Differences in distances ramify throughout the hyperbolic plane. The Euclidean geometer will observe from well beyond the disk that within the disk, distances contract as his hyperbolic colleague advances on a line toward the boundary. That poor fat fool is shrinking before his eyes. This the hyperbolic geometer does not see at all. He is in his advance toward the boundary precisely as he was reposing at its center. Everything is as it was.
THE DEFINITION OF distance by which the hyperbolic world is ruled brings about the failure of Playfair’s axiom. The curved lines throughout the disk are straight because they are geodesics, and they are geodesics because they represent the shortest distance between two points. The Euclidean geometer can cast a cold eye into the Poincaré disk and see why Euclid’s parallel postulate fails.
THE POINT P lies somewhere on the disk, and two lines m and n pass through P, both of them approaching the circumference at two different points A and B, the place—the very place—at which AB itself passes through the circumference on its way toward strictly Euclidean glory.
The chord PA is parallel to l because PA and AB do not intersect within the Poincaré disk. How could they? In A, they have a common point of intersection at the boundary. Strictly Euclidean, that point of intersection is located beyond the interior of the disk. Considered as hyperbolic lines, both AB and PA are infinitely far from A.
Thus AB has in PA one parallel line meeting the conditions of Playfair’s axiom.
But AB meets the circumference twice, once at A and again at B. Two lines pass through P, and both are parallel to AB.
POINCARÉ WAS A powerful mathematician and a subtle philosopher. He had induced the pendulum to swing wide, but he knew all about pendulums, and knew perfectly well that having swung wide, they have a tendency to swing back.
“Let us consider a certain plane,” Poincaré wrote in a brilliant little book, Science and Hypothesis, “which I shall call the fundamental plane, and let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries the words in two languages which have the same signification correspond to one another.” The words contained in this dictionary are space, plane, line, sphere, circle, angle, distance, and the like:
Let us now take Lobatschewsky’s [sic] theorems and translate them by the aid of this dictionary, as we would translate a German text with the aid of a German-French dictionary. We shall then obtain the theorems of ordinary geometry [italics in original]. For instance, Lobatschewsky’s [sic] theorem: “The sum of the angles of a triangle is less than two right angles,” may be translated thus: “If a curvilinear triangle has for its sides arcs of circles which if produced would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles.”
With these remarks, Poincaré invited an affronted common sense to reacquire a say in the Euclidean Joint Stock Company. The theorems of hyperbolic geometry are theorems of Euclidean geometry; they are theorems of Euclidean geometry disguised by a new, radical definition of distance. On back translation, the disguise drops away. Familiar old faces appear again. Euclidean and hyperbolic geometry are not two entirely different theories. They coincide in their assessment of the truth. There is amity between them.
And Euclid? What might der Alter have said? He might have said that finding an interpretation of distance under which the parallel postulate fails is an interesting exercise in misdirection, but one remote from his own concerns. A philosopher disposed to doubt that snow is white, on the grounds that “snow” might mean bauxite is not commonly understood to have discovered anything about snow.
Euclid might have said this.
IN 1872, THE German mathematician Felix Klein delivered a lecture at the University of Erlangen under the title “Vergleichende Betrachtungen über neuere geometrische Forschungen” (A comparative consideration of recent developments in geometry). His lecture was very much a manifesto. Klein had just joined the faculty at the university, Herr Professor and Herr Klein sharing a podium, an office, and a space on many of the same dotted lines; and their manifesto became known as the Erlangen program.
The Erlangen program was a call to classification and so a call to arms. Strange geometries were proliferating throughout European mathematical circles. Their significance, Klein argued, could never be assessed until their relationships were understood. The beloved Euclid of myth and memory was not so much demoted as absorbed, Euclidean geometry finding a place in Klein’s scheme, but one place among many.
The classification that Klein imposed on the unruly world of nineteenth-century geometry was both geometrical and algebraic. The chief business of the law is, as Charles Dickens observed, to make business for itself. It is a principle not commonly observed to fail within mathematics. If there are two systems at work in the classification of various geometries, their analysis might occupy mathematicians for more than a century. And it did.
Klein’s own analysis featured a reversion to certain familiars from family life in which Euclidean geometry is like a cherished and pampered son surrounded, especially on ceremonial occasions, by a curious constellation of uncles: fond, jovial, exuberant elliptical geometry; dark, scowling, saturnine hyperbolic geometry, his visits the occasion to remind everyone that family is, after all, family; and balanced, lucid, wise projective geometry, der Onkel, in Klein’s eyes, an uncle among uncles but chief among them anyway.
Projective geometry came to its flowering in the high Renaissance as a painterly method, a way of capturing in the two dimensions of stretched canvas a world that insists on conducting its affairs in three. In the real world, railway tracks receding into the distance maintain a fixed distance from one another, but in the painter’s world, they converge toward a distant, soundlessly spinning point. Chinese artists did not bother with perspective, and young children do not notice it.
The projective plane is very much like the Euclidean plane; existing in two dimensions, it stands between the human eye or the artist’s canvas and an object or a landscape in three dimensions. Nineteenth-century art schools often encouraged students to master perspective by drawing directly on a flat plane of glass held directly before a scene.
THE CLASSIFICATION OF uncles is one part of the Erlangen program. The other part is, as other parts so often are, more interesting because more algebraic and so more abstract. Évariste Galois, shortly to die in some dismal duel—rival, romance, revolver—had, one hundred years before Steinitz composed his work on fields, introduced mathematicians to the greatest and most powerful of algebraic abstractions in the idea of a group. Mathematicians thereafter did what Steinitz urged them to do by separating the group from its examples. A group G is a set of objects G = {a, b, c, . . . }. These objects are closed under an associative operation, a o b. An operation, meaning that a is imposing itself on b. Closed, meaning that whatever the operation, the result is still in G. Associative meaning that (a o b) o c = a o (b o c). There is an identity element e in the group such that for every element a, a o e = a. And for every element a, an inverse a−1, such that a o a−1 = e. The positive and negative integers form a group under the familiar operation of addition. Whenever two integers are added together, the result is yet again an integer. Summing integers is an operation indifferent to temporal order, and so associative. It hardly matters whether 3 and 5 are first added together and then added to 12, or whether 5 and 12 go first, with 3 tacked on afterward. Zero is an identity element for this group. When added to any integer, it does absolutely nothing. And every integer has an inverse in its negative mirror: 5 plus −5 returns sullenly to zero.
There is nothing more, although it may, perhaps, be acknowledged that this is quite enough.
What, then, are the groups to which a geometry might be attached? This is Klein’s question. What are the Euclidean groups? Ours.
Having stared so long at the Euclidean blackboard, the geometer, let us say, undertakes to pene
trate its surface and move around. Once blackboard bound, he may move by translation, rotation, or reflection. The geometer is himself hardly necessary to the ideas that follow. He may be allowed decently to disappear. The idea of a permissible move remains as his mathematical trace.
The Euclidean plane comprises points and exists in two dimensions. In translation, the geometer goes from one point to another along a straight line. The translation remains as a transformation, or mapping, of the plane back on to itself, a point-to-point mapping such that the starting place is mapped to the stopping place. Everything else remains the same. Rotations and reflections are also mappings of the Euclidean plane onto itself, an abstract way of recording what the geometer has done without ever bringing about his resurrection.
These transformations, or mappings, are the elements of a group. The group operation is the succession of transformations. It is easy enough to contrive identities and inverses among the transformations. Mathematicians have been doing it for more than two hundred years.
If the transformations preserve distances, so that under their action, things that were far apart remain to the same degree far apart, the transformation is called an isometry, and the result is a famous old group in mathematical physics, the Euclidean group E(n). It is a group with as many arms as Vishnu, describing the plane if n = 2, and ordinary space if n = 3. The transformations themselves are called Euclidean moves.
If those isometries happen to preserve orientation as well as distance, clockwise going to clockwise, and the reverse, the Euclidean group becomes the special Euclidean group, SE(n). This is an important group in analytical mechanics; it describes the behavior of rigid objects. For this reason, transformations are called rigid body moves, rather a spastic designation, all things considered, and the rigid body moves are precisely the old familiars of translation, rotation, and reflection.
THE CLASSIFICATION OF various geometries by means of their groups embodied the heavy cutting edge of a large and generous research program. It embodied, as well, the last stage in an ongoing drama within Euclidean geometry. Writing so long ago, Euclid had retained a vital connection between his geometrical structures and some purely human power of getting things to move around in space. In group theory, that power is promoted to an abstract pantheon and then disappears in favor of the group’s transformations, their actions, as mathematicians sometimes say, the link that is severed in theory returning in etymology. The promotion to abstraction is today general in Euclidean geometry: the shapes to Platonic forms, incomprehensible but irreplaceable; the constructions to derivations; the ruler and compass to numbers; Euclidean motion to transformations.
And the Euclidean Joint Stock Company?
Ownership has been diluted. There are additional members on the board. A new feeling prevails. But the old Euclidean Joint Stock Company is still known by its proper name: it is the Euclidean Joint Stock Company.
1. Omar Khayyám, The Rubáiyat of Omar Khayyám, translated by Edward FitzGerald (San Francisco: W. Doxey, 1898).
2. “Die Paralleln auf jenem Wege sollst Du nicht probieren: ich kenne auch jenen Wege bis zu Ende, auch ich habe diese bodenlose Nacht durchmessen: jedes Licht, jede Freude meines Lebens sind in ihr ausgelöscht worden. Ich beschwöre Dich bei Gott! Lass die Paralleln in Frieden.” Pretty strong stuff.
3. T. S. Eliot, “Little Gidding,” Selected Poems of T. S. Eliot (New York: Harcourt Brace Jovanovich, 1991).
Chapter X
EUCLID THE GREAT
Whatever withdraws us from the power of our senses, whatever makes the past, the distant, or the future predominate over the present, advances us in the dignity of thinking beings.
—SAMUEL JOHNSON
CLASSICAL EUCLIDEAN GEOMETRY is, in a narrow sense, an exhausted discipline. No student of mathematics is occupied in adding to the theorems that Euclid demonstrated ones that he might have overlooked.
The sturdy old oak, having weathered so many winter storms, occasionally puts forth a few resplendent new leaves. In 1899, the American mathematician Frank Morley discovered an exquisitely beautiful theorem. At the point of intersection of three angle trisectors, there is always an equilateral triangle.
Beautiful as Morley’s theorem is, much that might have been discovered by means of the Euclidean system has been discovered. With so little left to learn, the study of hard Euclidean problems has become something of a recreational obsession. These are problems that are easy to state but difficult to prove. The Steiner-Lehmus theorem is an example. Is any triangle whose angle bisectors are equal isosceles? Every now and then, an accomplished mathematician, having assured himself that he might knock over this problem quickly, tries to knock it over quickly, emerging days or weeks later, saying, if he is truthful, I did it, but it almost killed me.
TO MATHEMATICIANS, EUCLID offered a method of proof and so a way of life. That this method should have remained as an ideal for more than two thousand years is remarkable. An educated Greek, Euclid’s sensibilities must have been formed by the Homeric epics, as familiar to men of his time or place as Shakespeare is to us. But if the Homeric epics survived in the vault of Greek memory, the Homeric style had declined entirely into desuetude by the time that Euclid scratched his first diagram into the dust. The manner vanished with its maker. This is surely true today. No one but a lunatic would think to compose an epic poem.
The Euclidean style endures. It is vital, an ideal, a moral advantage, a corrective to whatever is spongy, soft, indistinct, slovenly, half-hidden, half-formed, half-baked, or only half-right, the mind in full possession of its powers, straight as an arrow, hard as a stone, uncompromising as a bank. “Pre-Scientific man,” observed the superbly original French mathematician René Thom, “must have had an implicit knowledge of the geometry of space and time.” Prescientific man obviously knew his way around: he would not have otherwise survived. But “only with the advent of Greek geometry,” Thom adds, “was this knowledge to attain an explicit, hence a deductive form” (italics added).1
In modern life, to be explicit is to be frank and thus willing to be tactless in the discussion of odious sexual details. This sense of the word is secondary. Derived from the Latin explicare and the French explicite, the word means to unfold; it carries the connotation of progressive revelation. The slow and painful undertaking by which the theorems of Euclidean geometry are derived from its axioms is an unfolding. The world of the senses recedes. The mind expands. A complex new figure emerges in thought, one expressing the relationship between the axioms of a system, its theorems, and its illustrations. The relationship cannot be seen at once; it must be understood. It is not immediate; it must be acquired. An axiomatic system is like the sonata or the nineteenth-century novel. Where the listener first hears a succession of melodies, the mathematician hears a theme and its development. A sense of coherence must be earned. It cannot be granted.
And it does not come easy.
HOW LONG? HOW long is it destined to last, the severe Euclidean ideal?
It is the purpose of a proof to compel belief. Violence often works to compel assent, and if not violence, then its threat. But a belief that is not freely given cannot easily be extorted. Mathematicians know this. They have reposed their confidence in their proofs.
Writing in the December 2005 issue of the Notices of the American Mathematical Society, the mathematician Brian Davies saw doubt creeping into all the sacred places on its little rat’s feet. In 1931, Kurt Gödel demonstrated that whole-number arithmetic is incomplete. No matter the axiomatic system of arithmetic, it had limits beyond which certain propositions of the system could not be demonstrated in the system. Gödel argued for his results by proving them. This did not by itself undermine the certainty of mathematics. Proof is, after all, proof. It did nothing to enhance the certainty of mathematics, either.
The decision by mathematicians to allow certain proofs to be completed (or verified) by the computer was undertaken in the 1970s. It has provoked skepticism ever since. In 1852, Franc
is Guthrie asked whether any map could be colored using just four colors to mark its distinct regions. No one knew. Simple to state, the problem is difficult to solve. In 1976, Kenneth Appel and Wolfgang Haken offered the mathematical community a proof of the four-color theorem. It demanded that a great many separate cases that could never be verified by hand be verified by the computer.
No one thought that Appel and Haken’s proof was mistaken. No one was completely convinced that it was not. No one knew quite what to think. To this day, no one does.
Davies raised another point, sadder than the others and more poignant. The finite simple groups are scattered throughout mathematics. Classifying them has been a considerable project, one involving many mathematicians. Proofs now run to tens of thousands of pages, but, says a reviewer of “Whither Mathematics?” Davies’s paper on this issue, “no one knows for certain whether this body of work constitutes a complete and correct proof. . . . [S]o much time has now passed that the main players who really understand the structure of the classification are dying or retiring, leaving open the possibility that there will never be a definitive answer to the question of whether the classification is true.”2
The King of Infinite Space Page 10