**Monomorphic and **

**K-Morphic Tilings**

So
far, we have considered the question of whether or not a given prototile admits
a monohedral tiling of the plane.
A related question is the following: how many distinct monohedral
tilings does a given prototile admit?
(Recall that two tilings are said to be equal if it is possible to scale
one so that it is congruent to the other.
Two tilings are distinct if this is not possible, i.e. if they are not
equal.)

For example, here are two distinct monohedral tilings admitted by the same prototile (a square):

However
there are some tiles which *only*
admit one tiling. If we modify the
square tile in the example above by adding a "zigzag" to each side, there is
now only one way to use it to tile the plane.

A
tile is called **monomorphic** if it
is the prototile of a *unique* (i.e.
only one) tiling of the plane. For
example, this tile

is monomorphic; it is clear
that there is only one way to tile the plane with it.

Here
are a few more examples of monomorphic prototiles:

**Question**: Given an example (not one of the ones above) of a
monomorphic prototile.

**Question**: Show that no triangle can be a monomorphic prototile
(i.e. show that you can make more than one distinct monohedral tiling using any
triangle.)

Some prototiles admit *precisely
two* distinct monohedral tilings; such
tiles are called **dimorphic**. A prototile is said to be **trimorphic
**if it admits precisely three distinct
monohedral tilings, and **n-morphic**
if it admits precisely n distinct monohedral tilings, where n is a positive
integer.

An
example of two tilings admitted by a dimorphic prototile:

More
examples of dimorphic and trimorphic prototiles can be found in "Tilings and
Patterns," section 1.5, by Grünbaum and Shephard.

** **

The following two questions
are "open" questions, meaning that not even professional mathematicians know
the answer yet! (Care to give them
a try?)

****Open Question****: for every positive integer n, is it possible to find
a tile which is n-morphic?

The next open question uses
the idea of countable infinity. An infinite set is said to be countably
infinite if the elements of that set are in one-to-one correspondence with the
positive integers. In other words,
a set is countably infinite if you can make an (infinitely long) list of all
the elements of that set.

****Open question****: Does there exist a tile that admits a countable
infinity of distinct tilings (but not uncountably many)?

(The square, for example,
does not satisfy this question, as it admits __un__countably many distinct
tilings)

** **