# Regular Maps – Index

This is an index to some pages showing regular maps (non-sphere analogues of regular polyhedra) drawn out on their 2-manifolds, which include the sphere, the torus, the projective plane, and the orientable manifolds of genera 2 and 3.

• The sphere. The five regular polyhedra; and two infinite series of regular things not always considered polyhedra.
• The torus. 68 of infinitely many regular maps, in three infinite series.
• The genus-2 manifold. Ten regular: five dual pairs and two self-duals. Possibly complete.
• The genus-3 manifold. 19 (of at least 20) regular: nine dual pairs and four self-duals. Not complete.
• The genus-4 manifold. Just one shown, of many.
• ( The genus-5 manifold. )
• ( Etc. )
• The projective plane. The four regular hemi-polyhedra, and two infinite series.
• The Klein bottle. No regular maps. Possibly complete.
• ( three crosscaps )
• four crosscaps. Just one dual pair of regular maps so far.
• ( five crosscaps )
• six crosscaps. Three dual pairs of regular maps so far.
• ( seven crosscaps )
• ( Etc. )

In these pages, regular polyhedra are regarded as existing in the 2-sphere, and regular maps as being their analogues in other compact 2-manifolds. The relationships between "graphs", "regular maps", and "regular polyhedra" are discussed further in the philosophy page. A regular map is defined with respect to the 2-manifold in which it exists. The possibility of embedding one in a space of higher dimension is not considered. I prefer to think of regular maps as being regular polyhedra like the cube, dodecahedron, etc., but in manifolds other than the sphere. However it is normal usage to call them "regular maps", and I have tried to do so.

Regular maps embed graphs which are often "named graphs". To search for a named graph on this site, use the page named graphs.

My diagrams for manifolds other than the sphere use pink lines and labels to show how the manifolds are constructed from the diagrams. This is explained in the page representation of 2-manifolds – "sewing instructions".

There is a glossary for some of the terms used here. In particular, the term antipodes is used in a non-standard way, to highlight a property of regular maps of genus greater than 0, and the Sylow subgroups of their rotational symmetry groups.

Some questions I cannot answer about regular maps. The methods I have used for finding these maps.

Here are a few links to other pages and sites about regular maps.

Some Cayley diagrams drawn on orientable 2-manifolds
Some pages on groups