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reed magazine logoMarch 2010

Capturing Infinity continued

The following diagram illustrates the Principle of Parallels in Euclidean geometry. The rectangle E represents the conventional model of the Euclidean plane: a perfectly flat drawing board that extends, in our imagination, indefinitely in all directions. The point P and the straight line L appear in red. The straight line M appears in blue.

Euclidean parallels

Euclidean parallels

In the beginning, all mathematicians regarded the postulates of Euclid as incontrovertibly true. However, they observed that the fifth postulate offered nothing “constructive” and they believed that it was redundant. They sought to prove the fifth postulate from the first four. In effect, they sought to prove that the existence of the parallel M entailed its own uniqueness. To that end, they applied the most flexible of the logician’s methods: reductio ad absurdum. They supposed that the fifth postulate was false and they sought to derive from that supposition (together, of course, with the first four postulates) a contradiction. Succeeding, they would conclude that the fifth postulate followed from the first four. For more than two millennia, many sought and all failed.

At the turn of the 18th century, the grip of belief in the incontro­vertible truth of the fifth postulate began to weaken. Many mathematicians came to believe that the sought contradiction did not exist. They came to regard the propositions that they had proved from the negation of the fifth postulate not as absurdities leading ultimately to a presumed contradiction but as provocative elements of a new geometry.

Swiftly, the new geometry acquired disciples, notably, the young Russian mathematician N. Lobachevsky and the young Hungarian mathematician J. Bolyai. They and many others proved startling propositions at variance with the familiar propositions of Euclidean geometry. The German savant K. Gauss had pondered these matters for 30 years. In 1824, he wrote to his friend F. Taurinus:

The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible. For example, the three angles of a triangle become as small as one wishes, if only the sides are taken large enough; and the area of a triangle can never exceed a definite limit.

However, the specter of contradiction, once sought by all but now by many feared, continued to cast its shadow over the planes. Fifty years would pass before mathematicians found a method by which they could, decisively, banish the specter: the method of models.

Let me explain the method in terms of a case study. At the turn of the 19th century, the French savant H. Poincaré suggested a novel interpretation of the points and the straight lines of the new geometry, using the elements of the Euclidean plane E itself. He declared that the points of the new geometry shall be interpreted as the points of the unit disk H, the same disk that would, in due course, serve Escher in his plans for the Circle Limit Series. He declared that the straight lines of the new geometry shall be interpreted as the arcs of circles that meet the boundary of H at right angles.

“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

These interpretations can be justified, in a sense, by introducing an un­usual method for measuring distance between points in H, with respect to which the shortest paths between points prove to be, in fact, subarcs of arcs of the sort just described. Moreover, the lengths of the various straight lines prove to be infinite. The same is true of the area of H.

Poincaré then proved that H served as a model for the new geometry. That is, he proved that the first four postulates of Euclid are true in H and the fifth postulate is false. He concluded that if, by a certain argument, one should find a contradiction in the new geometry, then, by the same argument, one would find a contradiction in Euclidean geometry as well. In turn, he concluded that if Euclidean geometry is free of contradiction, then the new geometry is also free of contradiction.

By similar (though somewhat more subtle) maneuvers, one can show the converse: if the new geometry is free of contradiction, then Euclidean geometry is also free of contradiction.

Hyperbolic parallels

Hyperbolic parallels

The following diagram illustrates the Principle of Parallels in the new geometry:

For any point P and for any straight line L, if P does not lie on L then there are many straight lines M such that P lies on M and such that L and M are parallel.

The disk H represents the model of the hyperbolic plane designed by Poincaré. The point P and the straight line L appear in red. Various parallels M appear in blue while the two parallels that meet L “at infinity” appear in green.

After more than two millennia of contentions to the contrary, we now know that the Euclidean plane is not the only rationally compelling context for the study of plane geometry. From a logical point of view, the Euclidean geometry and the new geometry, called hyperbolic, are equally tenable.

reed magazine logoMarch 2010