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 http://www.geometrygames.org/KaleidoTile/ 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.college, 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," geometry.college, 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 mathforum.org in the /pictures/articles/museum.exhibit directory. If you have Mosaic,you can also read this article with figures included in the WWW document http://www.geom.uiuc.edu/docs/forum/forum.html . If you are interested in trying the software, which only works on an SGI, a version is available by anonymous ftp from geom.umn.edu as priv/munzner/tritile.tar.Z . The exhibit can easily be duplicated at other science museums. If interested, contact Munzner (firstname.lastname@example.org). -Doctor Sarah, The Geometry Forum
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