KaleidoTile
Tilings and Symmetry
Questions for small group investigation using KaleidoTile 3
Warm-Up
Play with KaleidoTile. Experiment with different symmetries and different colors or pictures. Let each person in your group make a tiling that s/he thinks is especially beautiful. You can use the built-in pictures on KaleidoTile’s picture palette or you can paste in your own pictures by clicking one of the little clipboard-shaped buttons. Write up your answers to these questions on the computer, and copy-and-paste each person’s warm-up tiling into your report.
Question #1
Go to the View menu and experiment with Show Plain Images, Show Reflected Images and Cut Along Mirror Lines. Explain the relationship between the polyhedron or tiling (in the main part of the window) and the fundamental triangle (in the side panel on the right).
Question #2

Each symmetry pattern — or symmetry group — is named according to the angles of its fundamental triangle. Say, for example, the fundamental triangle is a 30–60–90 right triangle. Each angle evenly divides 180°:

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

and so the symmetry group is called a (6,3,2) triangle group. Usually, though, people list the numbers in ascending order, and call it a (2,3,6) triangle group. You can call up the (2,3,6) triangle group in KaleidoTile either by clicking the 6-petal flower on the control panel or by selecting Choose Symmetries… from the View menu and setting 2, 3 and 6.

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 is the name of the triangle group?

Call up that triangle group in KaleidoTile and paint the faces to get a good looking tiling. Copy-and-paste the tiling into your report.

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 Choose a Style on the control panel and click on the round symbol.

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. Copy-and-paste a picture of your soccer ball into your report.

Which (pqr) triangle group did you use?

Question #6

Go to the Move the Control Point section of the Control Panel 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, copy and paste a picture of the tiling into your report. 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, copy and paste a picture of the tiling into your report. 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?