**What is a Tiling?**

** **

**Tilings in the World
Around Us **

Tilings are all around
us. Given a bunch of little
pieces, it is human nature to wonder how they fit together. In the most general sense of the
word, a tiling is just a way of decomposing some space into lots of little
pieces (tiles) that fit together without gaps or overlaps. For example, even our bodies can be
tought of as tilings in which the "tiles" are cells. Land ownership gives a tiling of the surface of the earth!
Tilings are literally everywhere.

Some examples of tilings:

Farm fields "tiling" the surface
of the earth
Cement road tiles

A stone wall Honeycomb

A brick walkway
Onion root tip

**Question**: Identify ten tilings in the world around you.

**The
Mathematical Definition of Tilings**

As
we have seen above, it is possible to "tile" many different types of spaces;
however, we will focus on tilings of the plane. By "the plane," we mean the 2-dimensional Euclidean plane
-- i.e. a flat, infinitely large
surface. From this point on, we
will use the word "tiling" to refer to a tiling of the plane.

In
mathematics, a **tiling** (of the
plane) is a collection of subsets of the plane, i.e. **tiles**, which cover the plane without gaps or overlaps. There is one more detail to add to this
definition – we want a tile to consist of a single connected "piece" without "holes" or "lines" (for example, we don't want to think of two
disconnected pieces as being a single tile). Thus, we require that each tile be
a **topological disk** (this is
math lingo for "consists of a single connected piece without holes or
lines").

We can't draw an entire
tiling of the plane (it is of infinite size!), so when we give an illustration
of a tiling, it is actually a small piece of the whole tiling.

**When
Are Two Tilings Equal or Congruent?**

Two
tiles are said to be **congruent** if
one can be transformed into the other by a rigid motion of the plane, i.e. by a
combination of rotations, translations, and reflections (but not
stretching/shrinking).

Similarly,
two tilings are said to be **congruent**
if one can be transformed into the other by a rigid motion of the plane. Two tilings are said to be **equal** or **the same**
if one can be changed in scale (magnified or contracted equally throughout the
plane) so as to be congruent to the other.

For
example, these two tilings are equal (but not congruent since a change of scale
is required to make them coincide).