Change one rule of Euclid and parallel lines multiply without end
Euclid's geometry says that through a point off a line, exactly one parallel can be drawn. Hyperbolic geometry keeps every other rule but allows at least two, and therefore infinitely many. The result is a strange, consistent world where every point is a saddle and circles grow faster than they should.
The only axiom that separates hyperbolic from Euclidean geometry is the parallel postulate. Remove it altogether and you get absolute geometry, whose theorems, including the first 28 propositions of the first book of Euclid's Elements, hold in both worlds. So single lines behave normally here: two points fix a line, segments extend forever, and two distinct lines cross at most once. Differences appear only when a third line enters. Given two crossing lines, for instance, infinitely many other lines can miss both.
Lines that never meet a given line come in two kinds. Two of them are limiting parallels, creeping ever closer to it in each direction without touching. All the rest are ultraparallel: they have one point of closest approach and spread apart on both sides, and any pair of such lines shares exactly one common perpendicular. The angle a limiting parallel makes depends on distance, which gives this geometry an absolute scale linking lengths to angles, something flat geometry lacks.
Curvature explains the oddities. The hyperbolic plane has constant negative curvature, like a surface made entirely of saddle points, and surfaces called pseudospheres share its geometry. A circle's circumference is always larger than in flat space relative to its radius, though small circles come arbitrarily close. And no line stays a fixed distance from another; points equidistant from a line trace a curve called a hypercycle.
Early workers used many names before Felix Klein settled on hyperbolic, pairing it with elliptic geometry for spheres and parabolic for Euclid's flat plane. In the former Soviet Union it is usually called Lobachevskian geometry, after the Russian Nikolai Lobachevsky, one of its discoverers. One of its models even describes the space of velocities in special relativity.
Source: Hyperbolic geometry
