In this exercise, you get to design your own tiles. Tessellation and tiling are words used to describe flat, planar patterns consisting of a limited set of tiles, not unlike those embedded in your bathroom walls.

Mathematicians are interested in tessellations for several reasons:

• How many colours do you need to paint a tessallation without using the same colour for adjacent tiles?
• Is it possible to create infinitely repeating patterns by using a set of tiles?

A special kind of tessellation, called the Penrose tessallation, is featured beside this text. It can also be found decorating Keskuskatu in Helsinki, next to Svenska Teatern and Stockmann.

What makes the Penrose tiling so special? It is one of few tessallations with no repeating patterns. As in the figure, there are some repeating flower-like structures, but it is impossible to combine several of these indefinitely.

1. How many colours do you need to paint this Penrose tiling without two tiles of the same colour touching?
2. Design a blueprint for a Penrose tiling whose tiles are of side length 2 cm.