by Ellen Kaplan
path at A with the first of the terms remaining. This truncated harmonic series will still diverge (you’ll still be adding up an infinite number of halves), so the path will be infinitely long—but the sum of the areas on its segments will be less than the you required.
If you think of mathematics as housed in a gleaming cube far away, where a grid city rises on a pale horizon, don’t such wares as these suggest rather that its awnings spread in a murmurous souq, or in the compiled jumble of gables and chimneys where
Alleys John Winthrop’s cattle planned
Meander like an ampersand4
or in Lusitania, a shadowy Nowhere of catacombs and caverns, where custodians of truths and secrets keep the torches lit?5 Such was the name that the followers of Nikolai Nikolaevitch Luzin, whom you met in Chapter Three, gave to their society—which brings us to . . .
. . . the Staircase. Earlier in this chapter, you recall, we better and better approximated a straight line by an ever more rapidly vibrating sine wave. Couldn’t we modify that approach here by taking a flight of stairs instead, developed from the original triangle’s tread and riser, and replace the hypotenuse by it?
That is, make a sequence of functions of ever shorter verticals and horizontals, starting at A and going to B, and clinging more and more closely to the hypotenuse. This lets us fantasize that doing so infinitely will give us the hypotenuse itself, as the limit of these staircases, which are diminishing past shark teeth to sawteeth, and beyond. Didn’t we after all do something like this before, with our little triangles, in finding the length of a curve?
What an appealing fantasy—turned appalling. For were this possible, we might as well give up mathematics on the spot as hopelessly paradoxical: those staircases—no matter how short and numerous their treads and risers—would always have a total length equal to the original triangle’s two sides, while at the same time being shorter than them (by the fundamental triangle inequality).ae
Luzin was plagued, in his student days, by this staircase—as many another has been, before and since. “Boleslav Kornelievich!” he protested to his professor, “instead of a finite set I take a denumerably infinite set, for example, all points on the diagonal that are a rational distance from the beginning of the diagonal. Next I take all straight lines passing through the points of the set parallel to the X-and Y-axes respectively, and finally, I eliminate all ‘superfluous’ parts of the straight lines. What will I obtain? A saw, but one with actually infinitely small teeth! In other words, there is an individual, fixed curve that is infinitesimally different from the diagonal.”6
“Listen—what nonsense you are dishing out!” Professor Mlodzeevskii explained, and strode away.
Where did this leave Luzin—where does it leave us? Luzin had disturbed a kraken in the ocean trenches of mathematics, which is now swimming toward us with its lurid mouth agape. Around it are monsters with teeth on their teeth and yet finer teeth on those, ad infinitum, their series converging toward a function continuous everywhere and differentiable nowhere: all teeth.af Luzin’s creature may not be as definitively devastating—but it is menacing enough, since its irresolvable paradox threatens to devour mathematics.
Why should sine waves of higher and higher frequency converge to a straight line, and not these ever more rapidly vibrating square waves? What you may think a negligible difference between them plays an important role: the sine waves are smooth (differentiable), and length—as you saw—is defined in terms of the derivative. If a sequence of functions, which converges point by point to a limit, has its derivatives converging uniformly to that limit function’s derivative too, then the lengths of the graphs of the functions in the sequence will converge to the length of the limit’s graph.
Unfortunately this sufficient condition isn’t also necessary: we can have a sequence of functions not everywhere differentiable converging to a smooth limit, with its lengths converging to that limit’s length. So Archimedes famously made better and better approximations to the circumference of a circle by a sequence of inscribed polygons, whose sides shortened as they grew in number. Adjacent sides did indeed meet at sharp corners that therefore had no derivative at them, but the angles at those corners grew flatter and flatter, unlike Luzin’s constant right angles.
Our difficulty is growing increasingly angular itself. To smooth it away, keep before your mind’s eye the image of a saw laid across a knife: no matter how many minute teeth the one may have, it isn’t the other; they cut according to different principles. The functions in a convergent sequence will have many properties, not all—or even most—of which their limiting function may share (you can divide by each of the fractions that converge to 0 as n increases, but you can’t divide by 0). Here, picture the staircase fitted within a rubber band, which tightens and narrows as the steps diminish in size. This elastic boundary
encloses the hypotenuse too, of course, and shrinks down to it, compressing the staircase as it does so—but that doesn’t mean every property of the staircase—such as its length—persists with it. Haven’t you just seen, in fact, an example of length going haywire as a coastline crinkles?
This cages Luzin’s monster, making mathematics safe again—at least until the next kraken wakes.
Esprit d’escalier. A friend of ours, both a carpenter and a mathematician, told us that he had been faced with the problem of measuring a banister’s length for a spiral staircase he was building: a formidable-looking task, studying the drawing—what was the formula for the length of such an arc, and what arc, after all, among all those that cycle through space, was it?
And then the mathematician in him spoke to the carpenter: slice this cylinder open:
APPENDIX
(A) How follows from the law of cosines: You saw in Chapter Six that
Extract from this diagram the triangle in question,
then rethink and relabel it in terms of vectors:
In this new language, what is the length of the old side c? Complete the parallelogram and notice that its fourth vertex is (x1 − y1, x2 − y2)
so that the vector to it is X − Y.
Clearly the side we want has the same length | X − Y | as this, so that the law of cosines, translated into vector language, reads:
Writing all but the last term out in terms of the coordinates:
Expanding,
So, canceling and simplifying,
so that
Now just dress this up in our vector finery, with for the inner product, and we have
Looking at the two vectors X and Y in n-space, you’d see
The converse clearly holds, if neither X nor Y is the 0 vector.
(B) The inner product for Fourier series.
The inner product we had, , was a sum of the products of these vectors’ respective coordinates. Now it will be a sum of the multiplied products of f and g on the chosen interval [a, b]—where the analogue of sum is the integral from a to b (its symbol, after all—the elongated eighteenth-century S, —was meant to remind readers of this analogy). So
The norm drawn from this inner product is (just like the norm)
This tailor has killed two at a blow: establishing the perpendicularity of the axes , and that each is of unit length. These are the e1, e2, . . . we had hoped for.
New technology, however, inevitably raises new technological problems, which new techniques will solve. The tight little helices of infinite Fourier series taper down (like waterspouts tilted this way and that in a da Vinci notebook) to their limits—but what assurance have we that the limits themselves will also lie in this infinite dimensional space? Might not the tips of these screws just fail to engage in the vectors they tunnel toward?
Oh, the little more, and how much it is!
And the little less, and what worlds away!7
Fortunately the way our inner product behaves kills this problem dead too: how these series converge, and subtle inequalities applied to them, show that their limits are also in the space, which is therefore called complete: a Hilbert sp
ace (after its twentieth-century inventor). We have carried the natural bent of our thought, via induction, to its limit: at the edge, perhaps, of understanding, but within reason.
CHAPTER NINE
The Deep Point of the Dream
Every dream, Freud once wrote, has a deep point beyond which analysis cannot probe. Whether or not this is true of dreams, there are cyclones in mathematics that funnel down and down even below the ground level of apt connections and on past their intertwining roots.
We’ve watched the Pythagorean Theorem take shape in ways that were often starkly different from what reason would have predicted, then acquire a variety of proofs from as great a variety of people, and afterwards spread in the most far-reaching and unexpected directions. All along, however, while this theorem may have seemed an increasingly significant ganglion in Euclidean geometry, nothing suggested that it was Euclidean geometry itself: the geometry of flat planes, characterized by the Parallel Postulate. Now we have come to the deep point of this dream: PT, the Pythagorean Theorem, is equivalent to the Parallel Postulate, PP! Each, that is, implies the other, so that either could be taken as the Ancient of Days in this world of congenial insights and crisp proofs.
Hard to believe? As you saw in Chapter Five, when the forty-year-old Thomas Hobbes first came on the Pythagorean Theorem in a book lying open at a friend’s house,
“By God!”, said he, “this is impossible!” So he read the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. And so in order that at last he was demonstratively convinced of that truth.1
Let’s follow Hobbes’s example and demonstratively convince ourselves of its equivalence to the Parallel Postulate. You probably expect that a revelation this profound must be so refined or abstruse as to be all but ungraspable. In fact the story will fall out in just eight short encounters. And you’d certainly think that so deep a result, showing that what characterizes Euclidean geometry is as much the PT as the PP, would be the end if not the beginning of any introductory plane geometry course. Yet it is hardly known, and while it is evident that PP PT (look at any proof of the latter and you’ll find the PP used somewhere along the way—as in Euclid’s VI.31, where the key use of similar triangles depends on it), proofs that complete the equivalence by showing PT PP don’t exactly abound. You can hear a few faint whispers in stray late twentieth-century journals (“That Mr. Pythagoras living down at the end of the street? Well, they say that he . . .”).2
All but one of the steps in the proof we’ll follow are due to the renowned early nineteenth-century French mathematician Adrien-Marie Legendre (who, said his colleague Poisson, “has often expressed the desire that, in speaking of him, it would only be the matter of his works, which are, in fact, his life”—he was born rich, died poor, made outstanding contributions to algebra, statistics, and number theory, and left a still unproven conjecture behind: that there is a prime number between n2 and (n + 1)2, for every positive integer n). One step is the work of a Dr. Brodie.3
We’ll sketch the sequence of proofs, then outline the strategy of each, and finish with tactical details. The postulates of Euclidean geometry are in the appendix to this chapter. You may want to check that the PP hasn’t been used covertly in any of the steps along the way—all the other postulates are available.
This brings up a subtle issue. The postulates without the PP constitute what has come to be called ‘neutral geometry’: a geometry where through a point not on a line there is at least one parallel to that line. Exactly one gives us Euclid’s flat plane—but more than one would be possible (hence violating the PP) on a surface with negative curvature, like a whirlpool (‘hyperbolic geometry’). If, however, you think of a surface with positive curvature, such as a sphere, and on it take great circles for lines (we’ll soon justify this arbitrary-seeming move), then again the PP would be false, but this time because there would be no parallels to a line through a point not on it. Here the postulates of neutral geometry fail: specifically, the four betweenness postulates, which have to be replaced by seven ‘separation’ postulates; and the notions of segment and triangle have to be redefined. These modifications constitute another non-Euclidean geometry, called elliptic. So what we’ll now prove is that when the PT is added to the rest of the neutral geometry’s postulates, the curvature of the surface becomes zero, and we find ourselves returned to Euclid’s flat plane.
You see, then, that what we’re about to launch into is the one sort of generalization we hadn’t yet touched on: varying the kind of surface—flat or positively or negatively curved—on which our right triangle might lie. Once we complete the proof that zero curvature depends on the PP or the PT (by showing their equivalence), we will be able to look at what can be no more than analogues to the PT on a sphere and on a hyperbolic (negatively curving) surface. The proof has eight pieces (proposition I.16 of Euclid; four lemmas, numbered 1 through 4; three theorems, A—whose premise is the PT—B, and C, whose conclusion is the PP), and the proofs linking them together into the overall deduction: PT PP. The chain looks like this:
We’ll begin with a proof of I.16:
I.16 (EUCLID): In any triangle, the exterior angle is greater than either of the interior and opposite angles.
So here, we want to prove that .
STRATEGY: Show that each of the angles in question is congruent to another angle, which lies wholly within the exterior angle.
PROOF:
LEMMA 1 (LEGENDRE) : In any triangle ABC, the sum of any two angles must be < 180°.
STRATEGY: Apply Euclid I.16.
PROOF:
LEMMA 2 (LEGENDRE): The sum of the angles in any triangle ABC is 180°.
STRATEGY: This pivotal lemma is proved by assuming it false, then chasing around some angles and repeating the process until Lemma 1 is contradicted.
PROOF:
LEMMA 3 (LEGENDRE): If the angle sum of a triangle, ABC, is 180°, and a line from some vertex (say A) is drawn to meet the opposite side at D, then the angle sum of each of the two new, smaller triangles (BAD, DAC) is also 180°.
STRATEGY: Did the assumption and construction not force the conclusion, Lemma 2 would be contradicted.
PROOF:
LEMMA 4 (LEGENDRE): Given a line l and a point P not on it, there is a line m through P, intersecting l, so that the base angle formed is less than any given positive magnitude ε.
STRATEGY: Dropping a vertical from P to l, successive ‘hypotenuses’ from P to l make successively shallower angles with l.
PROOF:
You may feel as if you’ve been holding your breath while building a miniature bridge out of toothpicks. You will soon be rewarded by seeing how sturdy the finished structure is.
THEOREM A (BRODIE): PT The angle sum of every isosceles right triangle is 180°.
STRATEGY: Drop the altitude from the right angle, thus making two right triangles. Repeated use of the PT (making the proof long but not complicated) shows these are isosceles too. Equality of angles, and the initial right angle, give the desired result.
PROOF:
1. In right triangle ABC, right-angled at C, drop the altitude to BA, meeting it at F, and label the lengths and angles as shown.
THEOREM B (LEGENDRE): If the angle sum in any isosceles right triangle is 180°, then so is the angle sum in any right triangle.
STRATEGY: Constructing an isosceles right triangle containing the given right triangle, followed by one application of Theorem A and two of Lemma 3, yields our result.
PROOF:
1. Extend CA to D and CB to E so that CE CD; hence DCE is isosceles and (by Theorem A) its angle sum is 180°.
2. Construct EA.
3. By Lemma 3, the angle sum of EAC is 180°.
4. By Lemma 3 again, the angle sum of ACB is 180°.
THEOREM C (LEGENDRE): The angle sum of any right triangle is 180° PP.
STRATEGY: A proof by contradiction, via Lemma 1 to get one par
allel to l through P, no others via Lemma 4 (relying heavily on betweenness properties).
PROOF:
Oh you enthusiasts of the Flat Earth Society—if any of you are still contriving with us—what a marvelous undertaking was yours! How subtle your reasoning, how intricate your works, how far—even to infinity—your conclusions stretched! The lively culture of Euclidean geometry endorses your dedication, and you have here seen your faith strengthened by adding the authority of Pythagoras to it. Even more glorious, however, is the civilization of which that culture is a part: for as with everything in mathematics, what was taken to be the whole inevitably comes to be seen as involved in a greater, filled with similar, then ever more diverse cultures, evolving together harmoniously toward comprehension: Mind thinks One.
Here then is a universe parallel to Euclid’s: a sphere. What creatures aping the Pythagorean inhabit it? It models the postulates for the ‘elliptic geometry’ we spoke of before, if we interpret lines as great circles (circles on the sphere’s surface, that is, centered at the sphere’s center, O), and agree to think of two antipodal points as the same (pasting them together makes these postulates work, although visualizing this is impossible for our Euclidean eyes).ag
On the sphere there will be no lines parallel to a given line (great circle) through a point not on it—any two great circles intersect (“once”—in those two antipodal points). The angle-sum of a triangle, as you might guess, won’t be less but more than 180°.