## Non-Euclidean Geometries

The peculiar warp observed in the Bubble World perspective and its inversive counterpart in the conformal sphere is reminiscent of non-Euclidean geometry. Three mathematicians, Karl Friedrich GaussNicolai Lobaschefsky, and Janos Bolyai, each independently wondered while studying Euclid’s “Elements” why Euclid had not bothered to prove his “Fifth Postulate” with the same rigor as he had with the rest of his postulates. Essentially the “Fifth Postulate” states that if two lines that cross a third line form internal angles that sum to less than 180 degrees, then those lines must cross somewhere.

This is equivalent to the “Parallel Postulate” that parallel lines never meet, and it is also equivalent to the rule that the internal angles of a triangle sum to 180 degrees. Each of those three mathematicians set out to prove the Fifth Postulate, and all three of them failed, because although the postulate seems self-evident, it is in fact impossible to prove.

That in turn opened the possibility for non-Euclidean geometries, i.e. that it is possible to define a whole non-linear equivalent to Euclidean geometry that works in a space with positive curvature like the Bubble World perspective.