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.
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.
What different sorts of symmetry did you find? What different sorts of symmetry do you think a finite planar picture can have?
(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.)
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?
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? Can you think of other polygons that might also generate such a pattern?
(This is meant to be an open ended question. Some things to think about are: the relationship between the number of copies of an image seen in two mirrors and the angle between the mirrors, the pattern that results when you use the interior of your polygon as your motif, the reflected images of the mirrors, and the plane patterns shown by KaleidoTile.)
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.
Author: Chaim Goodman-Strauss, revised and edited by Heidi Burgiel
Comments to: firstname.lastname@example.org
Created: Dec 7 1995 --- Last modified: Jul 31 1996
Copyright © 1995-1996 by The Geometry Center All rights reserved.