Ramble, Part 44: A Little Hyperbolic

[stoutfellow]

Even though Carl Friedrich Gauss appears to have been the first to seriously grapple with the idea of non-Euclidean geometry, credit for the discovery goes to Nikolai Lobachevsky and Janos Bolyai. Under the cut, a couple of quasi-biographical notes and some comments on the distinctive features of the geometry they discovered.

It was Lobachevsky who first published on the subject, in 1829. Unfortunately for his fame, he published in the Kazan Bulletin, which was not a well-known journal, and was in Russian besides. His articles attracted little attention in Russia, and none outside it. He later published rather longer treatments in German (1840) and French (1855), but died before his work achieved wide recognition. (Gauss first noticed it in 1840, and Bolyai not before 1848.) It's a shame; from what I've read, he was a genuinely good man, a first-rate mathematician and a skilled administrator as well. (He was responsible for reorganizing the University of Kazan and making it, if not a powerhouse, at least a respectable school.)

As an aside: Tom Lehrer's choice of Lobachevsky as the mouthpiece for his "plagiarism" song annoys me. I'm inclined to think he chose the name purely for its metrical properties:

Nikolai I-
vanovich Loba-
chevsky is my
name!

Understandable, but not entirely forgiveable.

Janos/Johann Bolyai published his researches in 1832, in an appendix to a semiphilosophical book by his father, Farkas/Wolfgang Bolyai. (Hungary was, at this time, part of the Austrian Empire, and many Hungarians took German use-names; hence the doubling.) Farkas, impressed with the work, sent a copy to Gauss. Gauss responded less than graciously, saying that he could not praise it, for in so doing he would be praising himself: he had discovered most of the results himself, twenty or thirty years earlier.

I admire Gauss as a mathematician, but I find this reply reprehensible. By not publishing, he had abandoned any claim on the ideas, and in any case for the premier mathematician of Europe to score points off a young and extremely promising colleague is... childish. Janos Bolyai's career effectively ended after this, and I blame Gauss for it.

Enough. Let's do some mathematics. Below are some of the interesting ways in which the new ("hyperbolic") geometry differs from the Euclidean kind.

1. The sum of the angles of a triangle is always less than 180 degrees.


2. In both geometries, the size and shape of a triangle are completely determined by the lengths of its sides (the SSS theorem), by the length of one side and the magnitudes of the flanking angles (the ASA theorem), or by the magnitude of one angle and the lengths of the flanking sides (the SAS theorem). In Euclidean geometry, the magnitudes of the three angles determine the shape, but not the size, of a triangle; in hyperbolic geometry, they determine the size as well.


3. Combining and refining the previous two: the area of a hyperbolic triangle is proportional to its "defect", the amount by which its angle-sum falls short of 180 degrees.


4. Let l be a line and P a point not on l. Let Q lie on l so that PQ is perpendicular to l. There is an angle t, always less than 90 degrees, so that, if a line through P makes an angle less than t with PQ, then that line intersects l. If the angle is greater than or equal to t, the line does not intersect l.


5. Let's call a line through P parallel to l if its angle with PQ is exactly t, and ultraparallel to l if the angle is greater than t. There are two lines through P parallel to l, one on either side of PQ, and infinitely many which are ultraparallel.


6. In Euclidean geometry, parallelism is transitive: if l is parallel to m and m parallel to n, then l is parallel to n. This is not true in hyperbolic geometry; l and n may intersect, may be parallel, or may be ultraparallel. (Nor can the situation be improved by replacing "parallel" with "ultraparallel" one or more times in the Euclidean statement.)


7. In Euclidean geometry, if l and m are both perpendicular to n, then l and m are parallel to each other. In hyperbolic geometry, they are not parallel but ultraparallel, and n is the only line perpendicular to both. (Parallel lines, note, do not have any common perpendiculars.)

There was still one problem. Gauss, Bolyai, and Lobachevsky had each concluded that hyperbolic geometry is not self-contradictory, but none of them actually proved this. They had found no contradictions, but that does not guarantee that there were none. The resolution of that problem was left to the next generation, as we shall see.

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Comments

[countrycousin.livejournal.com]

Thanks for rescuing Lobachevsky from Lehrer's tar (for me, at least). Yes, I suspect it was for the meter.

[ndrosen.livejournal.com]

I recall from liner notes that Lobachevsky's name was chosen purely to fit the meter (and match someone's song about Stanislavsky). I had read about Lobachevsky (in E.T. Bell's Men of Mathematics, and possibly a little elsewhere before that) before I heard the Tom Leher song. Some time afterward, I commented to a college acquaintance that it wasn't really fair of Lehrer, since Lobachevsky had been not only a great mathematician, but apparently a very good man. He was surprised to learn that there really had been a Lobachevsky. I was surprised at him for that. We're all parochial in some way . . .