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
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
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.
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 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.