Basic concepts

A tiling is a subdivision of a region into non-overlapping subregions (called tiles) each of which is a copy of one of a finite set of shapes. Tilings have been used as artworks throughout known human history, and thus should qualify as one of the oldest mathematical subjects. The concept of a tiling is also a basic paradigm for organized behavior, and consequently the feel of the subject permeates many other areas of knowledge. It has only been recognized in the last twenty or thirty years that there are many dynamical aspects to the theory of tilings. The view of tilings as dynamical systems is the subject of this chapter.

Before proceeding to the dynamics, we must first make a few basic definitions. Much of this material is borrowed in simplified form from the massive book Tilings and Patterns, by Grünbaum and Shephard [GS87], which despite the age of the subject was really the first work to try to systematically organize and axiomatize the subject of tilings. The material on self-similar tilings is taken from the work of Thurston ([Thu89], unpublished, of course) and his student R. Kenyon. A (two-dimensional) tiling consists of the following:

1. a subset , usually a region (connected open set) possibly with some of its boundary attached. U may be all of .
2. A finite set of subsets , , called the prototiles (``proto'' as in ``prototype,'' meaning all other tiles are copies of these).
3. A decomposition

   

   of U into tiles such that each is congruent to some prototile and any two different tiles and intersect at most in a subset that has zero area.

Some classic examples of tilings are the usual tilings of the plane by squares, by equilateral triangles, and by regular hexagons, all commonly seen on bathroom floors throughout the world.

  
Figure 20: Tilings by squares, equilateral triangles, and regular hexagons

The usual method to create a tiling is to start with a huge collection of copies of the prototiles and lay them down tile-by-tile next to each other. We are interested in a different approach: a dynamical process emphasizing the self-similarity in the tiling. The tilings we describe have the following properties:

1. There is an integer k>1 such that the original tiles may be grouped into groups of k tiles each.
2. Each group considered as a single piece is similar to one of the original prototiles.
3. The decomposition is a tiling which is similar as a whole to the original tiling.

This sort of tiling is called a similarity tiling ([GS87], p. 520) or a self-similar tiling (see [Thu89]). The process of grouping the tiles together into larger copies of the same prototiles is called deflation. The opposite process of splitting the original tiles into smaller versions is called inflation. To make any sense of this definition, we cannot wait any longer before dealing with the details of congruences and similarities.