Symmetry Web

Symmetry Web is an exploration of the symmetries of geometric figures by means of the World Wide Web. When you enter the web (a truly appropriate term) for the figures listed below, you will see the figure together with choices of symmetries to apply to the figure. When you select a symmetry, you will see the new position of the figure together with a new list of choices of symmetries to apply. Repeatedly selecting symmetries moves you through the web of possible positions of the figure.

Some exercises:

  1. In some of the webs, a permutation is associated to the positions of the figure.
    1. Confirm that this mapping from the positions into the group of permutations is a homomorphism.
    2. What are the groups of permutations generated by the symmetries?
  2. After you have found all the possible positions, write them down as vertices. Connect the vertices by edges representing the generators stated in the web that take you from one position to another. Use arrows to make the directions clear. Try to present these graphs as symmetrically as possible. What you will have written down is called the Cayley graph for that group of symmetries and the stated generators.

Some projects:

In case you want paper models of these figures to manipulate while you navigate through Symmetry Web, we provide below FIG (for use in Xfig) and PostScript files of the geometric objects to print out and assemble.

The best course of action is to print the PostScript files on a color laser printer. Then glue the shapes onto stiff paper using a glue stick. Cut the shapes with little tags along the edges that need to be attached. Use a pen and ruler to etch a little groove in each line, making it easier to produce a nice crisp fold. Then fold and glue together. Good luck!

  1. Square FIG file
  2. Square PostScript file
  3. Equilateral Triangle FIG file
  4. Equilateral Triangle PostScript file
  5. Tetrahedron FIG file
  6. Tetrahedron PostScript file
  7. Cube FIG file
  8. Cube PostScript file
  9. Dodecahedron FIG file
  10. Dodecahedron PostScript file
wrightd@math.okstate.edu
Last modified: Thu Oct 31 14:36:08 CST 1996