by Ellen Kaplan
So here is a spherical triangle, CBA—think of those vertices as the
arrow-ends of vectors from the sphere’s center, O. We’ll call the triangle’s sides a, b, and c, as shown, with C opposite side c. We’re after the analogue of the Pythagorean Theorem—a relation among the sides a, b, and c, when C is a right angle—and all our previous work with vectors will help us now to find the ‘spherical law of cosines’, which is the tool for this job.
Finding, polishing, and using this tool is in the workshop of the appendix. What it will uncover is this remote likeness of the theorem we have come to know so well: on a sphere of radius 1, for a triangle with sides a, b, and c, right-angled at C,
Should you want the analogue of the PT on a sphere of radius R, then the radian measures of the spherical triangle’s sides are no longer a, b, and c but , and the theorem becomes
You may walk away disappointed after so much work, thinking that this looks very little like the Pythagorean Theorem. Well, we’re no longer on the plane—or are we? Here is the first of two great surprises.
What if we let the sphere’s radius R increase, so that its surface becomes ever less curved? Recall (from Chapter Five) the Taylor series for cosine:
Now subtract 1 from both sides, then multiply by 2R2:
The second great surprise lies around a couple of corners: the first takes us to the hyperbolic plane, the second to complex space.
Planes bending away from the Euclidean—the spherical with positive, the hyperbolic with negative curvature—there must be the sweetest of symmetries here. Since we’ve just found that the spherical analogue of the Pythagorean Theorem is
we might hope that for cosh, the hyperbolic cosine function, we would have
We won’t be disappointed. But while the distance from here to there isn’t a year and a half, there is considerable ground to cover: what in fact does the hyperbolic plane look like? How should distance be measured on it, so that ‘a’, ‘b’, and ‘c’ make sense? And exactly what are these analogous trig functions?
To say for a start what the plane looks like will take some doing. Just as any map of the sphere onto a flat surface entails unavoidable distortion (think of Mercator’s generous polar regions), so will picturing the hyperbolic plane—even in three dimensions. We have several models of it, which don’t look anything like one another.ah A significant distortion all share is this. Like the Euclidean plane and the sphere, the hyperbolic plane is highly homogeneous: which implies, for example, that in a right triangle anywhere on it, two of its sides will determine the third. This is, after all, the core of the Pythagorean Theorem, but you might not have guessed it was true on the hyperbolic plane by looking at any one of its incarnations, like the pseudosphere, where triangles down toward the bell will look so different from those farther up the tapering pipe.
To speak of triangles on the pseudosphere means understanding the lengths of their sides. Since using the Pythagorean ruler works only on a flat plane, we can’t expect to apply it here. Distance will still be measured along the shortest path from point to point—the geodesic—but on the pseudosphere this path is called a tractrix, and is just the pseudosphere’s profile:
Think of it as traced by a weight starting out at a table’s upper left-hand corner and tugged along it by a string hanging over the table’s bottom edge, pulled slowly but steadily to the right. The sides ‘a’, ‘b’, and ‘c’ of the inward-curving triangle on a pseudosphere are segments of such tractrices, and some ingenuity yields a metric on them (a measure satisfying those axioms for distance we talked about in Chapter Eight).ai
What will the trigonometric functions on a pseudosphere be? Since the standard trig functions are defined on a circle,
the analogous functions here will be defined on a hyperbola:
Letting ‘sinh’ abbreviate ‘hyperbolic sine’, just as ‘cosh’ did ‘hyperbolic cosine’, if we relate them to ex(as you saw on Euler’s hybrid orchid did with sine and cosine), by cross-fertilizing their Taylor series with that for the exponential function, we will see blossom
And look what this tells usaj:
Just as our diagram led us to expect—and eerily reminiscent of
It is as if we were looking in some sort of a transfiguring mirror. What we want is to see in it that for a triangle on the pseudosphere with sides a, b, and c, and right-angled at C,
While we could certainly do this by imitating the proof we had on the sphere, it would take longer, only because the geometry on such an unusual surface as this isn’t as clearly fixed in our thought as is the geometry on the Euclidean plane or the sphere.ak But by looking in this mirror, we will see not only the hyperbolic analogue of the Pythagorean Theorem, but the whole of the marvelous duality between sphere and pseudosphere, elliptic and hyperbolic geometry, and the movement here from the first to the second. The whole: we won’t therefore see the engineering—how the parts articulate, or even the shapes of those parts—but will catch the grand gist of this transformation. So a question answered in terms of why rather than how satisfies our need for meaning if not knowledge. Here then we round that second corner, bringing us back, as promised, to the garden of complex space in Chapter Six.
For this mirror is the imaginary numbers, whose reality—as gauged by their ubiquity, profundity, and power—is in fact greater than that of the reals they underlie.
You remember that after finding the spherical Pythagorean relation, we generalized from a unit sphere to one of any radius R—so that eventually became a factor when we multiplied everything out. In fact, is the constant of curvature k of a sphere of radius R. Since the curvature of a pseudosphere is negative, its curvature constant . As a magician would say, reaching into his hat, watch carefully. To go from k to k', multiply R by i:
This transforms the sphere into the pseudosphere, and at the same time metamorphoses the spherical into the hyperbolic trig functions! How?
Take Euler’s exotic orchid once again, but this time inject his formula for cosine with the complex variable z. This is what will grow:
And now, multiplying throughout by i, we have our transformation:
So the spherical Pythagorean relation, cos c = cos a · cos b, morphs into its hyperbolic double: cosh c = cosh a · cosh b.
The sphere, when inflated, with , approached the Euclidean plane and its Pythagorean relation, c2 = a2 + b2. Here, as and the hyperbolic triangles look smaller and smaller, the hyperbolic plane’s ever-lessening negative curvature approaches the zero curvature of the Euclidean, whose Pythagorean relation is likewise approached by its hyperbolic analogue.
Our having stepped back has a well-known advantage for better leaping. While for spherical geometry there is only one sphere (its constant positive curvature is all that matters, not the varying radii of its models), you may rightly be puzzled by what seem hyperbolic surfaces with many different negative curvatures—why might these not affect the geometry on them? And how, after all, do cosh and sinh really relate to the underlying curvature?
Instead of puzzling that out here, we could relish our backward step as out of embodiments and into form, saying: there is only one sphere which shows itself as the hyperbolic plane; its radius is i, and the formulas on it for cosh and sinh build an analogous geometry every bit as intricate, revelatory, and consistent as that on the sphere of radius 1.
With this sign you can go forward again, and conquer.
It may have been true for our grandparents that Gert’s poems were punk, Ep’s statues were junk, and nobody understood Ein. Ninety years on, however, it would be hard to navigate the universe without grasping the theory of relativity.
Einstein suggested—and numerous readings and experiments have since confirmed—that two observers moving uniformly with respect to each other, and momentarily coinciding, won’t agree on the time t or the place (x, y, z) where an event they both see occurs—butal they will agree on the interval, I, that replaces distance:
Were it not for that minus sign, how Pythagorean
their agreements would be—but the hyperbolic curvature of spacetime accounts for this difference. If we simplify for a moment to a single spatial dimension x, there’s that minus sign showing up, as before, with a hyperbola in the x-t plane:
Now turn the hyperbola back into a four-dimensional hyperboloid, spun around the time axis t, and you have our universe.
We can’t help being unsettled by this at first because we’re used to seeing and moving in our small, flat corner of spacetime—but physics, on its armature of mathematics, urges us on to think hyperbolically, act Euclideanly.
THE RETURN OF THE VASISTDAS
What’s that scrabbling behind the panels in the attic? We now know that the Pythagorean Theorem holds only on the flattest of surfaces—yet didn’t we see it at the end of Chapter Five clinging to the doubled bend of a torus? Will this, therefore, be the moment when mathematics self-destructs?
Look back at that torus. It was made by scrolling the square on the hypotenuse into a cylinder, and then pulling that around in the other direction, joining the cylinder’s top and bottom circles. Try it. What went wrong? There wasn’t enough cylindrical length actually to carry out this second manoeuvre. The 1 × 1 proportions of the original square wouldn’t allow it.
Perhaps the carpenter’s insight at the end of Chapter Eight will help us out. There we unrolled a spiral on a cylinder to find it was just a right triangle’s hypotenuse. Here, without even making a cylinder from our square, we can look at the pattern on it: Thabit’s proof of the Theorem, which we saw iterated infinitely left and right, up and down, in that other bizarre proof taking up the whole of the plane. We’ve known since Heraclitus that the way up and the way down are the same, so it does no violence to this plane with neither top nor bottom to equate them, and likewise to equate their missing left-and right-hand margins—and in our after-dinner’s sleep, dreaming on both, we see that this flat torus is the unbounded plane. Does it live in notional space?
A POINTING
Here is a curious point raised by two concerns a few pages back—or more aptly put—a pointing beyond the deep point of this dream.
We said that the hyperbolic plane has been variously modeled, with the same relations holding among parts that look so different, making the models hardly seem to be reflecting a common object.
And we passed from the sine and cosine functions to their hyperbolic equivalents via manoeuvres with complex numbers, which allowed us to write (with the complex variable z)
This was an aside we made in a footnote, even though its implications are momentous: for this equation can’t represent the Pythagorean Theorem, since on the complex plane it has nothing to do with a right triangle and the areas of squares on its sides!
What, then, has it to do with? Can we say: just as hyperbolic geometry—or indeed any geometry, and any province of mathematics—is a system of relationships showing itself in this model or that, one medium (numbers, shapes) or another, so the Pythagorean form is an invisible attunement of parts, made perceptible (like harmony in an instrument) through different embodiments?
A consequence of this view—as perplexing to us as it was to the Pythagoreans of Plato’s day4—is that form doesn’t exist independent of its incarnations, nor is it somehow passed on from one to another, but (as with all things adjectival) belongs with what can be named.am We wrote in Chapter Eight of the form remaining, no matter how various the content. Remaining where? Putting the same thought more vaguely now better reflects the condition of our comprehending: for we are trying to look at what we are looking with.
APPENDIX
(A) Hilbert’s Postulates for Euclidean Geometry (the first four sets constitute the postulates for Neutral Geometry).
INCIDENCE POSTULATES:
I-1. If P and Q are different points, then there is a unique line l through them.
I-2. There are at least two points on every line l.
I-3. There are three distinct points that don’t all lie on the same line.
BETWEENNESS POSTULATES (let A *B *C stand for: “Point B is between points A and C”):
B-1. If A*B*C, then A, B, and C are distinct points all on the same line, and C*B*A.
B-2. If B and D are different points, then there are points A, C, and E lying on line BD such that A*B*D, B*C*D, and B*D*E.
B-3. If A, B, and C are different points on a line, then one and only one of the points is between the other two.
B-4. For every line l and any three points A, B, and C not on l,
(i) if A and B are on the same side of l, and B and C are on the same side of l, then A and C are on the same side of l;
(ii) if A and B are on opposite sides of l, and B and C are on opposite sides of l, then A and C are on the same side of l.
CONGRUENCE POSTULATES:
CONTINUITY POSTULATES:
Archimedes: If AB and CD are any segments, then there is a number n such that if segment CD is laid off n times on the ray AB emanating from A, then a point E is reached where n · CD AE and A*B*E.
Dedekind: Suppose that the set of all points on a line l is the union of two non-empty subsets such that no point of is between two points of , and vice versa. Then there is a unique point O lying on l such that P1*O*P2 if and only if P1 is in and P2 is in (this says that if the points on a line fall into two non-overlapping sets, there will be a unique point between these two sets).
THE PARALLEL POSTULATE (Playfair’s version):
Through a given point P not on a line l, only one line can be drawn parallel to l.
(B) THE “CROSSBAR” THEOREM
The Crossbar Theorem states, reasonably enough, that you can’t imprison a ray from a triangle’s vertex inside the triangle: it has to escape through the opposite side. That is:
If a point D is in the interior of CAB, then ray AD will somewhere intersect the line segment CB.
To prove this, we need a prior theorem (which follows, with a proof by contradiction, from the betweenness postulates): If D is in the interior of CAB, and if C*A*E, then B is in the interior of DAE.
We can now prove the Crossbar Theorem by assuming it false, so that B and C would lie on the same side of AD. Now choose E on CA so
that C*A*E. By the theorem above, B is in the interior of DAE, putting B and C on opposite sides of ray AD. This contradiction proves the Crossbar Theorem.
(C) THE PYTHAGOREAN THEOREM ON A SPHERE
Take O as the center of a sphere of radius 1, with ABC, right-angled at C, on its surface. Its sides are arcs a, b, c as shown, and their lengths are the same as the measures, in radians, of the angles at O that subtend them (so arc a, for example, subtends BOC).
An angle, like C, in a spherical triangle is measured by the angle between the tangents to the sphere along the arcs of the sides meeting at that vertex: so here, tangents to a and b, meeting at C.
We will get the Pythagorean Theorem on a Sphere via the “Spherical Law of Cosines”, which in turn will follow from Euclid’s law of cosines applied to two different planar triangles in our diagram. We therefore need to determine the lengths of all their sides.
Giving a central angle the same name as the arc it subtends, we have in OCP:
Here is our measured setup:
the Spherical Pythagorean Theorem.5
CHAPTER TEN
Magic Casements
Charmed magic casements, opening on the foam
Of perilous seas, in faery lands forlorn.
—KEATS, “ODE TO A NIGHTINGALE”
The begottens and begets of mathematics never end—not because of some dry combinatorial play, but because curiosity always seeks to justify the peculiar, and imagination to shape a deeper unity. Jefferson stood at the windows of his Palladian Monticello and looked out at the untamed frontier. These are some of the windows opening from the House of Pythagoras.
SIERPINSKI’S QUERY
In 1962 Waclaw Sierpinski, the Polish mathematician whose excessively leaky gasket you saw in Chapter Six (a patriot who had refused to pass exa
minations in the Russian language, worked with Luzin, taught in the Underground Warsaw University during World War II, and remained cheerful for all that half his colleagues and students had been murdered by the Nazis; he disliked any corrections to his papers) asked if there were infinitely many Pythagorean triples which were all triangular numbers. Are there any? Yes: we know—such is human ingenuity—that 8778, 10,296, and 13,530 are each triangular, and the sum of the squares of the first two equals the square of the third. And that’s all we know.
THE PERFECT BOX
You’re aware from your circle of acquaintances how different 6 or 60 or 600 people can be—or 60,000, if you look around a football stadium on a fall Saturday. But 6 million, or 6 billion? Well, all the people in our world must be different from one another, and in many ways, though it’s hard to imagine the details. Harder to see how 60 billion beetles could really differ—and how, other than just in their names, the elements of yet greater collections of mere numbers could each have significant traits.