1Three Types of SymmetryWhen part of a design is repeated to make a balanced pattern, we say thedesign has Artists use symmetry to make designs that arepleasing to the eye. Architects use symmetry to produce a sense of balancein their buildings. Symmetry is also a feature of animals, plants, andmechanical objects.The butterfly, fan, and ribbon below illustrate three kinds of symmetry.•What part of each design is repeated to make a balanced pattern thatallows us to say the three figures have symmetry?•How do the figures suggest different kinds of symmetry?symmetry.Investigation 1 Three Types of Symmetry 5Getting Ready for Problem1.18cmp06se_KH1.qxd 6/8/06 7:16 AM Page 56 Kaleidoscopes, Hubcaps, and Mirrors1.1Reflection SymmetryYou have probably made simple heart shapes by folding and cutting paperas shown below.The resulting heart shape has which is sometimescalled mirror symmetry or line symmetry. The fold shows the A line of symmetry divides a figure into halves that are mirror images.If you place a mirror on a line of symmetry, you will see half of the figurereflected in the mirror.The combination of the half-figure and its reflectionwill have the same size and shape as the original figure. You can use amirror to check a design for symmetry and to locate the line of symmetry.You can also use tracing paper to check for reflection symmetry.Trace thefigure and the possible line of symmetry. Then reflect the tracing over thepossible line of symmetry. If the reflected tracing fits exactly on the originalfigure, the figure has reflection symmetry.What happens to the line of symmetry when you reflect the tracing and matchit with the original figure? Does its location change?symmetry.line ofreflection symmetry,8cmp06se_KH1.qxd 6/8/06 7:16 AM Page 6Investigation 1 Three Types of Symmetry 7Problem1.1Reflection SymmetryUse a mirror, tracing paper, or other tools to find all lines of symmetry ineach design or figure.A.B.C. D.E. F.G. H.Homework starts on page 15.8cmp06se_KH1.qxd 6/8/06 7:16 AM Page 78 Kaleidoscopes, Hubcaps, and Mirrors1.2Rotation SymmetryThe pinwheel design at the right does not have reflection symmetry. However, it can be turned less than a full turn around its center point in acounterclockwise direction to positions in which it looks the same as it does in its original position.Figures with this property are said to haveThe windmill, snowflake, and wagon wheel pictured below also have rotation symmetry.Which two of the three objects pictured above also have reflection symmetry?To describe the rotation symmetry in a figure, you need to specify twothings:•The center of rotation. This is the fixed point about which you rotate thefigure.•The angle of rotation. This is the smallest angle through which you canturn the figure in a counterclockwise direction so that it looks the sameas it does in its original position.There are several rotation angles that move the pinwheel design above to aposition where it looks like the original. In this problem, you will considerhow these angles are related to the angle of rotation.rotation symmetry.8cmp06se_KH1.qxd 6/8/06 7:16 AM Page 8Investigation 1 Three Types of Symmetry 9Problem1.2Rotation SymmetryA. List all the turns of less than 360° that will rotate the pinwheel designto a position in which it looks the same as what is pictured. What is theangle of rotation for the pinwheel design?B. In parts (1)–(3), list all the turns of less than 360° that will rotate theobject to a position in which it looks the same as what is pictured. Thengive the angle of rotation.1. the windmill 2. the snowflake 3. the wagon wheelC. Look at your answers for Questions A and B. For each object or figure,tell how the listed angles are related to the angle of rotation.D. The hubcaps below have rotation symmetry. Complete parts (1) and(2) for each hubcap.1. On a copy of the hubcap, mark the center of rotation. Then, find all the turns of less than 360° that will rotate the hubcap to aposition in which it looks the same as what is pictured.2. Tell whether the hubcap has reflection symmetry. If it does, draw all the lines of symmetry.E. Draw a hubcap design that has rotation symmetry with a 90° angle of rotation but no reflection symmetry.F. Draw a hubcap design that has rotation symmetry with a 60° angle of rotation and at least one line of symmetry.G. Investigate whether rectangles and parallelograms have rotationsymmetry. Make sketches. For the shape(s) with rotation symmetry,give the center and angle of rotation.Homework starts on page 15.Hubcap 1 Hubcap 2For: Hubcap MakerVisit: PHSchool.comWeb Code: apd-51028cmp06se_KH1.qxd 6/8/06 7:16 AM Page 910 Kaleidoscopes, Hubcaps, and Mirrors1.3Symmetry in Kaleidoscope DesignsAkaleidoscope (kuh ly duh skohp) is a tube containing colored beads or pieces of glass andcarefully placed mirrors. When you hold akaleidoscope up to your eye and turn the tube,you see colorful symmetric patterns.The kaleidoscope was patented in 1817 by theScottish scientist Sir David Brewster. Brewster was intrigued by the science of nature. He developed kaleidoscopes to simulate the designs he saw in the world around him.Five of the designs below are called kaleidoscope designs because they aresimilar to designs you would see if you looked through a kaleidoscope.MATHMATHIMEI M EMATHMATHIMEI M EMATHMATHIMEI M EABCDEF8cmp06se_KH1.qxd 6/8/06 7:16 AM Page 10Investigation 1 Three Types of Symmetry 11Problem1.3Analyzing SymmetriesUse what you know about reflection and rotation symmetry to analyze thesix designs.A. Locate all the lines of symmetry in the designs.B. Give the angles of rotation for the designs with rotation symmetry.C. 1. Make a table showing the number of lines of symmetry and theangle of rotation for each design.2. What relationship, if any, do you see between the number of lines ofsymmetry and the angle of rotation?3. Analyze the kaleidoscope design below to see whether it confirmsyour relationship.D. Each of the designs can be made by repeating a small piece of thedesign. We call this piece the For each design,sketch or outline the basic design element.E. One of the designs is not a kaleidoscope design. That is, it is not similarto a design you would see if you looked through a kaleidoscope. Whichdesign do you think it is? Why?Homework starts on page 15.basic design element.8cmp06se_KH1.qxd 6/8/06 7:16 AM Page 1112 Kaleidoscopes, Hubcaps, and Mirrors1.4Translation SymmetryThe next three