Homework: Introduction to Symmetries - Combining Symmetries Homework
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Introduction to Symmetries
Combining Symmetries Homework
Solutions Manual

SOLUTIONS ARE IN BOLD

Using The Geometer's Sketchpad, draw a motif (for example, construct a polygon interior); we will call this motif F. Draw two lines, R1 and R2. Look at R1 R2 (F) = R1(R2(F)) and R2 R1 (F) = R2(R1(F)), where R(F) is the figure generated by reflecting F across line R. Does R1 R2 equal R2 R1? How do you know?
R1 R2 does NOT equal R2 R1. Reflections are not commutative in general. The exceptions to this (when R1 R2 = R2 R1) are when R1 and R2 are perpendicular or the same line. (See the picture below.)
An interactive sample sketch showing this is "problem1.gsp". [GSP Help]
Find three or fewer reflections that transform the motif into the image shown in this Sketchpad sketch [GSP Help]. Check your work as suggested. You may wish to refer to the theorems mentioned in the previous section.
Your answer should take the form of a Sketchpad sketch showing mirror lines for all the mirrors you used labeled R1, R2, R3, etc. and with the intermediate images labeled R1(motif), R2 R1(motif), etc. Feel free to change the size and position of the motif, and to hide the text surrounding it.
You need to use exactly three reflections to transform the motif onto the image.
The sketch of this is "problem2.gsp". [GSP Help]
Explanation:
A theorem from transformational geometry says:
It can be seen that the image and the motif have reverse orientation. Since each mirror reflection reverses the orientation of an object, two reflections would produce an image with the same orientation as the original motif. Thus we need to use one or three reflections.
Line R1 (shown in diagram) reflects point B of the motif onto the corresponding point b of the image. Thus R1 is the perpendicular bisector of segment Bb. Suppose we wish to use only one reflection to transform the motif onto the image. Then R1 would also be the perpendicular bisector of segment Aa, where A maps onto a. You can see this is not the case. Thus using only one reflection would not be enough. Thus exactly three reflections are needed.

Construction:
Given points A and B on the motif and corresponding points a and b on the image, the first mirror line, R1, is constructed to be the perpendicular bisector of segment Bb. Let A' = R1(A). Note that b = R1(B). Then the second mirror line, R2, is the line perpendicular to segment A'a and going through point b. (R2 is also the perpendicular bisector of segment A'a.) Now R2 R1(A) = a and R2 R1(B) = b. Thus the third mirror line, R3, is the line containing segment ab.
a) Find, draw, or construct a picture of a finite object (snowflake, house, flower, pinwheel, quilt square, polygon) on a piece of paper. (If your computer can run Java programs, check out this neat snowflake drawer, written here at the Center!) What symmetries does your object have? Discuss your findings with your classmates.
b) Identify the different symmetries illustrated in the following sample images. Each image consists of more than one part -- each object or group of linked objects should be considered separately. The images should be considered as pictures on a plane, not 3D ojbects. In your answer, state whether or not you're considering different colored objects to be similar.
- Three groups of triangles
  If you consider the shading to be important, there is no symmetry in any of the triangle groups. However, you should ignore the shading.
  The upper left image has 90 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 right 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 lower image has 120 degree rotational symmetry, but no lines of reflection. You could also view the three triangles as translations of each other. If you look only at the "shadow" of the image, it has no additional symmetries.
- Images from the movie "Outside In"
  For reference, number the images 1-15 from upper left to lower right. Ignore all shadows and assume the images are 2-dimensional.
  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.
  
  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.
  
  Objects numbered 5, 6, 7, 8, and 9 have only the 45 degree rotational symmetry.
What different sorts of symmetry did you find? What different sorts of symmetry do you think a finite planar picture can have?
The only symmetries that can exist in a finite planar picture are reflections and rotations. (Translations and glide reflections require an infinite plane.)
(Once you know the answer to this question, you can categorize all finite planar pictures. Although this may not be the best classification system for museum curators, it could be useful to mathematicians, biologists, or chemists. Our goal in these chapters is to come up with a similar classification system for infinite planar patterns.)
Using the Geometer's Sketchpad, construct three or four lines so that they meet at angles of 30-60-90, or 45-45-90. Construct a motif inside the triangle bounded by the mirrors.
Reflect that motif across one of the mirrors, then reflect the motif and its image across another mirror. Reflect all four images across a third mirror. Continue to reflect copies of the motif across the mirrors until you have a distinguishable pattern formed by at least ten images, without any gaps in it. Examine this pattern -- can you find any rotational symmetries in it? Move your motif until copies of it form a ring around one of the corner points of your triangle. How many copies appear in that ring?
30-60-90 Triangle:
A sketch of this is "30_60_90.gsp". [GSP Help]

Around the 30 degree angle is a ring of 12 copies (6 reversed). A rotational symmetry of 60 degrees exists here. Around the 60 degree angle is a ring of 6 copies (3 reversed). A rotational symmetry of 120 degrees exists here. Around the 90 degree a ngle is a ring of 4 copies (2 reversed). A rotational symmetry of 180 degrees exists here.
45-45-90 Triangle:
A sketch of this is "45_45_90.gsp". [GSP Help]
Around each 45 degree angle is a ring of 8 copies (4 reversed). A rotational symmetry of 90 degrees exists here. Around the 90 degree angle is a ring of 4 copies (2 reversed). A rotational symmetry of 180 degrees exists here.
The angles between the mirrors were carefully chosen so that you would get a neat tiling of the plane. What properties must be satisfied for a triangle to generate such a nice pattern?
For any polygon to tile the plane by reflecting across the sides, all angles must of the form 180/(whole number). For a triangle, the sum of the angles must equal 180 degrees. These two conditions allow for only the following 3 possible triangle m irror tilings. The angles are:
60-60-60 Triangle:
A sketch of this is "60_60_60.gsp". [GSP Help]

Around each vertex is a ring of 6 copies (3 reversed). A rotational symmetry of 120 degrees exists at each vertex.
Can you think of other polygons that might also generate such a pattern?
The only other type of polygon that will tile the plane by reflecting across the sides is the rectangle. A sample sketch using a square is "90_90_90_90.gsp". [GSP Help]
1. Construct two lines that intersect at an angle of 360/n degrees, for some choice of n. Create a motif in the angle between the two lines. Construct the n-1 images generated by reflecting the motif in the mirrors, then reflecting its image, then reflecting the images' images...
  For example, you could create the pattern using n=5, giving a 36 degree angle between the mirrors. A sample sketch of this is "mirror36.gsp". [GSP Help]
2. Now imagine repeating the process by reflecting a picture in three dimensional space with three mirrors. Kaleidotile shows some of the patterns you might get.
  Build a model of one of the solids or the plane shown by KaleidoTile. Draw your own motif on the base triangle. The step command under the help menu shows what happens when you reflect the base triangle across the planes of the mirrors. What happens when you reflect your pattern across those mirror lines? Draw the resulting pattern on your model.
  A sample polyhedra net of the cuboctahedron with a reflected pattern is available as the sketch "cubocta.gsp". [GSP Help]
Think about teaching from the materials you just read. Are the instructions clear enough? Why or why not? When would you print them out? When would you have students read them on the computer? Why? What would you change to make them more useful to you?