Spherical Geometry (idea) by ariels

The parallel postulate (aka the fifth axiom) doesn't work in spherical geometry, in the most dramatic way: the "lines" (geodesics) are exactly the great circles, or intersections of planes going through the centre of the sphere with the sphere. So any two great circles belong to two planes sharing a point (the centre of the sphere!). But two planes in R³ sharing a point must share a straight line, which intersects the sphere at two antipodal points. Thus every two great circles intersect at two antipodal points; parallels are simply impossible here!

Actually spherical geometry just manages to violate another of Euclid's axioms. Between every two points there's a "line" (a shortest path), but it's not always unique! Consider two antipodal points (again). Then there exist infinitely many geodesics between them, and "both ways" on each geodesic have the same length. There's definitely more than just one shortest path between antipodes!

Euclidean geometry	hyperbolic geometry	curved space	great circle
elliptic geometry	Taylor Series	Triangle sum rule	projective geometry
Selling Real Estate in Hyperbolic Space	Pythagorean Theorem	Information wants to be free	Non-Euclidean geometry
Drawing hands	Lorentz Equation	Eugenio Beltrami	deep venous thrombosis
summation convention	dpph	Euler characteristic	proof that V-E+F=2
Stereographic	Convex	geometry	curvature