2 1. BASIC NOTIONS

left by 2 units, up by 2 units, or diagonally by

√

2 units gives the exact same

tiling, and we say that this tiling is “periodic”. As we shall see, periodic

tilings have very little complexity; we will concentrate on tilings that are

not periodic.

Figure 1.2. Random tiling of

R2

by triangles.

Suppose we take a basic square tiling and cut each tile in half along a

diagonal, as in Figure 1.2. We flip a coin to decide whether to cut from the

bottom left corner to the top right corner, or from the top left corner to

the bottom right corner. With probability 1, the result will not be periodic.

Such a tiling is extremely complicated, but it has little structure. Knowing

what the tiling is like in one region says nothing about what it is like in

another region (beyond the fact that triangles fit back-to-back into squares

that repeat periodically). While the square tiling has too much structure

and not enough complexity, the random tiling has too much complexity and

not enough structure.

We are mostly interested in something in between. We want tilings that

have long-range order, but are not periodic. Knowing what the tiling looks

like at one point should not completely determine the tiling far away, but it

should say something about the tiling far away. As we shall see, there are

many tilings that exhibit such “aperiodic order”.

One such tiling is the “chair” tiling of Figure 1.3. There are four kinds of

tiles, each consisting of a 2 × 2 square with one 1 × 1 corner removed. Each

tile is part of a group of four tiles that form a larger chair-shaped region, a

group of 16 that forms an even larger chair-shaped region, and so on. This

is an example of a “substitution tiling”. Most of the examples in this book

will be substitution tilings.

The Penrose tiling [Pen] of Figure 1.4 is known for its 10-fold rotational

symmetry. Any pattern that is seen in this tiling is also seen, with the

same frequency, rotated by any multiple of 36 degrees. The Penrose tiling

comes in several equivalent forms. The most famous is the “kites and darts”

version. One can break the kites and darts into triangles. One can also bend

the boundaries of the kites and darts into chickens. In the triangle version,

there are four kinds of triangles, each of which can appear in 10 different

orientations, for a total of 40 species of tiles.