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.
Overview
Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.
Topological approach Topologically, a map is a 2-cell decomposition of a compact connected 2-manifold.
The genus g, of a map M is given by Euler's relation which is equal to if the map is orientable, and if the map is non-orientable. It is a crucial fact that there is a finite (non-zero) number of regular maps for every orientable genus except the torus.
Group-theoretical approach Group-theoretically, the permutation representation of a regular map M is a transitive permutation group C, on a set of flags, generated by three fixed-point free involutions r0, r1, r2 satisfying (r0r2)2= I. In this definition the faces are the orbits of F = <r0, r1>, edges are the orbits of E = <r0, r2>, and vertices are the orbits of V = <r1, r2>. More abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2,m,n>-triangle group.
Graph-theoretical approach Graph-theoretically, a map is a cubic graph with edges coloured blue, yellow, red such that: is connected, every vertex is incident to one edge of each colour, and cycles of edges not coloured yellow have length 4. Note that is the flag graph or graph-encoded map (GEM) of the map, defined on the vertex set of flags and is not the skeleton G = (V,E) of the map. In general, || = 4|E|.
A map M is regular if Aut(M) acts regularly on the flags. Aut(M) of a regular map is transitive on the vertices, edges, and faces of M. A map M is said to be reflexible iff Aut(M) is regular and contains an automorphism that fixes both a vertex v and a face f, but reverses the order of the edges. A map which is regular but not reflexible is said to be chiral.
Examples
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.
χ g SchläfliVert.EdgesFaces GroupOrder Graph Notes2 0 {p,2}pp2 C2 × Dihp4p Cp Dihedron2 0 {2,p}2pp C2 × Dihp 4p p-fold K2 Hosohedron2 0 {3,3}464 S4 24 K4 Tetrahedron2 0 {4,3}8126 C2 × S4 48 K4 × K2 Cube2 0 {3,4}6128 C2 × S4 48 K2,2,2 Octahedron2 0 {5,3}203012 C2 × A5120 Dodecahedron2 0 {3,5}123020 C2 × A5 120 K6 × K2 Icosahedron1 n1 {2p,2}/2pp1 Dih2p 4p Cp Hemi-dihedron1 n1 {2,2p}/22pp Dih2p 4p p-fold K2 Hemi-hosohedron1 n1 {4,3}/2463 S4 24 K4 Hemicube1 n1 {3,4}/2364 S4 24 2-fold K3 Hemioctahedron1 n1 {5,3}/210156 A5 60 Petersen graph Hemidodecahedron1 n1 {3,5}/261510 A5 60 K6 Hemi-icosahedron
The images below show three of the 20 regular maps in the triple torus, labelled with their Schläfli types.
Toroidal polyhedra + Example visualized as nets{4,4}1,0(v:1, e:2, f:1){4,4}1,1(v:2, e:4, f:2){4,4}2,0(v:4, e:8, f:4){4,4}2,1(v:5, e:10, f:5){4,4}2,2(v:8, e:16, f:8){3,6}1,0(v:1, e:3, f:2){3,6}1,1(v:3, e:9, f:6){3,6}2,0(v:4, e:12, f:8){3,6}2,1(v:7, e:21, f:14){3,6}2,2(v:12, e:36, f:24){6,3}1,0(v:2, e:3, f:1){6,3}1,1(v:6, e:9, f:3){6,3}2,0(v:8, e:12, f:4){6,3}2,1(v:14, e:21, f:7){6,3}2,2(v:24, e:36, f:12)
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}.Coxeter 1980, 8.3 Maps of type {4,4} on a torus. {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.Coxeter 1980, 8.4 Maps of type {3,6} or {6,3} on a torus. 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.
+ Regular maps with zero Euler characteristicCoxeter and Moser, Generators and Relations for Discrete Groups, 1957, Chapter 8, Regular maps, 8.3 Maps of type {4,4} on a torus, 8.4 Maps of type {3,6} or {6,3} on a torusχ g SchläfliVert.EdgesFaces GroupOrder Notes01{4,4}b,0n=b2n2nn[4,4](b,0)8n Flat toroidal polyhedraSame as {4,4 | b}01{4,4}b,bn=2b2n2nn[4,4](b,b) 8n Flat toroidal polyhedraSame as rectified {4,4 | b}01{4,4}b,cn=b2+c2n2nn[4,4] 4n Flat chiral toroidal polyhedra01{3,6}b,0t=b2t3t2t[3,6](b,0) 12t Flat toroidal polyhedra01{3,6}b,bt=3b2t3t2t[3,6](b,b) 12t Flat toroidal polyhedra01{3,6}b,ct=b2+bc+c2t3t2t[3,6] 6t Flat chiral toroidal polyhedra01{6,3}b,0t=b22t3tt[3,6](b,0) 12t Flat toroidal polyhedra01{6,3}b,bt=3b22t3tt[3,6](b,b) 12t Flat toroidal polyhedra01{6,3}b,ct=b2+bc+c22t3tt[3,6] 6t Flat chiral toroidal polyhedra 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.
Hyperbolic regular maps The map {6,4}3 can be seen as {6,4}4,0. Following opposite edges will traverse all 4 hexagons in sequence. It exists in the petrial octahedron, {3,4} with 6 vertices, 12 edges and 4 skew hexagon faces.
See also Topological graph theory Abstract polytope Planar graph Toroidal graph Graph embedding Regular tiling Platonic solid Platonic graph
References
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