Donald Coxeter's tiling game

The two rhombus shapes in this game are also called Penrose tiles named after the famous English mathematician Roger Penrose. Penrose discovered a curious thing. One rhombus has two angles of 36 degrees and two of 144 degrees. The other one has two angles of 72 degrees and two angles of 108 degrees. Penrose has shown that these shapes can be used to make an aperiodic tiling of the plane. That means that they make a pattern that never repeats. Try it. Try to make 3d shapes from these 2d rhomboids. How many can you make? 

Here is an amazing thing about these two shapes. Notice 36 x 2 = 72. 36 x 3 = 108. 36 x 4 = 144. In other words multiples of 1, 2, 3, and 4 times 36 degrees make up all the angles in the two shapes (36, 72, 108, 144)! If you love this and it gets you thinking and dreaming, you may have a future in geometry and math.

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