Math 308 Project
Ianiv Schweber

# What is a tiling?

A tiling consists of covering a surface using copies of shapes, called tiles, without overlapping or leaving gaps.
For example, as shown in the figures below, using a square or any other parallelogram as a tile we can create a tiling that will cover an infinite plane.

Periodic Tilings

# Periodic Tilings

A periodic tiling is one on which you can outline a region of the tiling and tile the plane by translating copies of that region (without rotating or reflecting). You can think of a piece of transparent paper covering the tiling which has the outline of each tile drawn on it. If the tiling is periodic you can shift the paper to a new position and the outlines will match the tiles underneath. As an example, the tilings in the previous figures are periodic.

Probably some of the most famous periodic tilings are the ones drawn by the Dutch artist M. C. Escher. He used tiles with the shapes of living things, such as fish, birds and people. For examples of Escher works visit http://www.mcescher.com/.

# Aperiodic Tilings

An aperiodic tiling is one where if we repeat the exercise with the transparent paper we will not find another position where the outlines of the tiles will match with the tiles underneath except for the starting position.
There is an infinity of shapes that tile periodically and aperiodically. For example, one can convert a checkerboard into an aperiodic tiling by splitting each square into 2 quadrilaterals, altering the orientations to prevent periodicity.

Aperiodic Tiling

There are also tiles that tile only aperiodically. In 1961 Hao Wang came up with sets of units squares whose edges where colored in various ways, called Wang dominoes. Wang  conjectured that there exists a procedure for deciding whether any set of dominoes will tile by placing them such that adjacent edges are the same color. If such procedure exists then any set of dominoes that tiles aperiodically will also tile periodically.

Wang Tiles

But in 1964 Robert Berger showed that Wang's conjecture was wrong, therefore there is a set of Wang dominoes that tiles only aperiodically. Berger constructed such a set using more than 20,000 dominoes, and later found a set of 104 pieces. Raphael M. Robinson reduced the set to 24 in 1976.

Index | Next