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 maps not regarded as polyhedra.
• The torus. 68 of infinitely many regular maps, in three infinite series.
• The genus-2 manifold. Ten regular maps: four dual pairs and two self-duals. Complete.
• The genus-3 manifold. Twenty regular maps: eight dual pairs and four self-duals. Complete.
• The genus-4 manifold. One of twenty regular maps.
• ( The genus-5 manifold. )
• ( The genus-6 manifold. )
• The genus-7 manifold. Just one shown, of many. The one shown is chiral.
• ( Etc. )
• The projective plane. The four regular hemi-polyhedra, and two infinite series.
• The Klein bottle. No regular map exists.
• The genus-3 non-orientable surface. No regular map exists.
• The genus-4 non-orientable surface. Four regular maps, in two dual pairs. Complete.
• The genus-5 non-orientable surface. Six regular maps: two dual pairs and two self-dual. Complete.
• The genus-6 non-orientable surface. Three of six regular maps.
• The genus-7 non-orientable surface. Two of four regular maps.
• ( The genus-8 non-orientable surface )
• ( 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 (in English) to call them "regular maps".

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.

Links and references about regular maps.