Tilings and Symmetry

Questions for small group investigation using KaleidoTile 2.1.

Warm-Up

Play with KaleidoTile.  Experiment with different Symmetries and different Images.  Let each person in your group make a tiling that s/he thinks is especially beautiful.  You can use the images that come with KaleidoTile or you can use your own images (see Preparing Images on the Help menu to learn how to convert your GIFs and JPGs to the required BMP format).  Write up your answers to these questions on the computer, and copy-and-paste each person’s tiling into your report.

Question #1

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).  Hint:  Click “start” under the Tiling option, and then experiment with “add tile” and “remove tile”.

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 by clicking “Euclidean” under Symmetries and choosing (2,3,6).

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

(a)   If the fundamental triangle is a 45-45-90 right triangle, what is the name of the triangle group?

(b)   Call up that triangle group in KaleidoTile and select some images you think look good in it.  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

Does KaleidoTile offer all possible triangle groups (p,q,r) that tile the Euclidean plane?  the sphere?  the hyperbolic plane?  Hint:  You should answer “yes” to one of those questions and “no” to the other two.  When you answer “yes”, explain why.  When you answer “no”, give an example of a specific (p,q,r) triangle group that KaleidoTile omits.

Question #5

Make a soccer ball using KaleidoTile.  Which (p,q,r) triangle group did you use?  Copy-and-paste your soccer ball into your report.

Question #6

When you drag the puck in the fundamental triangle, you move the triple point at which the three images meet.

(a)   Which positions of the triple point give tilings with all regular faces?  (A regular face is one whose sides all have the same length.)

(b)   Can you find a position of the triple point for which the faces showing one of the images are regular, while the faces showing the other two images are not?   If so, copy and paste an example into your report.  If not, explain why not.

(c)   Can you find a position of the triple point for which the faces showing two of the images are regular, while the faces showing the remaining image are not?   If so, copy and paste an example into your report.  If not, explain why not.

Bonus Question

The only legal angles for the fundamental triangle are

         180° / 2,   180° / 3,   180° / 4,    180° / 5,    180° / 6,    ¼

     =  90°,   60°,   45°,   36°,    30°,   ¼

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?