Hyperbolic and Spherical Tiling Gallery

Here are some of the hyperbolic and spherical tilings I created while working on my hyperbolic maze demo. The images are rendered from PostScript files which I created using a program that I wrote. The reason I made these was so that I could get comfortable with hyperbolic space.

Tiling #1

Hyperbolic tiling with 7 equilateral triangles meeting at each vertex, Poincaré projection. Here the tiling is aligned to the center of one triangle.

Tiling #2

Same as tiling #1, but aligned to a vertex.

Tiling #3

Same as tiling #2, but with 8 triangles meeting at a vertex. In hyperbolic space, these are slightly larger triangles.

Tiling #4

Same as tiling #1, but this is a Klein projection. In the Klein projection, straight lines in hyperbolic space are seen as straight lines, but angles are distorted by the projection.

Tiling #5

Same as tiling #4, but aligned to a vertex.

Tiling #6

This is seven equilateral triangles which have been subdivided uniformly in hyperbolic space. You might think of this as portion of a "hyperbolic geodesic dome", as the construction is completely analogous. The projection is Poincaré.

Tiling #7

Same as tiling #5, but I have added more triangles. Looks kind of like a fly's eye.

Tiling #8

Here is the same hyperbolic geodesic dome, but under Klein projection.

Tiling #9

7-triangle Poincaré tiling, but with thickened edges.

Tiling #10

This is the hyperbolic equivalent of an icosidodecahedron. In this tiling, each vertex is surrounded by two 7-gons and two triangles.

Tiling #11

Same as tiling #10, but in wireframe.

Tiling #12

This is supposed to be a hyperbolic tiling where each vertex is surrounded by a 7-gon and two hexagons. The hexagons, unfortunately, aren’t quite regular. (They can be made regular by enlarging the 7-gons slightly.)

Tiling #13

This is a spherical tiling of an icosahedron subdivided into a geodesic dome. The projection is stereographic and clamped to a radius of 5 units.

Tiling #14

Same as tiling #13 but with more subdivisions.

Tiling #15

Same as tiling #14, but under orthogonal projection.

Tiling #16

Same as tiling #15, but with more subdivisions.