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 . Load the KaleidoTile program. 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.