![]() ![]() What different sorts of symmetry did you find? What different sorts Objects numbered 5, 6, 7, 8, and 9 have only the 45 degree rotational symmetry. The objects numbered 2, 3, 4, 10, 11, 12, 13, and 14 have a 45 degree rotational symmetry and eight lines of reflective symmetry through each object's center. ![]() The 1st and the 15th objects have infinitely many rotational symmetries of any angle of rotation and infinitely many lines of reflective symmetry through each object's center. Ignore all shadows and assume the images are 2-dimensional. If you look only at the "shadow" of the image, it has no additional symmetries.įor reference, number the images 1-15 from upper left to lower right. You could also view the three triangles as translations of each other. The lower image has 120 degree rotational symmetry, but no lines of reflection. If you look only at the "shadow" of the image, it has the additional symmetry of 3 lines of reflective symmetry intersecting at a 60 dergee angle. The upper right image has 120 degree rotational symmetry, but no lines of reflection. If you look only at the "shadow" of the image, it has four lines of reflective symmetry intersecting at a 45 dergee angle as well as the rotations. The upper left image has 90 degree rotational symmetry, but no lines of reflection. If you consider the shading to be important, there is no symmetry in any of the triangle groups. In yourĪnswer, state whether or not you're considering different colored Should be considered as pictures on a plane, not 3D ojbects. Group of linked objects should be considered separately. Each image consists of more than one part - each object or Object have? Discuss your findings with your classmates.ī) Identify the different symmetries illustrated in the following sample (If your computer can run Java programs, check out thisĭrawer, written here at the Center!) What symmetries does your (snowflake, house, flower, pinwheel, quilt square, polygon) on a piece a) Find, draw, or construct a picture of a finite object.Third mirror line, R3, is the line containing segment ab. (R2 is also the perpendicularīisector of segment A'a.) Now R2 R1(A) = a and R2 R1(B) = b. Then the second mirror line, R2, is the line perpendicular to The image, the first mirror line, R1, is constructed to be the Given points A and B on the motif and corresponding Thus exactly three reflections are needed. Thus using only one reflection would not beĮnough. Perpendicular bisector of segment Aa, where A maps onto a. Suppose we wish to use only one reflection to Thus R1 is the perpendicularīisector of segment Bb. Line R1 (shown in diagram) reflects point B of the motif onto theĬorresponding point b of the image. Thus we need to use one or three reflections. Reflections would produce an image with the same orientation as the SinceĮach mirror reflection reverses the orientation of an object, two It can be seen that the image and the motif have reverse orientation. You need to use exactly three reflections to transform the motifĪ theorem from transformational geometry says:Īny isometry may be formed as the composition of at most three Feelįree to change the size and position of the motif, and to hide the The intermediate images labeled R1(motif), R2 R1(motif), etc. Lines for all the mirrors you used labeled R1, R2, R3, etc. Your answer should take the form of a Sketchpad sketch showing mirror Image shown in this Sketchpad sketch [GSP Find three or fewer reflections that transform the motif into the.(See the picture below.)Īn interactive sample sketch showing this is " problem1.gsp". The exceptions to this (when R1 R2 = R2 R1) are when R1 and R2Īre perpendicular or the same line. Motif (for example, construct a polygon interior) we will call this Math5337: Symmetry Solutions Homework: Introduction to Symmetries - Combining Symmetries Homework
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