KaleidoTile
Tilings and Symmetry
Warm-Up
Play with KaleidoTile. Experiment with different symmetries and different colors or pictures. Go to Decoration > Face A/B/C > Image from … to use pictures from your Photo Library or take new pictures to apply to the polyhedron's faces.
Question #1
Go to the Options menu and experiment with Show Plain Images, Show Reflected Images and Cut Along Mirror Lines. What's the relationship between the polyhedron or tiling (in the main view) and the fundamental triangle (which you see when you tap Control point)?
Question #2

Select Options > Cut Along Mirror Lines and Symmetry > Style > Tiling to clearly see the triangles that comprise the tiling. Each symmetry pattern — or symmetry group — is then named according to the fundamental triangle’s angles. Say, for example, the fundamental triangle is a 30–60–90 right triangle. Each angle evenly divides 180°:

90° = 180° / 2
60° = 180° / 3
30° = 180° / 6

and so the symmetry group is called a △(2,3,6) triangle group (pronounced two-three-six triangle group, the △ symbol being silent). You can call up the △(2,3,6) triangle group in KaleidoTile from the Symmetry menu.

Definition. If you start with a triangle with angles ( 180°/p, 180°/q, 180°/r ), the resulting symmetry group is called a △(pqr) triangle group.

If the fundamental triangle is a 45–45–90 right triangle, what’s the name of the triangle group? Can you see how to call that group up in KaleidoTile?

Question #3

Sometimes KaleidoTile produces a tiling of a sphere, sometimes a tiling of a Euclidean plane, and sometimes a tiling of a hyperbolic plane. Find a simple rule that lets you predict which it will be for a given △(p, q, r) triangle group.

Hint: Which sets of angles ( 180°/p, 180°/q, 180°/r ) can be the angles of a Euclidean triangle?

Question #4

Make a list of all possible △(pqr) triangle groups that tile the Euclidean plane.

Make a list of all possible △(pqr) triangle groups that tile the sphere. To make your sphere look nice and round, go to the Symmetry menu and choose Style > Tiling.

List three different △(pqr) triangle groups that tile the hyperbolic plane. Altogether, how many different △(pqr) triangle groups tile the hyperbolic plane?

Question #5

Make a soccer ball using KaleidoTile. Which △(pqr) triangle group did you use?

Question #6

Tap Control point and experiment with the little round control point, where the three colors meet.

Which control point positions give tilings with all regular faces? A regular face is a face whose sides all have the same length and whose angles are all equal.

Can you position the control point so that the faces of one color are regular, while the faces of the other two colors are not? If so, do it. If not, explain why not.

Can you position the control point so that two sets of faces (of different colors) are regular, while the remaining set of faces is not? If so, do it. If not, explain why not.

Bonus Question

The only legal angles for the fundamental triangle are

180°/2180°/3180°/4180°/5180°/6
=90°60°45°36°30°

What would happen if you took a fundamental triangle with illegal angles, say 37°, 42°, and 101°, and started reflecting it across its sides to make a tiling?