Regular tilings

The images below picture regular tilings of the sphere, the plane and hyperbolic space. Regular tilings are tilings using a single type of regular polygon, they can be labeled by the number of polygon around each vertex and by the tile's number of sides: the n-p tiling is a tiling with n p-gons around each vertex.

As the angles formed by the sides of the polygons at any vertex must sum to 2*pi, this strongly constrains the regular tilings in the plane: we can only have 6 equilateral triangles, 4 squares or 3 hexagons around each vertex, yielding the 3 regular tilings of the plane.

On the sphere, the sum of the angles of a regular polygon has an excess, which makes possible to tile the sphere with less polygons at each vertex than on the plane. The regular spherical tilings are in bijection with the Platonic solids: 3, 4 or 5 triangles (tetrahedron, octahedron, icosahedron), 3 squares (cube) or 3 pentagons (dodecahedron).

While the plane has constant zero curvature and the sphere constant positive curvature, hyperbolic space has constant negative curvature. As a result, the angles of triangles don't sum to pi radians like in the plane, but to less than pi radians, with the deficit growing larger as the triangle grows larger. This angle deficit allows for an infinite number of tilings. A few of them pictured below in the Poincaré disk model for hyperbolic space.

The following picture displays the regular tilings arranged by their tile's number of sides (3, 4, 5, 6, 7, 8, 20, vertically) and by the number of tiles around each vertex (3, 4, 5, 6, 7, 8, 20, horizontally). The spherical tilings are displayed on the plane through a stereographic projection, and only parts of the planar tilings are shown.

The regular spherical tilings projected to the plane through a stereographic projection. From left to right: 3-3 (tetrahedron), 3-4 (cube), 4-3 (octahedron), 3-5 (dodecahdron) and 5-3 (icosahedron).