Non-Euclidean Geometries
So my first encounter with the term "non-Euclidean Geometry" was in the works of H.P. Lovecraft where it is repeatedly used as a shorthand for the alien "wrongness" of various spaces or creatures. I was always fascinated by the phrase, but in time, I learned that there really were non-Euclidean geometries invented in the 19th century. I was also disappointed to discover that they weren't particularly alien or horrific, they were just geometries derived from a slightly different set of postulates than Euclid's geometry.
Recently, I've been reading a book (Douglas Hofstadter's Godel, Escher, Bach) that discusses the origin and development of those non-Euclidean geometries. Apparently, several mathematicians had been struggling fruitlessly to prove Euclid's fifth postulate (sometimes called the "parallel postulate"), and were, unknowingly, near the brink of discovering non-Euclidean geometry. One such frustrated mathematician (Farkas Bolyai) sent the following letter in an attempt to dissuade his son (Janos Bolyai) from looking into the matter at all:
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone [. . . .] I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction [. . . .] I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all-mankind [. . . .] I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness.
Godel, Escher, Bach at 92
And we're right back to Lovecraft! Clearly, I failed to appreciate the true horror lurking inside Euclid's fifth postulate.
Humorously, Janos ignored his father's warnings and, undeterred, went on to actually invent a non-Euclidean geometry. Less clear is whether that newly-minted non-Euclidean geometry did, in fact, extinguish "all light and joy" from young Janos's life, or perhaps simply consumed him bodily (cosmic horror style)? I, for one, find it suspicious that no contemporary depictions of Janos Bolyai survive. At minimum, we can't rule out the possibility that he vanished into an "infernal Dead Sea" at some point.