This page is obsolete. See the current version of Regular Maps on the C⁶ non-Orientable Manifold

Regular Maps on the C⁶ non-Orientable Manifold

This page shows some of the regular maps that can be drawn on the genus-C⁶ (a sphere plus six crosscaps) non-orientable manifold. For the purpose of these pages, a "regular map" is defined here.

To draw these regular maps, we need a way of portraying this surface in 2-space. We can use the diagram shown to the right: note the regular but non-obvious arrangement of the labels on the edges of this icosagon, which seems well-suited for drawing regular polyhedra in theis manifold. The surface itself is shown in white, the pink letters show how the "cut edges" are to be joined up, and the light pink regions are not part of the surface. This is further explained by the page Representation of 2-manifolds.

Schläfli symbol	V+F-E=Eu	thumbnail (link)	dual Petrie dual	Rotational Symmetry Group	comments	qy
... more polyhedra to be listed here ...
{10,3}5	20+6-30=-4		{3,10} dodecahedron	S5	Exists	3
{3,10}5	6+20-30-4		{10,3} S4:{5,10}
{10,3}10	20+6-30=-4		{3,10}	S5	Exists This is the Desargues graph. Its girth is 6.	3
{3,10}10	6+20-30-4		{10,3}
{6,5}	6+5-15=-4		{6,5}		Irregular	2½
{5,6}	5+6-15-4		{5,6}
{20,4}	5+1-10=-4		{4,20}	D40	Exists	½
{4,20}	1+5-10=-4		{20,4}
... more polyhedra to be listed here ...