# 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
C&N no.
V+F-E=Euthumbnail
dual

Petrie dual

Symmetry
Group
{5,4}
N6.3′
20+16-40=-4 {4,5}

?

A group of order 80 ? 8
{4,5}
R6.3
16+20-40=-4 {5,4}

?

{10,3}5
N6.2′
20+6-30=-4 {3,10}

dodecahedron

S5 3
{3,10}5
N6.2
6+20-30-4 {10,3}

S4:{5,10}

{10,3}10
N6.1′
20+6-30=-4 {3,10}

S5 This is the Desargues graph. Its girth is 6.

3
{3,10}10
N6.1
6+20-30-4 {10,3}

The things listed below are not regular maps.

 {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 Irregular ½ {4,20} 1+5-10=-4 {20,4}

Index to other pages on regular maps;
indexes to those on S0 C1 S1 S2 C4 C5 S3 C6 S4.
Some pages on groups