6 1. BASIC NOTIONS
Figure 1.7. Two chair tiles. Do they meet full-edge to full-edge?
1.2. Tiling spaces
Now that we know what tilings are, where’s the topology? The plane
is contractible, as is
There’s not much topology there! Instead, from
each tiling T we will construct a space ΩT of tilings and study the topology
of ΩT . As noted in the preface, mathematical properties of ΩT are closely
related to physical properties of materials modeled on T .
The first step is to define a metric on tilings. Given two tilings T and
T of the same
we say that T and T are -close if they agree on a ball
of radius 1/ around the origin, up to a translation of size or less. More
precisely, let R(T, T ) be the supremum of all radii r such that there exist
vectors x and y with |x| 1/2r and |y| 1/2r and such that T − x and
T − y agree on Br, the ball of radius r around the origin. Then the distance
d(T, T ) between two tilings is defined to be the smaller of 1 and 1/R(T, T ).
Figure 1.8. Three one-dimensional tilings. The first two
are aligned but have different sequences, while the last is a
translate of the first. The vertical line shows the location of
For instance, consider a tiling T and its translate T − x. If |x| is small,
then d(T, T −x) = |x|. However, tilings that are close to T are not necessarily
translates of T . If a tiling T is the same as T on Br, but has a different tile
at a distance r from the origin, then d(T, T ) = 1/r.
In Figure 1.8, the first two tilings have distance at most 1/4 since they
agree out to distance 4 from the origin. (The actual distance is slightly less