Unless, like me, you have the misfortune to be in a room with painted or paneled walls, there is a curious fact of nature staring you in the face: There are precisely seventeen different types of wallpaper. Yours is one of them. Most likely you have never considered your wallpaper, except perhaps to admire its beauty. Yet a man’s home is his castle, and it is incumbent upon him to look closely at how he adorns it.
Maybe my statement seems patently false. After all, isn’t wallpaper just some paper with a picture on it, which we happen to put on our walls? And how can there be any limit to the number of pictures in the world? Surely this quantity is as unbounded as human imagination itself. But in fact my claim is both meaningful and deep; to unpack it we must distill wallpaper to its essence.
Wallpaper is sold to eager homeowners in the form of rolls. Such rolls, for economic reasons, have a certain fixed width and contain a certain fixed length of paper. As a result of these facts, wallpaper cannot have just any picture on it. It must have a pattern that repeats in two directions in the plane, once along the length and once across the width of the sheet. In some sense, the exact nature of the pattern is unimportant – it could be whatever you like. Some people are partial to flowers, others like more geometric figures, and an occasional nutcase prefers the logo of his favorite professional sports team. But we cannot ignore the directions in which the pattern repeats and the orientation of the pattern in each repetition. Say I wanted to tessellate my walls with a loved one’s face. She would look positively ridiculous sideways or upside-down, so my wallpaper had better not force her into such positions! I need to know whether my wallpaper pattern can accommodate the picture I want with the repetitions and orientations I desire to look at. In technical parlance, my aesthetic decisions concerning the pictures I put on my wallpaper are constrained by the symmetry group of the plane crystallographic lattice associated with my wallpaper pattern. And it turns out that there are only 17 different possibilities for this object; hence, only 17 essentially different wallpaper patterns.
Personally, I find this obscure result – from the mysterious intersection between mathematics and interior design – both deeply disturbing and profoundly exhilarating. Long ago, Descartes drew two crossed lines and discovered the Cartesian plane, and suddenly a new and seemingly infinite realm of decorating possibilities was opened to man. But 20th century techniques of abstract algebra have reduced this to a mere handful, restricted our act of creativity to picking one of seventeen things out of a hat.
That number seventeen seems particularly suspicious. Where did it come from anyway? The number theorist Daniel Shanks writes of the incidence of such bizarre “Pythagoreanism” – when seemingly arbitrary integers pop up for no discernible reason. (The name derives from the ancient Greek mathematician Pythagoras, who was such a fan of integers that he believed there to be nothing else. His followers, the Pythagorean brotherhood, evolved into a cultish group willing to kill in defense of this fact.)
Lately this sort of thing has been happening a lot in theoretical physics. For a long time we thought that the universe was smooth, but now it seems as though it is discrete: everything, from matter to energy to time, comes in integer multiples of miniscule quantum chunks. And what is more, the fabric of the universe itself has random integers woven into it. For instance, one promising unifying theory of physics holds that there are 10 dimensions to the universe.
Some friends and I once asked a physicist to explain where the 10 came from. “That’s easy,” he replied. “10 is two thirds of 15.” But where does the 15 come from (to say nothing of the two thirds)? He had an answer for this too: “Negative 15 is negative 26 plus 11. And negative 26 is 1 minus 3 times 3 squared. Whereas negative 11 is 1 minus 3 times 2 squared.” All these numbers! What we really wanted to know was what do they mean?! But knowing how way leads on to way, and terrified of what we might learn, we wisely broke off the conversation at this point.
Let me get something straight. It’s not the fact that 17 is a number that bothers me. It’s the fact that it’s such an arbitrary number – an integer of all things. If you were to randomly throw a dart at a board with all the numbers on it, it is a mathematical fact that the probability of hitting an integer would be zero – no chance at all. So what business does 17 have dictating my wallpapering decision?
I suppose, all things considered, that there are worse numbers than 17. A great feat in geometry – one that eluded even Euclid and the other Greeks – is the construction of a perfectly regular 17-sided polygon with just a straightedge and a compass. In honor of his discovery of this process, the great German mathematician Karl Friedrich Gauss had a regular 17-gon inscribed upon his tombstone. 17 is prime in its primacy, pronounced in its preeminence, a jolly good number, all things considered.
But I don’t like the fact that Nature (or perhaps something greater than Nature – God, maybe, or Mathematics) has partitioned us unawares into 17 equivalence classes, dividing neighbors and family members from one another by the symmetry groups of the plane crystallographic lattices on their walls. When Armageddon breaks loose and the inevitable clash of wallpaper patterns strikes the Earth like an asteroid, who knows which side shall prevail? And without this foreknowledge, how are we to make an informed decision about what to put on our walls? Perhaps it is best to stick with paint after all.
Yet at the same time the 17 bothers me, it also excites me. For although the appearance of an integer seems disturbingly random, it is suggestive of a deeper structure to the world, to comprehend which is within the grasp of human intellect. Integers are unusual, but we actually understand them pretty well -- how they relate to one another through addition and subtraction, multiplication and division. So even though the 17 constrains our freedom, in a sense, it also reveals the presence of a heretofore unimagined architecture to our existence.
The fact that mathematicians can classify wallpaper patterns means that Mathematics, and hence the universe, “knows” about the symmetry – the beauty – of our wallpaper in a deep way. This odd little theorem suggests that at least mathematically, there is some truth to our notion of beauty. Unlike so much of what we experience in life, it lends some credence to Keats’ assertion that “Beauty is truth, truth beauty,--that is all / Ye know on earth, and all ye need to know.” Again, modern physics backs up this assertion even further. It seems that the root of all the physical Pythagoreanism I described earlier may lie in the fact that space and time, matter and energy, forces and interactions are beautiful. That is, if all the stuff of the universe looks the same in some way, if it has some symmetry, then we can classify it, deconstruct it, and discover integers hidden in its deeper structure – just as we can classify the 17 types of wallpaper. “17” is therefore symbolic of a greater fact about how we fit into the universe: We find beauty in symmetry, and symmetry is true.