Self-similar self-similarity

Let us explore the nature of symmetry, beginning simply, but moving steadily to abstraction, eventually to the symmetries of the symmetries themselves, and their symmetries and so on transfinitely.

Let me tell a mathematician's tale about symmetry. We begin with playful curiosity about a concrete elementary case—the symmetries of the letters of the alphabet, for instance. Seeking the essence of symmetry, however, we are pushed toward abstraction, to other shapes and higher dimensions. Beyond the geometric figures, we consider the symmetries of an arbitrary mathematical structure—why not the symmetries of the symmetries? And then, of course, we shall have the symmetries of the symmetries of the symmetries, and so on, iterating transfinitely. Amazingly, this process culminates in a sublime self-similar group of symmetries that is its own symmetry group, a self-similar self-similarity.

In the light of symmetry consider a capital letter A.

Well, this particular A in this particular font, unfortunately, is not quite perfectly symmetric. The uprights at left and right differ in thickness, for example, and the serif on the right foot is ever so slightly larger than the serif on the left. Let us try to draw a somewhat more symmetric letter A, even though it may be less graceful.

That's better. This A exhibits a vertical line symmetry—the vertical line down the center.

If we reflect the letter across that line—like Alice through the looking glass—it lands perfectly upon itself. We might fold the paper on that line to realize the symmetry.

The letter B also exhibits symmetry.

Well, again, this particular B in this particular font is not perfectly symmetric—the upper curved half is slightly smaller, and the two curves are not exactly the same shape. But we can try to draw a more symmetric B, if less graceful, so as to exhibit a horizontal line symmetry.

We might similarly consider all the letters of the alphabet, drawing each of them as symmetrically as possible. Get some paper and try!