then unfold line 1 line 2 line 3

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.

then unfold

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 2 line 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

D C

A B

D C A'

A'

B' = A C' = B

B' B'

D' D' D' = C

C'

C'

A' = D

A B

D C

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…

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 figureis any movement that fits the figure onto the same set of points as it started with.

Activity: Symmetries of an Equilateral Triangle

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

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 after a 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

F N

E D

C B A

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

C M B

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 J K

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.

1 2

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

D C

A

B B A

C

D C 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

people find a model of a shape helpful in counting all the symmetries of a 3D shape.

Learning Exercises for Section 18.2

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 view

for 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

D C

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 Measurement Are 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.

Isosceles triangle

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.

Learning Exercises for Section 19.1

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.)

z y x

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…

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.

Learning Exercises for Section 19.2

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.