INTRODUCTION TO KALEIDOTILE SOFTWARE

ACTIVITY SHEET

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 Downloadable Software archive.

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 right angles.

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:

Symmetry Group

What to Draw

Basepoint

Tiling

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.

Experiment!