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Date: 11/15/95 at 11:29:46
From: Anonymous
Subject: Kaleido Tile

Dr. Math,

I would appreciate it if you would tell me a bit about kaleido tile.

Date: 11/15/95 at 11:48:52
From: Doctor Sarah
Subject: Re: Kaleido Tile

Hello there -

Here's an answer to your question about KaleidoTile.  It was 
written by Evelyn Sander of the Geometry Center and posted to the 
newsgroup geometry.announcements in February 1995.  KaleidoTile
is easily downloaded and is a lot of fun!

Macintosh Version of Science Museum Software

This is an announcement that Jeff Weeks has written a Macintosh
version of the science museum math exhibit developed at the Geometry
Center last year.  The program is called KaleidoTile, version 1.0.  It
is available from   

I have included a copy of the article I wrote last June about the
museum exhibit.  Although the SGI version described below is not
precisely the same as the Macintosh version, it will give you an idea
of what KaleidoTile does.

Science Museum Math Exhibit, geometry.pre-college, Fri, 17 Jun 1994

Geometry Center staff collaborated with the Science Museum of
Minnesota to produce a museum exhibit on triangle tilings.  Starting
with a module for Geomview written by Charlie Gunn, staff members
Tamara Munzner and Stuart Levy, with assistance from Olaf Holt, worked
with exhibit developers at the museum to make software and
explanations which are accessible and interesting to the general
public.  This is an especially difficult task at a museum; the average
length of stay at the exhibit is only about five minutes.  Despite
this, the Geometry Center and museum collaborators managed to create
an exhibit which contains sophisticated concepts such as tilings of
the sphere and the relationship between tilings and the Platonic and
Archimedean solids.  Here is a brief description of the exhibit.

The sum of the angles of a planar triangle is always 180 degrees.
Repeated reflection across the edges of a 30,60,90 degree triangle
gives a tiling of the plane, since each angle is an integral fraction
of 180 degrees.  Figure 1 shows the exhibit's visualization of these
ideas.  What about a triangle whose angles add up to more than 180
degrees?  Such triangles exist on the sphere.  Whenever the angles of
such a spherical triangle are integral fractions of 180 degrees,
repeated reflections across the edges give a tiling of the sphere.  The
exhibit shows this for triangles with the first two angles always 30
and 60 degrees, and the third angle selected as 45, 36, or 60 degrees.
See figure 2.

A spherical triangle which tiles and a point of the triangle, called
the bending point, uniquely determine an associated polyhedron as
follows.  Repeated reflection through the edges of the triangle gives a
tiling of the sphere, each tile of which contains a reflected version
of the bending point.  These bending point reflections are the vertices
of the associated polyhedron. The edges are chords joining each
bending point and its mirror images.  The faces are planes spanning the
edges.  Associated to the spherical triangle which tiles the sphere,
there is a flattened triangle which tiles the polyhedron.  The bending
point is the only point of the flattened triangle which is still on
the sphere. See figure 3. Also compare the spherical tiling with
marked bending point in figure 2 with the associated polyhedron in
figure 4.

Tilings of the sphere and polyhedra visually demonstrate the idea of a
symmetry group.  Each choice of angles for the base triangle selects a
different symmetry group.  Reflections across the edges of the base
triangle are the generators of the group.  In the language of group
theory, the vertices are images of the bending point under the action
of the group.  The choices of group and bending point completely
determine the polyhedron.

The exhibit software allows the viewer to move the bending point to
see how the resulting polyhedron changes.  For particular choices of
bending point, the resulting polyhedra are Platonic and Archimedean
solids. See figure 4.  The software allows viewers to see the
relationship between these polyhedra more easily than would a set of
models.  Using the mouse, viewers can watch the polyhedron change as
they drag the bend point.  Thus they can begin to understand the idea
of duality of Platonic solids, as well as the idea of truncation to
form Archimedean solids.

The triangle tiling exhibit is currently on view at the Science Museum
of Minnesota.  In addition to the software, the exhibit contains books
and posters explaining the software, toys for constructing Platonic
and Archimedean solids, and other gadgets useful for understanding the
ideas of tilings of the sphere.  For example, the exhibit includes a
set of mirrored triangular tubes, each of which contains a spherical
or flattened triangle.  The mirrored walls make it appear as though
inside each tube there is a sphere or a polyhedron. This gives a
physical demonstration that repeated reflections of some spherical
triangles tile the sphere, and repeated reflections of certain
flattened triangles result in the Platonic and Archimedean solids.

The triangle tiling exhibit is successful with museum visitors; around
2500 people use it each week.  In addition, the exhibit has been
accepted for display at the annual meeting of the computer graphics
organization SIGGRAPH.  It will be part of graphics display called The
Edge.  (For more about SIGGRAPH, see Evelyn Sander, "SIGGRAPH Meeting,", 17 August, 1993.)

This article is based on an interview with Tamara Munzner and a visit
to the Science Museum of Minnesota.  Figures are available by anonymous
ftp from in the /pictures/articles/museum.exhibit
directory.  If you have Mosaic,you can also read this article with
figures included in the WWW document   . 

If you are interested in trying the software, which only works on an SGI, 
a version is available by anonymous ftp from as 
priv/munzner/tritile.tar.Z . The exhibit can easily be duplicated at
other science museums. If interested, contact Munzner 

-Doctor Sarah,  The Geometry Forum

Associated Topics:
High School Calculators, Computers
High School Geometry
High School Higher-Dimensional Geometry
High School Polyhedra
High School Symmetry/Tessellations

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