Mathematical Beauty and the K4 Crystal

In geek news today we came across Toshikazu Sunada‘s paper on something called the K4 Crystal, and his claim that it “looks no less beautiful than the diamond”. In this Sunada is consciously following in a very long aesthetic tradition.

The whole paper is available online as “Crystals That Nature Might Miss Creating“, a notice published by the AMS. The meat of the paper is technical, and won’t be our concern here, although it looks extremely interesting.

Sunada begins thus:

No one should have any objections against the claim that the diamond crystal, the most precious gem polished usually with the brilliant cut, casts a spell on us by its stunning beauty. The beauty would be more enhanced and its emotional appeal would be raised to a rational one if we would explore the microscopic structure, say the periodic arrangement of carbon atoms, which is actually responsible for the dazzling glaze caused by the effective refraction and reflection of light

The notion that beauty has a rational foundation in symmetry is very ancient, and has never really gone away in aesthetic theory either in the Western or Arabic traditions. Indeed, complex symmetries have long been a key feature of Islamic art in which figurative depictions of living things have in some contexts been forbidden or discouraged.

Sunada explicitly places himself in this tradition, referring obliquely to Pythagoras and Euclid:

We thus follow the Greek tradition in geometry that beautiful objects must be classified.

If you have a formula for beauty, of course, such a classification is perfectly possible and, for the rationalist, it’s therefore well worth doing. He refers explicitly to the classification of the Platonic solids, which was accomplished by Euclid and represented an early example of valuing a certain class of shapes for their symmetry and simplicity and then finding all possible shapes of that type.

The idea that “the more symmetry, the more beauty” is too simplistic, however, despite the occasional silly-season recurrences of this story. In the modern language of symmetry, group theory, it’s perfectly possible to count the number of symmetries a shape has. A square, for example, has 8 symmetries — 4 reflections, 3 rotations and an “identity” symmetry that does nothing, common to all shapes. A pentagon, on the other hand, has 10 symmetries, but most people would consider it daft to claim that a pentagon is 25% more beautiful than a pentagon. Instead the arrangement is more like the fuzzy orderings we talked about recently.

Still, the idea that symmetry, along with some kind of idea like “elegance” or “simplicity”, is a key part of what makes something beautiful is hard to shake. There’s a strong philosophical tradition at work here, too.

If beauty is an intrinsic, objective property of an object then we can say, with Kant, that when something is truly said to be beautiful then that statement is true for everyone, under all circumstances, necessarily. On that account, the beauty of an object is a fact about which one can be right or wrong.

But Kant described another aesthetic category, that of the “sublime”, and Nietzsche borrowed the distinction (via Schopenhauer) and developed it in his first book, The Birth Of Tragedy From The Spirit Of Music. There he named the distinction after Apollo and Dionysus. Mathematics is perhaps the ultimate Apollonian art: lucid, clear, tranquil, formal, self-contained.

I don’t want to claim that this distinction has any special merit, but it does have a powerful presence in the history of ideas, and particularly ideas about something we haven’t mentioned thus far: art.

It’s natural for many of us to think of artistic “beauty” in these terms, and to contrast it with the passionate, chaotic and dangerous “sublime” in art celebrated by the Romantics. Mathematicians, and other people who are interested in formal patterns, are rarely Romantics. Many consider themselves to be engaged in a creative activity — rightly, I think. But few, if any, call themselves artists.

Perhaps when people talk about “beauty” in mathematics they mostly mean simplicity, economy of means and perhaps a high degree of symmetry, which may be the abstract kind of symmetry defined by group theory rather than the familiar geometrical sort. It seems unlikely that they’re talking about the same thing as they are when talking about beauty in a work of art.

Getting a lot of symmetry into a simple object is a species of “doing a lot with a little”, a widespread human pastime. There are computer programmers who consider this to be beautiful:

#!/bin/perl -sp0777i<x+d*lMLa^*lN%0]dsXx++lMlN/dsM0<j]dsj
$/=unpack('H*',$_);$_=`echo 16dio\U$k"SK$/SM$n\EsN0p[lN*1

because it does something very complicated (RSA encryption) with very few keystrokes. Is the mathematician’s “beauty” any more than this, only with a much more refined subject?

That might sound like a disparaging question, but it isn’t intended to be. In fact, most modern accounts of aesthetic beauty are culturally relativistic, and even allow for idiosyncratic differences between individuals. We might agree that someone who thinks Banksy is a better artist than Da Vinci is wrong, but not in the same sense that someone who thinks the Sun goes round the Earth is wrong. We may also validly disagree about such statements, without one of us being wrong and the other right.

Put simply, “Beauty” is a many-faceted idea that carries rich layers of connotations. Yet when a mathematician uses an idea, it must stay in its place and do just what is wanted of it. I think it’s entirely appropriate for Sunada to refer to the K4 crystal as “beautiful”, with the special sense that word has in the tradition of mathematics. It would, however, be wrong to move from this to conclusions about art, culture, psychology or — heaven forfend — genetically-encoded preferences.

But hopefully we’re safe — this one’s most likely too geeky for the mainstream press to pick up anyway.

[UPDATE 20080126: Corrected a mistake in the numbers of symmetries of the square and pentagon. Thanks to Joshua for spotting it!]