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

Rd.

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

Rd,

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

the origin.

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