Basic notions
1.1. Tilings
A tiling is a subdivision of the plane (or more generally, of
pieces called tiles. Specifically, we have a set of tiles ti that intersect only on
their boundaries, and whose union is the entire plane. Technically, the tiling
is that set of tiles. Besides their geometric shapes, tiles may carry labels
think of them as the color of the tiles. A patch of a tiling is a finite subset
of the tiles in a tiling. If A is a bounded subset of
then [A] denotes the
patch consisting of all tiles that intersect A. Sometimes we will write [A]T ,
to emphasize that we are talking about a patch of the tiling T . If T is a
tiling and x
then T x is the same set of tiles shifted over by the
vector −x. This is equivalent to moving the origin by +x, and generally
results in a different tiling. In T there may be a tile near the point x, while
in T x that tile appears near the origin. Moving a tiling around like this
is one of the most important operations that we will consider.
Although tiles are usually polygons, sometimes they take shapes that
are artistically more interesting. Several interesting tilings appear on the
following pages.
Figure 1.1. Basic square tiling and basic checkerboard.
The basic square tiling, shown in Figure 1.1, is frequently found in
kitchens. There is only one kind of tile, and it is repeated over and over
again. Not very interesting. A checkerboard is only slightly more compli-
cated. There are now two kinds of tiles. Both are unit squares, but one is
called “black” and the other is called “white”. Translating this tiling to the
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