Mathematics and precedence rules

[warty goblin]

Assuming Einstein was right: then the universe has a parabolic geometry.

Euclidean geometry is hyperbolic. (See Euclid:postulate 5)

Rather by construction, hyperbolic geometry is non-Euclidean. Euclidean geometry extends neutral geometry with the parallel postulate: that for every line and every point not on that line, there exists a single unique line through the point parallel to the given line.

Hyperbolic geometry assumes the logical negation of this postulate given the axioms of neutral geometry: that there exists at least one line and at least one point not on that line such that there exists more than one line through the point parallel* to the given line. It's a fairly simple proof to show that in a hyperbolic geometry, for any line and any point not on the line, there exist infinite lines through the point parallel to the original line.

You can of course define pi as the limit of a sequence (and there are a lot of sequences that converge to pi) but I rather prefer the geometric definition. It's rather more aesthetically pleasing.

*In the sense 'does not intersect.' Maintaining the same distance across space, having common perpendiculars, etc turn out to be logically equivalent to the parallel postulate. As does the sum of interior angles of a triangle being 180 degrees, and the existence of rectangles.

(elliptic geometry assumes that parallel lines do not exist. Since the existence of parallel lines can be proven in neutral geometry, this makes it rather a separate beast than Euclidean/hyperbolic geometries, which simply extend the neutral axioms.)