Figure 1.10. The chair substitution
We then define the tiling space
Ωσ = {Tilings T | every patch of T is found in a supertile of some order}
Theorem 1.4. If the substitution matrix is primitive, then there exists
a fixed point of
for some n 0. (Such fixed points are called “self-similar
tilings”.) In particular, Ωσ is non-empty.
Proof. If σ is primitive, there exists an integer k 0 for which each
supertile of order k and type a contains a copy of the tile a. By taking k
large enough, we can assume that the supertile contains a copy of a in its
interior. Let t1 be the set of points in our supertile, and let t2 be the chosen
tile of type a inside t1. We will find a point x inside t2 in such a way that
(t1 x) =
x). This means that t2 x can be used as a seed for a
self-similar tiling.
To find the point p, we subdivide the tile t2 into smaller subtiles along
the same pattern by which the supertile t1 is subdivided into tiles. Let t3 be
the subtile that sits inside t2 in the same way that t2 sits inside t1. Repeat
the process indefinitely, subdividing tn and letting tn+1 sit inside tn the way
that tn sits inside tn−1. The intersection of the nested sequence of subtiles
tn is nonempty and has diameter 0, hence is a single point, which we call
This procedure is illustrated for the chair tiling in Figure 1.11. If a is the
chair in standard orientation (a square with the northeast corner missing),
then σ(a) contains two copies of a, but neither are in the interior. We have
to go to
to get interior tiles, and there are two copies of a in the
interior of
both of which are shaded. Using them as seed tiles, we can
get two different self-similar tilings, one of which is shown in Figure 1.12.
Why did we need interior tiles? If we had chosen t2 from σ(a), then the
point x would have been on the boundary of t2, as in Figure 1.13. Using
that as a seed would have yielded an infinite self-similar structure, but it
would have covered only part of the plane. To get the whole plane we must
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