Description

We live in 3 dimensional Euclidean space. Various transformations of the space around us preserve distance between points, for example rotations and translations. By defining a new distance, hyperbolic space is born. Each point above represents a transformation which preserves this new distance.

Each point in the drawing above represents an orientation preserving isometry of the hyperbolic plane. This space is otherwise known as PSL(2,R). Each colored surface represents a conjugacy classes of isometries. Points on the green surfaces are versions of rotation, the purple are translations, and the blue are an entirely new kind of transformation unique to hyperbolic space, called parabolic. For example, fix a single green surface. Each point on this surface represents an isometry that rotates the hyperbolic plane by the same angle, but with varying fixed point. Nearby green surfaces represent rotations by slightly different angles. Only finitely many conjugacy classes are drawn, but there are truly infinitely many, and they fill the interior of a solid torus.