museum query

[munzner]

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.
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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 user-friendly
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. 

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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