# Regular Maps on the Genus-4 Orientable Manifold

This page shows just one (so far) of the regular maps that can be drawn on the genus-4 orientable manifold. For the purpose of these pages, a "regular map" is defined here.

To draw these maps, we need a way of portraying the oriented genus-4 surface in flat 2-space. We use the diagram shown to the right, which can be regarded as an octahedron with a tunnel connected each pair of opposite faces.

An image in the "thumbnail" column of the main table is a link to a page with more information about the regular map. Where the thumbnail image is shown on a grey background, the link is to a map that is not regular.

Schläfli
symbol
C&D no.
V+F-E=Euthumbnail
(link)
dual

Petrie dual

Rotational
Symmetry
Group
commentsqy
{5,4}
R4.2′
30+24-60=-4 {4,5}

?

A group of order 120 ? 12
{4,5}
R4.2
24+30-60=-4 {5,4}

?

{6,4}
R4.3′
18+12-36=-4 {4,6}

?

A group of order 72 ? 6
{4,6}
R4.3
12+18-36=-4 {5,4}

?

{5,5}
R4.6
12+12-30=-4 self-dual

?

A group of order 60 ? 6
{12,3}
R4.1′
24+6-36=-4 {3,12}

?

A group of order 72 3
{3,12}
R4.1
6+24-36=-4 {12,3}

?

{6,6}
R4.8
6+6-18=-4 self-dual

?

A group of order 36 ? 3
{6,6}
R4.7′
6+6-18=-4 {6,6}

?

A group of order 36 ?
{6,6}
R4.7
6+6-18=-4 {6,6}

?

{10,4}
R4.4′
10+4-20=-4 {4,10}

?

A group of order 40 ? 2
{4,10}
R4.4
4+10-20=-4 {10,4}

?

{16,4}
R4.5′
8+2-16=-4 {4,16}

?

A group of order 32 ? 1
{4,16}
R4.5
2+8-16=-4 {16,4}

?

{12,6}
R4.9′
4+2-12=-4 {6,12}

?

A group of order 24 ? 1
{6,12}
R4.9
2+4-12=-4 {12,6}

?

{10,10}
R4.11
2+2-10=-4 self-dual

?

A group of order 20 ? 1
{18,9}
R4.10′
2+1-9=-4 {9,18}

?

A group of order 18 ? ½
{9,18}
R4.10
1+2-9=-4 {9,18}

?

{16,16}
R4.12
1+1-8=-4 self-dual

?

A group of order 16 ? ½

The things listed below are not regular maps.

Schläfli
symbol
V+F-E=Euthumbnail
(link)
{3,9} 24+8-36=-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

Copyright N.S.Wedd 2009,2010