237 Tiling of the Hyperbolic Plane. Copyright 1991 by the Geometry Center (www.geom.umn.edu) at the University of Minnesota. Used with permission.

In plane geometry, one of five assumptions about the plane is the “parallel” postulate which is attributed to Euclid. This states that given a line and a point not on the line, there is exactly one line through the point parallel to the given line. Later on, new geometries were discovered when other assumptions were substituted for Euclid’s “parallel” postulate. In hyperbolic geometry there are many parallel lines through the given point.