INTRODUCTION TO KALEIDOTILE SOFTWARE
The exercises contained in this activity require the
KaleidoTile program developed by Jeff Weeks. If you do not yet
have a copy of this program, it is available from the Geometry Center
The KaleidoTile software shows tilings and polyhedra that are
described by reflecting a vertex, edges, and faces in three mirrors.
KaleidoTile allows you to change the appearance of the tilings, and to
change the angles between the mirrors.
When the angles between the mirrors add up to Pi radians, or one
hundred and eighty degrees, they form a triangular tube. KaleidoTile
shows the kaleidoscope-like tiling of a plane that cuts the tube at
When the angles sum to greater than Pi radians, the mirrors form a
cone. KaleidoTile shows the intersection of the mirrors with a sphere
centered on the vertex of the cone, and the tiling of the sphere
generated by reflection in those mirrors. On a sphere, the angles of
a triangle can sum to greater than Pi radians.
When the angle sum is less than Pi radians, KaleidoTile shows a tiling
of hyperbolic space. Hyperbolic space is a space in which triangles
can have angles summing to less than Pi radians.
For more information regarding this activity and its objectives, see the
accompanying Teacher's Guide.
Time to Play!
Load the KaleidoTile program. Four different windows will appear
on the screen:
What to Draw
- In the Tiling window, a polyhedron is slowly spinning and
changing. Drag your mouse in this box to spin the polyhedron.
- Move the basepoint in the Basepoint window with the mouse.
This changes the vertex, edges, and faces that make up the object
shown in the Tiling window.
- Pressing the buttons in the Symmetry Group window changes
the angle between the mirrors. For example, the (2,3,4) button
corresponds to mirrors meeting at angles of Pi/2, Pi/3 and Pi/4.
- The What to Draw window changes what is shown in the
Tiling window. You can hide or show faces, edges, "dual
edges", and the triangle formed by the three mirrors.
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