Math 308 Project
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
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
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
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.
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
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
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