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Tiling Terminology
In the previous page, The Tilings Around Us, we decided a a tiling is just a way of covering a flat surface with smaller shapes or tiles that fit together nicely, without gaps or overlaps. But we also saw that tilings come in many varieties, both man-made ones, and ones in nature.

In order to make it easier to talk about tilings and their differences and similarities, it comes in handy to know some of the mathematical terminology for tilings and their properties.

Kinds of Tilings

To a mathematician, a tiling is just a collection of sets called tiles which cover some larger set. As usual, by cover we mean that the tiles fit together without gaps or overlaps. However, notice that we haven't bothered to say what the larger set is. This is because mathematicians love abstraction, and hate to rule things out before they have to!

We will usually talk about plane tilings, where the larger set is the 2-dimensional Euclidean plane, and the tiles are some collection of 2-dimensional shapes. There can be other kinds of tilings though. For example, the larger set might be a sphere, where the tiles are parts of a sphere instead of flat pieces.

For math experts: Mathematicians usually impose a couple other conditions on tilings. Suppose T is a tiling, i.e. a set of tiles. Then we require that T be countable. We also require that all the tiles in T are closed sets.

Kinds of Tiles

Tilings can have many different kinds of tiles. One of the most obvious ways in which tilings is in how many different kinds of tiles they have.

To classify tilings by their tiles, the first step is to say what we mean by "different" kinds of tiles. Say two tiles are congruent if they are of the same size and shape.

These three tiles are all congruent to each other
These two tiles are not congruent because they are not the same shape.
These two tiles are not congruent because they are not the same size.

Now we are ready to say precisely how many different kinds of tiles a tiling has. If T is some tiling, i.e. a collection of tiles, there is always some smallest collection of tiles, call it P, so that every tile in T is congruent to some tile in P. The tiles in P are called prototiles, and S, the whole collection of prototiles is called a generating set of prototiles, since you can "generate" the whole tiling from tiles in it.

For example, with the checkerboard, brickwall and beehive, T consist of an infinite number of tiles that are all the same size and shape. Thus, in these cases, the generating set contains only one tile: a square, a rectangular side of a brick, and a hexagon, respectively. However, the generating collection of prototiles for the mudflat is infinite, since there are an infinite number of different tile shapes in that tiling.

A tiling like the one shown on the left, with only one tile (a single rhombus shape, even though it appears in three colors) in its generating set is called a monohedral tiling. The checkerboard, brick wall and beehive are all examples of monohedral tilings.

A tiling with two tiles in its generating set (a thick rhombus and a thin rhombus) is called a dihedral tiling. The tiling shown on the right is a dihedral tiling.

For math experts: A more formal definition of prototiles and generating sets runs like this. Suppose T is a tiling, and suppose P is a subset of Q such that each no two tiles in P are congruent. Then we say each tile in P is a prototile. If every tile in T is congruent to some tile in P, then P is a generating set of prototiles. It is a fun exercise to show that if P and Q are two collections of generating prototiles, then they must contain the same number of tiles.

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Page last updated Wed Oct 19 23:11:18 CDT 2005
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