Hidden Harmonies

by Ellen Kaplan

  would equal the square of the area of the last, ‘hypotenuse’, face, D (i.e., XWY): that is,

  If not quite up there with 666, still pretty wonderful. Here’s a more precise statement of the theorem, and its proof.

  THEOREM: In a tetrahedron, three of whose edges (p, q, r) meet at right angles at vertex Z, if the areas of the three faces meeting at Z are | A |, | B |, and | C |, then with | D | the area of the fourth (‘hypotenuse’) face,

  PROOF:

  Good ideas are born over and over because, like Pythagoras, they would show their immortality through reincarnation. Descartes may have co-fathered Faulhaber’s discovery; then in late eighteenth-century France its new parents were d’Amondans Charles de Tinseau and—independently, it seems—Jean-Paul de Gua de Malves: the first editor of the great Encyclopédie, and the man whose name is usually attached to this theorem (The Return of Stigler’s Law). But the best ideas have progeny, not just doubles, and now we can let loose that numerical impulse so recently restrained.

  We said rather cryptically that Descartes might have shared with Faulhaber the parentage of this theorem. In the winter of 1619 the two may well have met in Ulm. Local legend has it that one night, dazzled by his brilliance, Faulhaber reached out and touched his guest to make sure that he was human and not an angel. Wouldn’t you have done as much, were your visitor to have mentioned that this theorem also held in four dimensions? Descartes wrote, in his Private Thoughts (Cogitationes privatae), dating from 1619–21:

  This demonstration comes from the Pythagoreans and can also be extended to quantities of four dimensions. There the square of the solid opposed to the right angle is the squares from the other four solids altogether. To this let there be the example of progression in numbers 1, 2, 3, 4; in right angles of two, three, four lines.4

  Yet what does “let there be the example” mean? Might faithful analogy not have been sent to damn us? The restraint we had exercised before came from rightly recognizing that fantasy can lead imagination astray; the informing particularity of things can vanish at a touch of abstraction. The architectural instinct carries us too easily past detail, as would makes could, could makes might, and might makes must grow hazy. Hilbert’s more formalist followers, and people like Dijkstra, help in making syntax abet semantics, so that how we say will enhance what we think (though then they err by narrowing correction to discipline—as if they had emptied the perspective grids in a Vermeer of their lively content, leaving behind mere Mondrian).

  We can go past Descartes and generalize this theorem to five—to six—to spaces of any dimension you like. We can deduce that the hypotenuse ‘face’ of such a ‘pyramid’ in, say, a thousand and one dimensions has an ‘area’ equal to the sum of that of the other ‘faces’ (where we need all those raised eyebrows of ‘so-called’ because the faces, for instance, are—as in two and three dimensions—one dimension down from that of the space the object lives in: so here, thousand-dimensional faces with their multi-dimensional equivalent of area). But we’ll first have to understand how to think about (rather than just be stunned by) all this, since seeing it is out of the question. We’ll have to insure that we have been entertaining an angel, not a dev il, unawares.

  The insurance comes from that cautious run-up called induction, which, like the high jumper’s measured steps, will hurl us over the bar (how it works is explained in the appendix to this chapter). It was born and reborn from at least the eleventh through the seventeenth centuries, but only in 1912 did the great French mathematician Henri Poincaré see how it would let us precisely grasp higher dimensions. The question with induction is always: have you a way of going uniformly from any stage to the next? Poincaré made use of a very simple observation: two points on a line will be separated by a cut at a point between them. Since the line is one-dimensional and a point has dimension zero, perhaps the line’s dimension depended on the dimension of what was needed to cause the separation: 1 was the successor of 0.

  Did this work for a two-dimensional plane? Given any two points on it, they could be separated from one another only by cutting a closed curve (which has dimension one) around one of them—and 2 is the successor of 1. Likewise in three dimensions, it would take a two-dimensional surface (like the skin of a sphere) to separate any two points—no lower-dimensional boundary would always suffice—and 3 is the successor of 2.

  Rather than a proof by induction, therefore, Poincaré made this definition by induction: a space is n-dimensional if any two points in it can be separated by an n − 1 dimensional subset of it, and no lower-dimensional subset would always work.

  What guarantees that this inductive definition is right? We’re past the child’s fiat “Anything I say three times is true.” As in all matters of religion, what’s left is justification by faith and by works. Our faith rests on uniformity: the picture Poincaré’s definition builds up makes the next dimension arise from the prior in such a way that the tools and insights we had before will extend smoothly. If you say this is all very well, but the higher dimensions we thus get are just the playing fields for our game, as arbitrary in their plan as are the game’s rules, we would agree—but add that this seems to be the only game in town.

  Justification by works comes from seeing how this understanding of dimension enlightens and predicts in physics: one degree less separated from what we think of as the real world. We need three dimensions to locate a particle in this world—locating its distance in height, length, and breadth from a fixed center. These three axes we conveniently think of as at right angles to one another. This picture accords with Poincaré’s, for he had three-dimensional Euclidean space in mind as his model, and then broadened it to achieve his topological viewpoint (which has evolved greatly since him). Once the particle moves, however, we need to keep track of the forces causing its motion—momenta in those three directions. Since these are independent of one another and of the directions, we’ll need three more axes, each at right angles to all the rest, in order to pin down the moving particle—altogether the six dimensions of ‘phase space’. You can imagine adding more criteria that you want to keep track of, and new dimensions, one at a time, for each. As in the moral realm, what was so recently unthinkable has become a matter of course.

  We’re prepared now for the modern generalization5 of the Pythagorean Theorem to any dimension n. Let’s first choose the right words for so overweening a project. What was a triangle in two and a tetrahedron in three dimensions is called a polytope in higher dimensions. We’ll let P stand for this orthogonal polytope—i.e., one with that right-angled vertex (though you might prefer to think of P as standing for our pyramid, no matter what dimension it is at home in).

  A triangle has legs, a pyramid faces—each therefore with its kind of boundary one dimension down from that of the figure itself. The n − 1 dimensional equivalents of these for an n-dimensional P are called facets. What was a right triangle’s hypotenuse, and the face opposite the pyramid’s right-angled corner its hypotenuse face, will now accordingly be the hypotenuse facet. And while we worked with a facet’s area in two dimensions and volume in three, the general term in any dimension is content.

  So the stunning generalization we hope to prove is that for P in any dimension n, the square of the hypotenuse facet’s content equals the sum of the squares of the contents of its other facets. This hope will become a reality in the chapter’s appendix, should you choose to stretch your legs—or really your mind—there. For while the ideas, tactics, and strategies are thoroughly human, the language that bears them has a concision that takes some getting used to, and abbreviations that are immensely helpful in the end, but intimidating when you begin. And if you choose not to go for that exhilarating run, your hope may reliably be replaced by faith.

  FORMS OTHER THAN SHAPES

  Pythagoras’s soul moved from body to body, the Pythagorean Theorem from plain to ever more rarefied triangular embodiments. The natural movement now is for shape to distill wholly away to the single malt of nu
mber: the ultimate generalization whose intoxicating fumes layer the mind’s upper air. But this will need a chapter of its own.

  APPENDIX

  (A) We promised a proof that if T1 and T2 are similar closed curves on straight line segments, then their areas are to one another as the squares of those segments.

  PROOF:

  (B) This is Matthew Stewart’s proof of his theorem, which you saw the chapter: b2m + a2n = c(d2 + mn).

  (C) Here is your kit for induction. If dominoes are set up vertically in a line on a table and you push over the first, all the rest will fall down in sequence—as long as you’ve set them up so that each is less than a domino-length from the next. Think of the natural numbers (1, 2, 3, . . .) as these dominoes, and suppose you want to prove that some statement is true for each and every one of them. You first prove it is true for the first number, 1—that’s like pushing over the first domino. Then (and this is the great idea), prove that if your statement is true for any one of the numbers, then it will be true for the next. That’s like making sure that the dominoes are close enough together to fall when the first is tipped. If you manage both steps (proving your statement for n = 1, then proving that if it holds for the number k, it must also hold for k + 1), you will have proved it true for all natural numbers: a proof by induction.

  Suppose, for instance, you want to prove that the sum of the internal angles of an n-sided polygon is (n − 2)180°.

  Well, this is certainly true for a triangle (where n = 3)—though this needs to be established ahead of time, and deductively. Now assume this statement is true for a k-sided polygon: i.e., assume that its interior angles add up to (k − 2)180°. Using this assumption, prove that the statement is also true for a (k + 1)− sided polygon: i.e., that its interior angles add up to (k + 1 − 2) 180°, or (k − 1)180°.

  You may object at this point that we’re assuming what we’re setting out to prove! Not really: We’re assuming only that it holds for a single number, k—and then somehow or other we’ll prove that it holds also just for the next, k + 1.

  So here’s the picture of our assumption: a k-sided polygon, with its angles adding up to (k − 2) 180°.

  Let’s draw a (k + 1)− sided polygon:

  and (here’s the cleverness) nest a k-sided one within it:

  Behold! (as the second Bhaskara would say): by our “inductive assumption”, the angles of the k-sided polygon add up to (k − 2) 180°; and perched on top of it, a solitary triangle, whose angle sum is 180°. Altogether, then, the angle sum of this composite figure is (k − 2)180° + 180° = (k − 1)180°. Now remove the interior line that separated this triangle from the shape below it. We lose one side but gain two, so have indeed k + 1 sides, and the required angle sum.

  The great advantage of an inductive proof is the relative ease with which it can often be made; its great drawback is that the proof substitutes how for why a statement is true, indubitably establishing that it is, by the domino effect. We never have nor ever will see a trillion-sided polygon, yet are as confident as about anything in life that its interior angles add up to (1012 − 2) 180°, for we understand the gearing. So, after the initial astonishment at the figures emerging on the hour from Munich’s Glockenspiel, there may be appreciation but there will be no surprises.

  This peculiar situation reflects the fundamentally structural nature of mathematics and its objects: abstraction, generality, and truth ripple through it, and them, because their meaning isn’t (some would argue) to be sought elsewhere.

  (D) We end this appendix with the proof of the Pythagorean Theorem in n dimensions—a generalization of Faulhaber’s. A word first about those notational conventions used to simplify the algebra.

  The way we write exponents is a fine example of notation and meaning leading each other on.

  The other notation we use here is the abbreviation (sigma, the upper-case Greek S) for ‘sum’. Instead of writing a1 + a2 we could write: “the sum of aj, as the index j goes from 1 to 2”—abbreviated

  This seems an absurd convention until you get to longer—and arbitrarily long—sums:

  is briefer and more transparent than a1 + a2 + a3 + a4 + a5 + a6 + a7, and certainly

  Now we’re ready for the proof itself.

  A triangle in 2-space has 3 sides, a tetrahedron in 3-space has 4 faces, and P in n-space has n + 1 facets, one of which is the hypotenuse facet Fh. So what we want to prove is that

  or, using the shorthand explained above,

  You know that the volume (area) of a triangle in 2-space is half the base times the height, as here:

  Looked at more generally, this base is the hypotenuse face, Fh, so

  Of course we could have gotten the same volume by taking half the content (i.e., length) of another side times the altitude to it:

  In n = 3,

  An inductive process generalizes both of these to

  Notice that we’ve used the subscripts to link a facet to the vertex it doesn’t contain, and from which therefore its altitude is dropped.

  And if you wonder about that (s), it isn’t only to decorate a meritorious achievement, but to give us a way to refer back to these equalities.

  Our strategy will be to say the same thing, P’s content, in two different ways, and reach our goal by sliding them together. Algebraic tactics along the way will be for the sake of this strategy.

  We begin by going back to n = 2 and recalling that OSA2 ~ A1OA2,

  We’ve expressed g in terms of the lengths of the triangle’s sides, which will turn out to be just what we need. First, though, we have to transform our formula through six small algebraic manoeuvres, in order to put it in a more usable form:

  CHAPTER SEVEN

  Number Emerges from Shape

  Amidst all the wreckage caused by , one Babylonian monument still towered up in the Pythagorean sky: those triples of whole numbers they had found which made sides of right triangles. At its base lay (3, 4, 5), with its multiples piled endlessly on. But you recall there were also (5, 12, 13), (7, 24, 25), and (11, 60, 61): in fact, the whole family of triples that came from fitting a pebbled gnomon to a pebbled square, so long as the number of pebbles in that gnomon was (despite its L-shape) itself a square number. So the gnomon around the square which has

  4 pebbles on a side has 9 = 32 pebbles in it, and the gnomon fitted to a 24 × 24 square has 72 = 49. As we would say, whenever a gnomon with 2a + 1 pebbles is pushed up against a square with a2 pebbles, the outcome (a2 + 2a + 1) is a square with (a + 1)2 pebbles, and the ones that interest us are those where 2a + 1 is a square number, though here arranged in an L (once again, it is mind that disembodies number).

  We conjectured that the Old Babylonians had also discovered a second family of these triples, made by attaching two gnomons to a square, at its opposite corners, so long as the sum of their pebbles was also a perfect square—as when two gnomons with 32 pebbles each surround a square with 152 = 225 pebbles, since 2 × 32 = 64 = 82, and 82 + 152 = 172. Putting this too in our terms, (a + 2)2 = a2 + 4a + 4 (this 4a + 4 is made up of the two gnomons each with 2a + 2 pebbles).

  It is awesome that our finite minds can encompass one, and then even two, such infinite families.

  No tower but excites the urge to out-top it, as the Empire State Building pushed higher than the Chrysler, the World Trade Center and then the Sears Tower beyond both, and then the succession of Petronas Twin Towers, Taipei 101, Burj Khalifa—and Babel, past them all. Perhaps the ravages of the irrational elsewhere increased the need, among the Greeks, to find more infinite families of Pythagorean triples (as these came to be called): here among whole numbers, at least, rationality could work securely with its ratios. Or it may have been, if such study preceded Hippasus, that this made his revelation of the irrationality of all the more devastating. But the skyscrapers of Manhattan are unmoved, though tunnels honeycomb the rock beneath them, and wonders are still to achieve. We now know that 2 isn’t alone: there are incomparably many more irrationals than rationals. But this makes th
ose whole number solutions blaze even more brightly among the dark numbers, as stars do in a universe made mostly of dark matter.

  Yet how can we find more infinite families of Pythagorean triples, when there’s nowhere else to attach gnomons to the inner square? By turning away from shapes, for all their nursery warmth, and letting mind loose on number per se. And when mind is let loose, its grasp tightens even as what it holds grows abstract. You see this already beginning to happen when a square number is distinguished from a square shape.

  The Pythagoreans wanted to find all triples of whole numbers, (a, b, c), such that a2 + b2 = c2. They knew that given one such, all multiples of it, (ma, mb, mc), would be triples too, so their search narrowed down to finding the ancestors of each family line: those triples that had no common factor (or were ‘relatively prime’, as they say in the trade). This is the time to sit under your favorite tree and close your eyes.

  On reflection, we can be sure that if any three numbers satisfying our equation have no common factor, no two of then can share a factor either. For if a and b, for example, were both multiples of some prime p, then a = pk and b = pm for some k and m. But since a2 + b2 = c2, this would mean that

  which means that c also has p as a common factor, and we agreed that the triple (a, b, c) had no factor in common.