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Symmetric Tilings
For most people, the idea of symmetry conjures up thoughts of Art and Nature, as well as mathematics. Our ideas of beauty are closely tied up with symmetry, and principles of symmetry manifest themselves in subtle ways throughout the natural world. Books like Design and Color in Islamic Architecture : Eight Centuries of the Tile-Maker's Art and Structure in Nature Is a Strategy for Design pursue these connections in greater detail.

What is Symmetry?

Even though we all understand and recognize symmetry intuitively, it is a little harder to say just what it is. However, in the plane, the basic idea is clear enough:

A figure in the plane is symmetric if you could pick up a copy of it, move it around to a new location somehow, and set it back down on the original figure so that it exactly matches up again.

The animations below illustrate this definition of symmetry in the case of a checkerboard tiling. On the left, the tiling is just being shifted or translated is such a way that it matches up with itself again. On the right, the tiling is being rotated a quarter turn to make it match up again.

The checkerboard tiling is a very symmetric tilings. There are lots and lots of ways of moving it on top of itself so that it matches up again in addition to the two shown above. Mathematicians say that different way of moving a tiling onto itself is a "symmetry of the tiling". How many symmetries can you think of for the checkerboard tiling?

Kinds of Symmetry

One of the first things to notice about symmetry is that there are several different kinds. This is because there are different ways of moving something in the plane. One way is to just translate it a little. Another is to rotate it. Yet another is to turn it over. As a consequence, there are different kinds of symmetry.

Consider the pictures above. The one on the left exhibits rotational symmetry. You could pick it up, rotate it a third of a circle, and set it back down so it would exactly match up. The figure on the right has mirror symmetry. In this case, you could pick it up, turn it over along the axis of the dotted line, and set it back down on top of itself.

There are actually four distinct kinds of symmetry, corresponding to four basic ways of moving a tile around in the plane, illustrated to the right. In mathematical language, these different ways of moving things in the plane are called isometries.

To get a quick idea about how isometries work, you can look at some animations of translations, rotations, reflections, and glide reflections, or you can read about isometries in more depth in Introduction to Isometries in the Science U library.

Symmetry Groups

If a tiling has any symmetries at all, it usually has lots of them. This is simply because if we do one symmetry followed by another, then we could have just move the tiling directly from its initial postion to its final position, and it would still match up. In other words, the net effect of doing two symmetries one after the other is itself a new symmetry. In mathematical language, we say that the composition of two isometries (i.e. symmetries) is again an isometry. For example, as you can see on the left, if we flip the checkerboard tiling twice, one in the x-direction, and once in the y-direction, it has the same net effect as rotating the tiling half a turn. (Focus on the purple square for a landmark.)

This pleasant fact makes it much easier to think about the symmetry group for a tiling. The symmetry group of a tiling is just the collection of all its symmetries. If we think of the checkerboard tiling continuing on forever in all directions, it is clear that it has infinitely many symmetries. Just for starters, we could tranlate any even number of squares in the x-direction or y-direction and the tiling would still match up. This doesn't even count all the rotations and other isometries which take it back onto itself.

In general, the more symmetric a tiling is, the larger its symmetry group is. Indeed, as our checkboard example shows, symmetry groups can be huge, infinite things. However, because we can compose symmetries to get other symmetries, instead of thinking about all symmetries, we need only think about a few basic ones that we can compose with each other to get all the rest!

In the case of the checkerboard tiling, for example, instead of thinking about all of the possible ways of translating it back on top of itself, we can concentrate on two basic translations, one in the x-direction and one in y-direction, as show above on the left. We can get any other translational symmetry just by doing these two enough times. For example, the diagonal translation on the right is the same as the net effect of the two vertical and one horizontal basic translations shown in the middle.

Mathematicians call such a collection of basic symmetries the generators of the symmetry group. As you will see, generators play an important role in classifying different kinds of symmetry groups.


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