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

**R**^{3} 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!