rotated around a


M L

K

fixed point until it fits exactly on the space it originally occupied. The

fixed turning point is called the center of the rotational symmetry. For

example, suppose square ABCD below is rotated counter-clockwise about

the marked point as center (the segment to vertex C is to help keep track

of the number of degrees turned; a prime as on B' is often used as a

reminder that the point is associated with the original location):

A B

CD

A B

CD

A'

A'

B' = A C' = BB' B'

D' D'D' = C

C'

C'

A' = D

A B

CD

Eventually, after the square has rotated through 90°, it occupies the same

set of points as it did originally. The square has a rotational symmetry of

90°, with center at the marked point. Convince yourself that the square

also has rotational symmetries of 180° and 270°. Every shape has a 360°

rotational symmetry, but the 360° rotational symmetry is counted only if

there are other rotational symmetries for a figure. Hence, a square has four

rotational symmetries. Along with the four reflection symmetries, these

give eight symmetries for a square, in all.

Think About…The rotations above were all counter-clockwise.

Explain why 90°, 180°, 270°, and 360° clockwise rotations do not

give any new rotational symmetries.

The symmetries make it apparent that they involve a movement of some

sort. We can give this general definition.

Definition: A symmetry of a figure is any movement that fits the

figure onto the same set of points as it started with.

Activity: Symmetries of an Equilateral Triangle

What are the reflection symmetries and the

rotational symmetries of an equilateral

triangle like KLM? Be sure to identify the

lines of reflection and the number of degrees

in the rotations.


Notice that to use the word “symmetry” in geometry, we have a particular

figure in mind, like the tree to the right. And

there must be some movement, like the

reflection in the dashed line to the right, that

gives as end result the original figure. Many

points have “moved,” but the figure as a

whole occupies the exact same set of points

after the movement as it did before the

movement. If you blinked during the

movement, you would not realize that a motion had taken place.

Rather than just trust how a figure looks, we can appeal to symmetries in

some figures to justify some conjectures for those figures. For example, an

isosceles triangle has a reflection symmetry. In an isosceles triangle, if we

bisect the angle formed by the two sides of equal length, those two sides

“trade places” when we use the symmetry from the bisecting line. Then

the two angles opposite the sides of equal length (angles B and C in the

figure) also trade places, with each fitting exactly where the other angle

was:

A A

B B'C C'

Triangle ABC after reflection

So, in an isosceles triangle the two angles opposite the sides of equal

length must have equal sizes. Notice that the same reasoning applies to

every isosceles triangle, so there is no worry that somewhere there may be

an isosceles triangle with those two angles having different sizes. Rather

than just looking at an example and relying on what appears to be true

there, this reasoning about all such shapes at once gives a strong

justification.


C B

A

Discussion: Why Equilateral Triangles Have to Be Equiangular

Use the result about isosceles triangles to

deduce that an equilateral triangle must

have all three of its angles be the same

size.

Notice that you used the established fact about angles in an isosceles

triangle to justify the fact about angles in an equilateral triangle, and in a

general way. Contrast this method with just looking at an equilateral

triangle (and trusting your eye-sight).

Discussion: Does a Parallelogram Have Any Symmetries?

Does a general parallelogram have any reflection symmetries?

Does it have any rotational symmetries besides the trivial 360˚

one?

Here is another illustration of justifying a conjecture by using symmetry.

Earlier you may have made these conjectures about parallelograms: The

opposite sides of a parallelogram are equal in length, and the opposite

angles are the same size. The justification takes advantage of the 180°

rotational symmetry of a parallelogram, as suggested in these sketches.

Notice that the usual way of naming a particular polygon by labeling its

vertices provides a good means of talking (or writing) about the polygon,

its sides, and its angles.

D C

A B

X

B'=D A'=C

C'=A D'=B

X

parallelogram the parallelogram aftera 180˚ rotation, center X

Activity: Symmetries in Some Other Shapes

How many reflection symmetries and how many rotational

symmetries does each of the following have? In each case, describe

the lines of reflection and the degrees of rotation.

a. regular pentagon PQRST b. regular hexagon ABCDEF

c. a regular n-gon


M

T

S R

Q

P

NF

E D

C

BA

Take-Away Message…Symmetries of shapes is a rich topic. Not only do

symmetric shapes have a visual appeal, they ease the design and the

construction of many manufactured objects. Nature also uses symmetry

often. Mathematically, symmetries can provide methods for justifying

conjectures that might have come from drawings or examples.

Learning Exercises for Section 18.1

1. Which capital letters, in a block printing style (e.g., A, B, C, D, E,

F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z),

have reflection symmetry(ies)? Rotational symmetry(ies)?

2. Identify some flat object in nature that has reflection symmetry,

and one that has rotational symmetry.

3. Find some human-made flat object that has reflection symmetry,

and one that has rotational symmetry. (One source might be

company logos.)

4. What are the reflection symmetries and the rotational symmetries

for each of the following? Note the lines of reflection and the

degrees of rotation.

a. an isosceles triangle with only two sides the same length

b. a rectangle PQRS that does not have all its sides equal in length

(Explain why the diagonals are not lines of symmetry.)

c. a parallelogram ABCD that does not have any right angles

d. an isosceles trapezoid

e. an ordinary, non-isosceles trapezoid

f. a rhombus EFGH that does not have any right angles

g. a kite


5. Shapes besides polygons can have symmetries.

a. Find a line of symmetry for an angle.

b. Find four lines of symmetry for two given lines that are

perpendicular (i.e., that make right angles). Find four

rotational symmetries also.

c. Find three lines of symmetry for two given parallel lines.

d. Find several lines of symmetry for a circle. (How many lines of

symmetry are there?)

e. How many lines of symmetry does an ellipse have?

6. Explain why this statement is incorrect: “You can get a rotational

symmetry for a circle by rotating it 1°, 2°, 3°, etc., about the center

of the circle. So a circle has exactly 360 rotational symmetries.”

7. Copy each design and add to it, so that the result gives the required

symmetry.

Design I Design II

a. Design I, rotational symmetry

b. Design I, reflection symmetry

c. Design I, reflection symmetry with a line different from the one

in part b

d. Design II, rotational symmetry

e. Design II, reflection symmetry

8. Pictures of real-world objects and designs often have symmetries.

Identify all the reflection symmetries and rotational symmetries in

these.


a. b. c.

d. e. f.

9. (Pattern Blocks) Make an attractive design with Pattern Blocks. Is

either reflection symmetry or rotational symmetry involved in your

design?

10. Suppose triangle ABC has a line of symmetry k.

A

BC M

k

x y

What does that tell you, if anything, about...

a. segments AB and AC? (What sort of triangle must it be?)

b. angles B and C?

c. point M and segment BC?

d. angles x and y?

11. Suppose that m is a line of symmetry for hexagon ABCDEF.


A

B

C

D

E

F

m

What does that tell you, if anything, about...

a. segments BC and AF? Explain.

b. segments CD and EF?

c. the lengths of segments AB and ED?

d. angles F and C? Explain.

e. other segments or angles?

12. Suppose that hexagon GHIJKL has a rotational symmetry of 180°,

with center X.

X

G H

I

JK

L

What does that tell you about specific relationships between

segments and angles?

13. a. Using symmetry, give a justification that the diagonals of an

isosceles trapezoid have the same length.

b. Is the result stated in part a also true for rectangles? For

parallelograms? Explain.

14. a. Using symmetry, give a justification that the diagonals of a

parallelogram bisect each other.

b. Is the result stated in part a also true for special parallelograms?

For kites? Explain.

15. Examine these conjectures about some quadrilaterals to see

whether you can justify any of them by using symmetry.


21

a. The "long" diagonal of a kite cuts the "short" diagonal into

segments that have the same length.

b. In a kite like the one shown, angles 1 and 2

have the same size.

c. All the sides of a rhombus have the same

length.

d. The diagonals of a rectangle cut each other into four segments

that have the same length.

18.2 Symmetry of Polyhedra

Earlier, congruence of polyhedra was informally linked to motions.

Because symmetry of 2D shapes was also linked to motions, it is no

surprise to find that symmetry of 3D shapes can also be described by

motions. This section introduces symmetry of 3D shapes by looking at

polyhedra and illustrating two types of 3D symmetry. Have your kit

handy!

Clap your hands together and keep them clapped. Imagine a plane (or an

infinite two-sided mirror) between your fingertips. If you think of each

hand being reflected in that plane or mirror, the reflection of each hand

would fit the other hand exactly. The left hand would reflect onto the right

hand, and the right hand would reflect onto the left hand. The plane cuts

the two-hands figure into two parts that are mirror images of each other;

reflecting the figure—the pair of hands—in the plane yields the original

figure. The figure made by your two hands has reflection symmetry with

respect to a plane. Symmetry with respect to a plane is sometimes called

mirror-image symmetry, or just reflection symmetry, if the context is

clear.

Activity: Splitting the Cube

Does a cube have any reflection symmetries? Describe the cross-

section for each one that you find. Does a right rectangular prism

have any reflection symmetries? Describe the cross-section for each

one that you find.


A figure has rotational symmetry with respect to a particular line if,

by rotating the figure a certain number of degrees using the line as an axis,

the rotated version coincides with the original figure. Points may now be

in different places after the rotation, but the figure as a whole will occupy

the same set of points after the rotation as before. The line is sometimes

called the axis of the rotational symmetry. A figure may have more than

one axis of rotational symmetry. As with the cube below it may be

possible to have different rotational symmetries with the same axis, by

rotating different numbers of degrees. Since the two rotations below, 90°

and then another 90° to give 180°, affect at least one point differently,

they are considered to be two different rotational symmetries. The cube

occupies the same set of points in toto after either rotation as it did before

the rotation, so the two rotations are indeed symmetries.

A B

CD

A

AB BC

CD D

After 90 degree rotation, clockwise (viewed from the top)

After 180 degree rotation, clockwise (viewed from the top)

axis

Similarly, a 270° and a 360° rotation with this same axis give a third and a

fourth rotational symmetry. For this one axis, then, there are four

rotational symmetries: 90°, 180°, 270°, and 360° (or 0°).

Activity: Rounding the Cube

Find all the axes of rotational symmetry for a cube. (There are more

than three.) For each axis, find every rotational symmetry possible,

giving the number of degrees for each one.

Repeat the above for an equilateral-triangular right prism (shape C

from Chapter 18).

Take-Away Message…Some three-dimensional shapes have many

symmetries, but the same ideas used with symmetries of two-dimensional

shapes apply. Except for remarkably able or experienced visualizers, most


people find a model of a shape helpful in counting all the symmetries of a

3D shape.


1. Can you hold your two hands in any fashion so that there is a

rotational symmetry for them? Each hand should end up exactly

where the other hand started.

2. How many different planes give symmetries for these shapes from

your kit? Record a few planes of symmetry in sketches, for

practice.

a. Shape A b. Shape D c. Shape F d. Shape G

3. How many rotational symmetries does each shape in Exercise 2

have? Show a few of the axes of symmetry in sketches, for

practice.

4. You are a scientist studying crystals shaped like shape H from

your kit. Count the symmetries of shape H, both reflection and

rotational. (Count the 360° rotational symmetry just once.)

5. Describe the symmetries, if there are any, of each of the following

shapes made of cubes.

b.a. c. Top viewfor c:

6. Copy and finish these incomplete “buildings” so that they have

reflection symmetry. Do each one in two ways, counting the

additional number of “cubes” each way needs. (The building in

part b already has one plane of symmetry; do you see it? Is it still a

plane of symmetry after your addition?)


a. b.

7. Design a net for a pyramid that will have exactly four rotational

symmetries (including only one involving 360°).

8. Imagine a right octagonal prism with bases like . How many

reflection symmetries will the prism have? How many rotational

symmetries?

9. The cube below is cut by the symmetry plane indicated. To what

point does each vertex correspond, for the reflection in the plane:

A –> ? B –> ? C –> ? ... H –>?

A B

CD

E

F G

H

10. Explain why each pair is considered to describe only one

symmetry for a figure.

a. a 180° clockwise rotation, and a 180° counterclockwise rotation

(same axis)

b. a 360° rotation with one axis, and a 360° rotation with a

different axis.

11. (In pairs) You may have counted the reflection symmetries and

rotational symmetries of the regular tetrahedron (shape A in your

kit) and the cube. Pick one of the other types of regular polyhedra

and count its reflection symmetries and axes of rotational

symmetries.


18.3 Issues for Learning: What Geometry and MeasurementAre in the Curriculum?

Unlike the work with numbers, the coverage of geometry in K-8 is not

uniform in the U.S., particularly with respect to work with three-

dimensional figures. Measurement topics are certain to arise, but often the

focus is on formulas rather than on the ideas involved.

The nation-wide tests used by the National Assessment of Educational

Progress give an indication of what attention the test-writers think should

be given to geometry and measurementii. At Grade 4, roughly 15% of the

items are on geometry (and spatial sense) and 20% on measurement, and

at Grade 8, roughly 20% on geometry (and spatial sense) and 15% on

measurement. Thus, more than a third of the examination questions

involve geometry and measurement, suggesting the importance of those

topics in the curriculum.

One statement for a nation-wide curriculum, the Principles and Standards

for School Mathematicsi, can give a view of what could be in the

curriculum at various grades, so we will use it as an indication of what

geometry and measurement you might expect to see in K-8. PSSM notes,

"Geometry is more than definitions; it is about describing relationships

and reasoning" (p. 41), and "The study of measurement is important in the

mathematics curriculum from prekindergarten through high school

because of the practicality and pervasiveness of measurement in so many

aspects of everyday life" (p. 44). In particular, measurement connects

many geometric ideas with numerical ones, and allows hands-on activities

with objects that are a natural part of the children's environment.

In the following brief overviews, drawn from PSSM, you may encounter

terms that you do not recognize; these will arise in the later chapters of

this Part III. PSSM includes much more detail than what is given here, of

course, as well as examples to illustrate certain points. Throughout, PSSM

encourages the use of technology that supports the acquisition of

knowledge of shapes and measurement. PSSM organizes its

recommendations by grade bands: Pre-K-2, 3-5, 6-8, and 9-12. Only the

first three bands are summarized here.


Grades Pre-K-2. The children should be able to recognize, name, build,

draw, and sort shapes, both two-dimensional and three-dimensional, and

recognize them in their surroundings. They should be able to use

language for directions, distance, and location, with terms like "over",

"under," "near," "far," and "between." They should become conversant

with ideas of symmetry and with rigid motions like slides, flips, and turn.

In measurement, the children should have experiences with length, area,

and volume (as well as weight and time), measuring with both non-

standard and standard units and becoming familiar with the idea of

repeating a unit. Measurement language like "deep," "large," and "long"

should become comfortable parts of their vocabulary.

Grades 3-5. The students should focus more on the properties of two- and

three-dimensional shapes, with definitions for ideas like triangles and

pyramids arising. Terms like parallel, perpendicular, vertex, angle,

trapezoid, etc., should become part of their vocabulary. Congruence,

similarity, and coordinate systems should be introduced. The students

should make and test conjectures, and give justifications for their

conclusions. The students should build on their earlier work with rigid

motions and symmetry. They should be able to draw a two-dimensional

representation of a three-dimensional shape, and, vice versa, make or

recognize a three-dimensional shape from a two-dimensional

representation. Links to art and science, for example, should arise

naturally.

Measurement ideas in Grades 3-5 should be extended to include angle

size. Students should practice conversions within a system of units (for

example, changing a measurement given in centimeters to one in meters,

or one given in feet to one in inches). Their estimations skills for

measurements, using benchmarks, should grow, as well as their

understanding that most measurements are approximate. They should

develop formulas for the areas of rectangles, triangles, and parallelograms,

and have some practice at applying these to the surface areas of

rectangular prisms. The students should offer ideas for determining the

volume of a rectangular prism.

Grades 6-8. Earlier work would be extended so that the students

understand the relationships among different types of polygons (for


example, that squares are special rhombuses). They should know the

relationships between angles, of lengths, of areas, and of volumes of

similar shapes. Their study of coordinate geometry and transformation

geometry would continue, perhaps involving the composition of rigid

motions. The students would work with the Pythagorean theorem.

Measurement topics would include formulas dealing with the

circumference of a circle and additional area formulas for trapezoids and

circles. Their sense that measurements are approximations should be

sharpened. The students should study surface areas and volumes of some

pyramids, pyramids, and cylinders. They should study rates such as speed

and density.

This brief overview of topics, as recommended in the Principles and

Standards for School Mathematics, can give you an idea of the scope and

relative importance of geometry and measurement in the K-8 curriculum,

with much of the study beginning at the earlier grades.

References

i. National Council of Teachers of Mathematics. (2000). Principles and

standards for school mathematics. Reston, VA: Author.

ii. Silver, E. A., & Kenney, P. A. (2000). Results from the seventh

mathematics assessment of the National Assessment of Educational

Progress. Reston, VA: National Council of Teachers of Mathematics.

18.4 Check Yourself

Symmetry with both 2D and 3D figures was featured in this chapter.

Along with being able to work exercises like those assigned, you should

be able to…

1. define symmetry of a figure.

2. sketch a figure that has a given symmetry.

3. identify all the reflection symmetries and the rotational symmetries

of a given 2D figure, if there are any. Your identification should

include the line of reflection or the number of degrees of rotation.


4. use symmetry to argue for particular conjectures. Some are given

in the text and others are called for in the exercises, but an

argument for some other fact might be called for.

5. identify and enumerate all the reflection symmetries (in a plane)

and the rotational symmetries (about a line) of a given 3D figure.


Chapter 19 Tessellations

Covering the plane with a given shape or shapes—a tessellation--is a topic

that appears in many elementary curricula nowadays. Tessellations give

an opportunity for explorations, some surprises, connections to topics like

area, and a relation to some artwork. The attention in elementary school

focuses on the plane, but the same ideas can be applied to space.

19.1 Tessellating the Plane

You have seen the above patterns in tiled floors. These tilings are

examples of tessellations, coverings of the plane made up of repetitions of

the same region (or regions) that could completely cover the plane without

overlapping or leaving any gaps. (The word “tessellation” comes from a

Latin word meaning tile.) The first tiling above is a tessellation with

squares (of course, it involves square regions), and the second with

regular hexagons. As in the examples, enough of the covering is usually

shown to make clear that it would cover the entire plane, if extended

indefinitely. The second example also shows how shading can add visual

interest; colors can add even more. In passing, you can see that a

tessellation with squares could be foundation work for a child for later

work with area.

Activity: Regular Cover-ups

Each tessellation pictured above involves regions from one type of

regular polygons, either squares or regular hexagons. A natural

question is, What other regular polygons give tessellations of the

plane? Test these regions to see whether each type will give a

tessellation of the plane.


It may have been a surprise to find that some regular polygons do not

tessellate the plane. Are there any other shapes that will tessellate?

Activity: Stranger Cover-ups

Another natural question is whether regions from non-regular shapes

can tessellate the plane. Test these regions to see whether any will

tessellate the plane. Again, show enough of any tessellation to make

clear that the whole plane could be covered.

Isoscelestriangle

Acute scalene triangle

Obtuse scalene triangle

Parallelogram Trapezoid

The last activity may have suggested that the subject of tessellations is

quite rich. If you color some of the tessellations, using a couple of colors

or even shading alone, you can also find a degree of esthetic appeal in the

result. Indeed, much Islamic art involves intricate tessellations (Islam

forbids the use of pictures in its religious artwork). Islamic tessellations

inspired the artist M. C. Escher (1898–1972) in many of his creations,

some of which you may have seen. The drawing below shows two

amateurish “Escher-type” tessellations, with features added; surprisingly

each starts with a simpler polygon than the final version might suggest.

Elementary school students sometimes make these types of drawings as a

part of their artwork, coloring the shapes in two or more ways.


How does one get polygons that will tessellate? One way is to start with a

polygon that you know will tessellate, and then modify it in one, or

possibly more, of several ways. For example, starting with a regular

hexagonal region, then cutting out a piece on one side, and taping that

piece in a corresponding place on the opposite side will give a shape that

tessellates. The final shape can then be decorated in whatever way the

shape suggests to you—perhaps a piranha fish for this shape.

The same technique—cut out a piece and slide it to the opposite side—

can be applied to another pair of parallel sides. Again, notice that you can

add extra features to the inside of any shape you know will tessellate.

Yet another way to alter a given tessellating figure so that the result still

tessellates is to cut out a piece along one side, and then turn the cut-out

piece around the midpoint of that side:


When you draw the tessellations with such pieces, you find that some

must be turned. This gives a different effect when features are added to the

basic shape. As always, coloring with two or more colors adds interest.

Take-Away Message…That tessellations, or coverings, of the plane are

possible with equilateral triangles, or squares, or regular hexagons is

probably not surprising. More surprising is that these are the only regular

polygons that can give tessellations. Even more surprising is that every

triangle or every quadrilateral can tessellate the plane. Some clever

techniques allow one to design unusual shapes that will tessellate the

plane, often with an artistic effect.


1. Test whether the regular heptagon (7-gon) or the regular

dodecagon (12-gon) gives a tessellation.


2. Verify that each of the following shapes can tessellate, by showing

enough of the tessellation to be convincing. Color or “decorate”

the tessellation (merely shading can add visual interest).

a.

b.

c.

3. a. Start with a square region and modify it to create a region that

will tessellate. Modify the shape in two ways; add features and

shading or coloring as you see fit.

b. Start with a regular hexagonal region and modify it to create a

region that will tessellate. Modify the shape in two ways; add

features and shading or coloring as you see fit.

c. Start with an equilateral hexagonal region and modify it to

create a region that will tessellate. Use the midpoint of each side

to modify the region. Add features and shading or coloring as

you see fit.

4. Trace and show that the quadrilateral below will tessellate. (Use

the grid as an aid, rather than cutting out the quadrilateral.) Add

features and shading as you see fit.


5. Which of the Pattern Blocks give tessellations?

6. More than one type of region can be used in a tessellation, as with

the regular octagons and squares below. Notice that the same

arrangement of polygons occurs at each vertex.

Show that these combinations can give tessellations.

a. regular hexagons and equilateral triangles

b. regular hexagons, squares, and equilateral triangles

7. Will each type of pentomino tessellate?

8. Tessellations can provide justifications for some results.

a. Label the angles with sizes x, y, and z in other triangles in the

partial tessellation below, to see if it is apparent that x + y + z =

180°.

b. Use the larger bold triangle to justify this fact: The length of the

segment joining the midpoints of two sides of a triangle is equal

to half the length of the third side.

c. How does the area of the larger bold triangle compare to the

area of the smaller one? (Hint: No formulas are needed; study

the sketch.)


zyx

9. Which sorts of movements are symmetries for the tessellations

given by the following? What basic shape gives each tessellation?

a.

b. c.

d.


19.2 Tessellating Space

The idea of tessellating the plane with a particular 2D region can be

generalized to the idea of filling space with a 3D region.

Think About…Why are these shaped the way they are: Most

boxes? Bricks? Honey-comb cells? Commercial blocks of ice?

Lockers? Mailbox slots in a business?

When space is completely filled by copies of a shape (or shapes), without

overlapping or leaving any gaps, space has been tessellated, and the

arrangement of the shapes is called a tessellation of space. You can

imagine either arrangement below as extending in all directions to fill

space with regions formed by right rectangular prisms. These “walls”

could easily be extended right, left, up, and down, giving an infinite layer

that could be repeatedly copied behind, and in front of, the first infinite

wall to fill space. Hence, we can say that a right rectangular prism will

tessellate space and will do it in at least these two ways.

or

Although it is rarely, if every, a part of the elementary school curriculum,

tessellating space with cubical regions is the essence of the usual

measurement of volume.

Activity: Fill ‘Er Up

Which of shapes A–H from your kit will tessellate space?


After your experience in designing unusual shapes that will tessellate the

plane, you can imagine ways of altering a 3D shape that will tessellate

space, but still have a shape that will tessellate. We do not pursue this

idea, but you might.

Take-Away Message…The idea of covering the plane with a 2D region

can be extended to the idea of filling space with a 3D region.


1. Are there arrangements of right rectangular prisms, other than

those suggested above, that will give tessellations of space? (Hint:

You may have seen decorative arrangements of bricks in

sidewalks, where the bricks are twice as long as they are wide.)

2. Which could tessellate space (theoretically)?

a. cola cans

b. sets of encyclopedias

c. round pencils, unsharpened

d. hexagonal pencils, unsharpened and without erasers

e. oranges

3. Which, if any, of the following could tessellate space? Explain

your decisions.

a. b. c.

d. each type of the Pattern Block pieces

e. each of the base b pieces (units, longs, flats)

4. How are tessellation of space and volume related?

5. (Group) Show that the shape I in your kit can tessellate space.

6. (Group) Will either shape J or shape K tessellate space?


19.3 Check Yourself

This short chapter about tessellations of a plane or of space opens up an

esthetic side of mathematics, in that intricate designs can be derived from

basic mathematical shapes.

Besides working with tasks like the exercises assigned, you should be able

to…

1. tell in words what a tessellation of a plane is.

2. determine whether a given shape can or cannot tessellate a plane.

You should know that some particular types of shapes can

tessellate, without having to experiment.

3. create an “artistic” tessellation.

4. tell in words what a tessellation of space is.

5. determine whether a given shape can or cannot tessellate space.

Date post:	12-Sep-2021
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Chapter 18 Symmetry 18.1 Symmetry of Shapes in a Plane

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