Tiling Terminology
One of the first thing that comes to mind when people think of tilings are repeating patterns. Tile floors, mosaics, and brickwork generally all display some kind of repeating pattern. In mathematical language, a pattern that repeats in a regular way is called periodic.

### Periodic vs. Repeating Tilings

Imagine walking in a straight line across a tile floor. As you walk, you notice that you keep seeing the same pattern again and again. In ordinary language, we might say that the pattern repeats.

In order to say that the tiling is periodic, as opposed to merely repeating, a mathematician would want to be sure the repetition happens in a regular way. If you were to walk from left to right across the the tiling shown on the right, you would keep seeing the thin tile again and again, but the repetition isn't at all regular, since the distance between occurences of the thin tiles keeps increasing.

On the other hand, if you were to cross the tiling from top to bottom, instead of left to right, the pattern would repeat regularly. So we also need to be sure that a tiling is as regular as possible in all directions, before declaring it is periodic. In general, it a hard to say what "as regularly as possible" means in a precise mathematical way. But for a plane tiling, it basically means that the tiling repeats in two independent directions, since the plane is 2-dimensional.

### A Test for Periodic Tilings

So how can we tell if a tiling is periodic? One way is to construct a lattice. A lattice is a grid consisting of two sets of evenly spaced parallel lines. In the image on the left the lattice is the black lines (the blue lines represent the cooridinate axes in the plane).

By the way it is made, you can see that a lattice repeats regularly in two directions. The parallelograms formed by the lattice are called period parallelograms. A tiling is periodic when we can lay a lattice over the tiling in such a way so that the period parallelograms contain idential pieces of the tiling. We call these pieces fundamental domains for the tiling.

To see how this works in a specific example, consider the sequence of images below. The picture on the left shows the tiling. The middle pciture shows the tiling with a lattice superimposed on it. The picture on the right is a close up of a fundamental domain.

You can easily see how given a single fundamental domain, we can recreate the periodic tiling by translating and pasting. We just translate copies of it in the two directions given by the lattice.

### Fun with Fundamental Domains

An interesting thing to notice about fundamental domains is that they are not unique. For example, given a lattice we can combine two adjacent fundamental domains to obtain a new fundamental domain. The rectangle on the right below is a closeup of the fundamental domain obtained by taking two copies of the original fundamental domain from above and combining them. It is the fundamental domain coming from the larger lattice shown below on the left.
We have produced two fundamental domains. Are there more? Indeed, yes! There are many more. Look at the period parallelograms of the lattices below the right and left. Can you convince yourself that these period parallelograms are also fundamental domains of the tiling? Can you see how to find other fundamental domains?

### Next on the Menu: Symmetry

In discussing periodic tilings, we encountered the idea of building up a tiling by translating copies of a fundamental domain all over the plane. In this case, the rules for how to move the copies of the fundamental domain are pretty clear -- just slide the copies in the directions parallel to the lattice lines.

However, this is just the tip of the iceberg when it comes to building up tilings from fundamental domains. Different sets of rules for shifting around fundamental domains lead to tilings with intricate and beautiful symmetries.

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