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=Eu	thumbnail (link)	dual Petrie dual	Rotational Symmetry Group	comments	qy
{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=Eu	thumbnail (link)
{3,9}	24+8-36=-4