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In fact, Kali was used to create the wallpaper patterns for the Geometry & the Imagination summer course notes. The groups are labelled in both Conway and crystallographic notation on the control panel. You can save a pattern with the "Save" button, and use the "kaliprint" program to create printable PostScript. It runs only on SGIs, as does the museum program.  The museum exhibit was adapted from Charlie Gunn's "trigrp" module, which you may have seen when you were here before. A base triangle is used to tile space. Two of the angles of the triangle are always pi/2 and pi/3. When the third angle is pi/6, 12 triangles fit at a vertex and the tiling is a flat plane. The third angle can be increased to pi/5, pi/4 or pi/3, which gives you a spherical triangle and thus a closed surface for the tiling. You can move around a point inside the base triangle, which leads to kaleidescopic color changes in the flat case. For the spherical cases, you can construct the convex hull of the images of the point under the action of the group (either *532, *432, or *332), which gives you a closed polyhedron. You treat the edges of the triangle as mirrors: \  \ + (handicapped by ascii art, these two  \ / lines are supposed to be perpendicular)  \/  /\ ++ \ _____\  + Your point determines the "bending point" of the triangle, and thus the shape of the polyhedron. At certain points of the triangle, the polyhedron is one of the Platonic or Archimedean solids. When you drag the point along the horizontal and vertical edges of the triangle, you see truncation. Dragging along the hypotenuse illustrates duality. For example, in the pi/4 case we have the 234 group: PI/4 * octahedron  \ the star in the middle is the  \ rhombitruncated cuboctahedron  \ trunc. * * rhombicuboctahedron octa.  * \  \ PI/3 PI/2 *** cube (obviously the triangle is not to scale) cuboct trunc ahedron ated cube You can't get to the 2 snub Archimedean solids. You can pick "true spherical" instead of "polyhedral" mode, so you see triangles where all the curvature is in the faces and the tiling is a sphere. The pi/7 hyperbolic case was left out of the interface, but you can still get at it through the keyboard by typing "7" in the window with the base triangle. If you hit "t" in the big window, you can translate the hyperbolic disc to show that all the triangles are indeed congruent. I'm somewhat familiar with the curriculum of the past versions of the course through the Center preprints. I think the museum program does a good job of illustrating some concepts in the symmetry/orbifold, spherical/hyperbolic geometry, and curvature, and polyhedra parts of the course. (I showed John Conway a version of the program at the Smith College Regional Geometry Institute this summer, after one of his lectures which touched on this very subject.) The glitzy interface which makes it accessible to the public shouldn't deter you from using it. Your students will be able to concentrate on the content without having to spend time figuring out the interface. (The interface assumes you only have access to a mouse, all mouse buttons are treated identically.) Also, Charlie's original version is still a part of the normal Geomview distribution. You can get at it by clicking on the "Triangle Groups" line of the Applications browser on the Geomview main panel. You can ftp it from geom.umn.edu as priv/munzner/tritile.tar.Z. To run, type "museum" from the directory into which it unpacks. I haven't put in in our public ftp directory yet because our collaborators at the Science Museum of Minnesota are still working on userfriendly documentation. The program itself is ready to go. You are more than welcome to use it in your class. If you do use it, let me know how it goes.  A few other tidbits you might find useful for this course: There's a Geomview animation of a square rolling up into a torus. The data files are in /u/gcg/ngrap/data/geom/rollup, and you can use the Animator module to look at them. John Sullivan wrote a 2D stereographic projection module, which will lets you play with stereographic projection of one of the 5 Platonics or a globe. Unfortunately it expects a data file for the globe to be in "/u/sullivan/stereop/globe", but the polyhedra should work. While it's untested outside of the Center, I packed up the binary and the globe file in case you want to try them (in priv/munzner/stereop.tar.Z). If you start up geomview in the directory you unpacked them into, it should start up when you click on "2D Stereographic Projection" in the Applications browser. A prompt will appear at the shell asking you to pick a polyhedron. Rotate the "ball" object in Geomview to see the projection change. 3D stereographic projection is perhaps beyond the scope of the course, but John also incorporated it into Geomview in the conformal model of spherical space. There are some spherical objects here in the /u/gcg/ngrap/data/geom/spherical subdirectory, if you want to take a look. Tamara


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