In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group.

Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.

• The great dodecahedron is a regular map with pentagonal faces in the orientable surface of genus 4.
• The hemicube is a regular map of type {4,3} in the projective plane.
• The hemi-dodecahedron is a regular map produced by pentagonal embedding of the Petersen graph in the projective plane.
• The p-hosohedron is a regular map of type {2,p}.
• The Dyck map is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph, can also form a regular map of 16 hexagons in a torus.

The following is a complete list of regular maps in surfaces of positive Euler characteristic, χ: the sphere and the projective plane.^[2]

More information χ, g ...

χ	g	Schläfli	Vert.	Edges	Faces	Group	Order	Graph		Notes
2	0	{p,2}	p	p	2	C2 × Dihp	4p	Cp		Dihedron
2	0	{2,p}	2	p	p	C2 × Dihp	4p	p-fold K2		Hosohedron
2	0	{3,3}	4	6	4	S4	24	K4		Tetrahedron
2	0	{4,3}	8	12	6	C2 × S4	48	K4 × K2		Cube
2	0	{3,4}	6	12	8	C2 × S4	48	K2,2,2		Octahedron
2	0	{5,3}	20	30	12	C2 × A5	120			Dodecahedron
2	0	{3,5}	12	30	20	C2 × A5	120	K6 × K2		Icosahedron
1	n1	{2p,2}/2	p	p	1	Dih2p	4p	Cp		Hemi-dihedron[3]
1	n1	{2,2p}/2	2	p	p	Dih2p	4p	p-fold K2		Hemi-hosohedron[3]
1	n1	{4,3}/2	4	6	3	S4	24	K4		Hemicube
1	n1	{3,4}/2	3	6	4	S4	24	2-fold K3		Hemioctahedron
1	n1	{5,3}/2	10	15	6	A5	60	Petersen graph		Hemidodecahedron
1	n1	{3,5}/2	6	15	10	A5	60	K6		Hemi-icosahedron

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The images below show three of the 20 regular maps in the triple torus, labelled with their Schläfli types.

• {6,4}
• {4,8}
• {8,4}

Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a duocylinder as a flat torus. These are labeled {4,4}_b,c for those related to the square tiling, {4,4}.^[4] {3,6}_b,c are related to the triangular tiling, {3,6}, and {6,3}_b,c related to the hexagonal tiling, {6,3}. b and c are whole numbers.^[5] There are 2 special cases (b,0) and (b,b) with reflective symmetry, while the general cases exist in chiral pairs (b,c) and (c,b).

Regular maps of the form {4,4}_m,0 can be represented as the finite regular skew polyhedron {4,4 | m}, seen as the square faces of a m×m duoprism in 4-dimensions.

Here's an example {4,4}_8,0 mapped from a plane as a chessboard to a cylinder section to a torus. The projection from a cylinder to a torus distorts the geometry in 3 dimensions, but can be done without distortion in 4-dimensions.

In generally regular toroidal polyhedra {p,q}_b,c can be defined if either p or q are even, although only euclidean ones above can exist as toroidal polyhedra in 4-dimensions. In {2p,q}, the paths (b,c) can be defined as stepping face-edge-face in straight lines, while the dual {p,2q} forms will see the paths (b,c) as stepping vertex-edge-vertex in straight lines.

• Topological graph theory
• Abstract polytope
• Planar graph
• Toroidal graph
• Graph embedding
• Regular tiling
• Platonic solid
• Platonic graph