Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional shapes, and then moves on to introduce symmetry of polyhedra (and of three-dimensional objects in general). 18.1 Symmetry of Shapes in a Plane Symmetry of plane figures can appear as early as Grade 1, where symmetry is restricted to reflection symmetry, or line symmetry, for a figure, as illustrated at 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 (for example, snowflakes or Valentine s Day hearts) by first folding a piece of paper, next 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 reflection lines, each of which gives a reflection symmetry for the square. Hence, a square has four reflection symmetries. 423
424 Chapter 18 Symmetry A second kind of symmetry for some shapes in a plane is rotational symmetry. A shape has rotational symmetry if it can be rotated around a 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 counterclockwise about the point highlighted as the center. The dashed segment to vertex C is used here to help keep track of the number of degrees turned. A prime mark is often used as a reminder that the point is associated with the original location. For instance, we denote B' as the point to which B would move after the rotation. 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 highlighted point. Convince yourself that the square also has rotational symmetries of 180 and 270. Every shape has a 360 rotational symmetry, but in counting the number of symmetries 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, the square has eight symmetries in all. THINK ABOUT The rotations in the figure above were all counterclockwise. 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 the following general definition. A symmetry of a figure is any movement that fits the figure onto the same set of points it started with. Activity 1 Symmetries of an Equilateral Triangle What are the reflection symmetries and the rotational symmetries of the equilateral triangle KLM? Be sure to identify the lines of reflection and the number of degrees in the rotations. M K L
Section 18.1 Symmetry of Shapes in a Plane 425 Notice that when we use the word symmetry with an object in geometry, we have a particular figure in mind, such as the tree shown at the right. And we must imagine some movement, such as the reflection in the dashed line, that gives the original figure as the end result. 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 trusting 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 bisecting line as the reflection line. Then the two angles opposite the sides of equal length (angles B and C in the figure) also trade places, with each angle fitting exactly where the other angle was. A A C B B' 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 looking at just one example and relying on what appears to be true in the example, this reasoning applies to all isosceles triangles and as a result, gives a strong justification for not having equal angles as part of a definition of isosceles triangle. Discussion 1 Why Equilateral Triangles Have to Be Equiangular Use the previous fact about isosceles triangles to deduce that all three angles of an equilateral triangle must be of the same size. C A B In Discussion 1 you used the established fact about angles in an isosceles triangle to justify in a general way the fact about angles in an equilateral triangle. Contrast this method with just looking at an equilateral triangle and trusting your eyesight.
426 Chapter 18 Symmetry Activity 2 Does a Parallelogram Have Any Symmetries? Trace a general parallelogram and look for symmetries. Does a parallelogram have any reflection symmetries? Does it have any rotational symmetries besides the trivial 360 symmetry? Here is another illustration of justifying a conjecture by using symmetry. Previously 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 the following 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. Activity 3 Symmetries in Some Other Shapes How many reflection symmetries and how many rotational symmetries does each of the following polygons have? In each case, describe the lines of reflection and the degrees of rotation. a. regular pentagon PQRST b. regular hexagon ABCDEF P A B T M Q F N C S R E D c. a regular n-gon TAKE-AWAY MESSAGE... Symmetries of shapes is a rich topic. Not only do symmetric shapes have a visual appeal, they make the design and construction of many manufactured objects easier. Symmetry is often found in nature. Mathematically, symmetries can provide methods for justifying conjectures that might have come from drawings or examples.
Section 18.1 Symmetry of Shapes in a Plane 427 Learning Exercises for Section 18.1 1. Which capital letters, in a block printing style (for example, 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 polygons? Describe the lines of reflection and give the number of degrees of rotation. a. an isosceles triangle with only two sides the same length b. a rectangle that does not have all its sides equal in length (Explain why the diagonals are not lines of symmetry.) c. a parallelogram that does not have any right angles d. an isosceles trapezoid e. an ordinary, non-isosceles trapezoid f. a rhombus that does not have any right angles g. a kite 5. Shapes other than 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 (that is, 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, and so on, 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 described in parts (a) (e). Design I Design II a. Design I, rotational symmetry Continue on the next page.
428 Chapter 18 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 the following pictures. a. b. c. d. e. f. 9. (Pattern Blocks) Make an attractive design using Pattern Blocks (or the paper ones found in Appendix G). Is reflection symmetry or rotational symmetry involved in your design? 10. Suppose triangle ABC has a line of symmetry k. What does that tell you, if anything, about the following objects? a. segments AB and AC (What sort of triangle must ABC be?) b. angles B and C c. point M and segment BC d. angles x and y
Section 18.1 Symmetry of Shapes in a Plane 429 11. Suppose that m is a line of symmetry for hexagon ABCDEF. What does that tell you, if anything, about the following objects. 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. G H L K X J I What does a 180 rotation tell you about specific relationships between segments and between angles in the hexagon? 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 the following conjectures about some quadrilaterals to see whether you can justify any of them by using symmetry. 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 at the right, angles 1 and 2 have the same size. 1 2 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.
430 Chapter 18 Symmetry 18.2 Symmetry of Polyhedra In Section 16.4, we informally linked congruence of polyhedra to motions. Because we linked symmetry of 2D shapes to motions also, 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 of shapes handy! Reflection Symmetries Clap your hands together and keep them together. 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 mirrorimage symmetry, or just reflection symmetry, if the context is clear. Activity 4 Splitting the Cube 1. Does a cube have any reflection symmetries? Describe the cross-section for each reflection symmetry that you find. 2. Does a right rectangular prism have any reflection symmetries? Describe the cross-section for each reflection symmetry that you find. Rotational Symmetries 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, or axis. A figure may have more than one axis of rotational symmetry. As with the cube pictured on the next page, it may be possible to have different rotational symmetries with the same axis, by rotating different numbers of degrees. Because the two rotations shown a 90 rotation and a 180 rotation 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.
Section 18.2 Symmetry of Polyhedra 431 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 different rotational symmetries: 90, 180, 270, and 360 (or 0 ). Activity 5 Rounding the Cube 1. 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 rotational symmetry. 2. Repeat Problem 1 for an equilateral-triangular right prism (see shape C in your kit of shapes). 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. 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 (besides a 360 one)? 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 Learning Exercise 2 have? Show a few of the axes of symmetry in sketches, for practice.
432 Chapter 18 Symmetry 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. a. b. c. Top view for c: 6. Copy and finish the following incomplete buildings so that they have reflection symmetry. Finish 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 additional cubes?) a. b. 7. Design a net for a pyramid that has 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 shown below is cut by the symmetry plane indicated. For the reflection in the plane, to what point does each vertex correspond? A > B > C > D > E > F > G > H >
Section 18.3 Issues for Learning: What Geometry and Measurement Are in the Curriculum 433 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. (Suggestion: Work with a classmate.) 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 grades K 8 is not uniform in the United States, 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 nationwide 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 measurement. i At Grade 4, roughly 15% of the items are on geometry (and spatial sense) and 20% on measurement. At Grade 8, roughly 20% of the items are 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. The book Principles and Standards for School Mathematics ii offers a view of what could be included 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 grades 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 it 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 later chapters of this book. 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. These brief overviews 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.
434 Chapter 18 Symmetry Grades Pre-K 2. 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, using terms such as over, under, near, far, and between. They should become conversant with ideas of symmetry and with rigid motions such as slides, flips, and turns. In measurement, the children should have experiences with length, area, and volume (as well as weight and time), measuring with both nonstandard and standard units and becoming familiar with the idea of repeating a unit. Measurement language involving words such as deep, large, and long should become comfortable parts of their vocabulary. Grades 3 5. Students should focus more on the properties of two- and threedimensional shapes, with definitions for ideas like triangles and pyramids arising. Terms like parallel, perpendicular, vertex, angle, trapezoid, and so forth, should become part of their vocabulary. Congruence, similarity, and coordinate systems should be introduced. Students should make and test conjectures, and they should give justifications for their conclusions. They 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 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 estimation 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 they should have some practice at applying these to the surface areas of rectangular prisms. Students should offer ideas for determining the volume of a rectangular prism. Grades 6 8. Earlier work should be extended so that students understand the relationships among different types of polygons (for example, squares are special rhombuses). They should know the relationships between angles, lengths, areas, and volumes of similar shapes. Their study of coordinate geometry and transformation geometry should continue, perhaps involving the composition of rigid motions. Students should 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. The students sense that measurements are approximations should be sharpened. They should study surface areas and volumes of some pyramids, pyramids, and cylinders. They should also study rates such as speed and density. A second document, Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, iii was published after PSSM. This document provides a list of three focal points for each grade. In this way, teachers can be sure that fundamental ideas are treated at each grade level when not everything in a textbook can be completed in the school year. For example, here is one Curriculum Focal Point for Grade 3:
Section 18.4 Check Yourself 435 Describing and analyzing properties of two-dimensional shapes. Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry (page 15). The Grade 4 Focal Point continues this work by focusing on developing an understanding of area and determining area, and in Grade 5 continues on to threedimensional shapes, including surface area and volume. 18.4 Check Yourself You should be able to work problems like those assigned and to meet the following objectives. 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 identifications 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 Learning 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. REFERENCES FOR CHAPTER 18 i 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. ii National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author. iii National Council of Teachers of Mathematics. (2006). Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: Author.