1.2. TILING SPACES 7

than 1/4, since translating both tilings a small distance to the left will cause

them to agree on a slightly larger ball.) The first and third have distance

1/4, since they agree everywhere, up to a translation by 1/4. The second

and third have distance at most 8/31, since after translating the first to the

right by 1/8 4/31 and the second to the left by 1/8, they agree on a ball

of radius 31/8 around the origin.

Definition. The orbit of a tiling T is the set O(T ) = {T − x|x ∈

Rd}

of translates of T .

Definition. A tiling space is a set of tilings that is (1) closed under

translation, and (2) complete in the tiling metric. That is, if T ∈ Ω then

T − x ∈ Ω, and every Cauchy sequence of tilings in Ω has a limit in Ω.

Definition. The hull ΩT of a tiling T is the closure of O(T ). This is

sometimes called the “orbit closure” of T .

The hull ΩT should be viewed as the set of tilings that locally look like

translates of T . In particular, a tiling T is in ΩT if and only every patch of

T is found in a translate of T . (Equivalently, a translate of every patch of

T is found in T .)

To see this, suppose that P is a patch in T , located somewhere in Br

for some r 0. If T is in the hull of T , then there are translates of T

that agree with T on arbitrarily large balls around the origin, hence that

contain P . Conversely, let Pr = [Br] in T . If every patch of T is found in

a translate of T , then there exist translates Tr of T that contain Pr. But

then T = limr→∞ Tr is in the orbit closure of T .

Examples of Hulls. To build our intuition, let’s look at the hulls of

some simple 1-dimensional tilings. Let T0 be a periodic tiling with just one

kind of tile: a white tile of length one. Since T0 − 1 = T0, the orbit of T

is topologically the circle R/Z. This is already a complete metric space, so

ΩT0 is just a circle.

Next consider the “one black tile” tiling T1. Every tiling with one black

tile is in O(T1), and hence in ΩT1 . However, tilings with no black tiles are

also in ΩT1 , since every patch of such a tiling can be found in T1, both

suﬃciently far to the right of the origin and suﬃciently far to the left. ΩT1

consists of two path-components: the circle ΩT0 , and the line O(T1), with

both ends of the line asymptotically approaching the circle, as in Figure 1.9.

Exercise 1.1. What is ΩT when T is the half-and-half tiling? How

many path components does it have?

Exercise 1.2. Let T be a 1-dimensional tiling, with the color of the

tiles (black or white) decided by fair and independent coin flips. What is

ΩT ? Your answer could in principle depend on T , since it’s conceivable

that your coin flips would always yield heads (T0) or yield tails exactly once