**Tilings with Just a Few Shapes**

Notice
that in our definition of a tiling there is no limit on the number of "shapes"
the tiles may be – there could be just a few shapes of tiles, or there
might be infinitely many! (Think, for example, of the stone wall and hexagonal
brick walkway shown on the first page.)

A **monohedral **tiling is one in which all the tiles are the same
"shape," meaning every tile in the tiling is congruent to a fixed subset of the
plane. This set is called the **prototile** of the tiling, and we say that the prototile **admits** the tiling. Here are some examples of monohedral tilings.

You might think at first that
monohedral tilings are necessarily somewhat uninteresting and easy to
understand. Not so! Here are two striking monohedral
tilings admitted by the same prototile.

In fact, we have no method__
__for determining whether or not a given prototile admits a tiling of the
plane! It is even unlikely that such a method could exist!

Of the 24 heptiamonds (a
heptiamond refers to seven equilateral triangles stuck together) pictured
below, only one is *not* the
prototile of a monohedral tiling.
Can you guess which one it is?

A **dihedral** tiling is a tiling in which every tile is congruent
to one or the other of two distinct prototiles. Similarly, **trihedral, 4-hedral, ..., n-hedral** tilings refer to tilings in which there are 3, 4, ...,
n distinct prototiles, respectively.

For example, here is a
dihedral tiling whose prototiles are two squares of different sizes:

**Question**: Show that every pentagon with 2 parallel sides and
hexagon with 3 pairs of pairs of equal parallel sides is the prototile of a
monohedral tiling.