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.
Topologically, a map is a 2-cell decomposition of a compact connected 2-manifold. [1]
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-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-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.
The following is a complete list of regular maps in surfaces of positive Euler characteristic, χ: the sphere and the projective plane. [2]
χ | g | Schläfli | Vert. | Edges | Faces | Group | Order | Graph | Notes | |
---|---|---|---|---|---|---|---|---|---|---|
2 | 0 | {p,2} | p | p | 2 | C2 × Dih p | 4p | Cp | Dihedron | |
2 | 0 | {2,p} | 2 | p | p | C2 × Dihp | 4p | p-fold K2 | Hosohedron | |
2 | 0 | {3,3} | 4 | 6 | 4 | S 4 | 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 × A 5 | 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 |
The images below show three of the 20 regular maps in the triple torus, labelled with their Schläfli types.
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.
χ | g | Schläfli | Vert. | Edges | Faces | Group | Order | Notes |
---|---|---|---|---|---|---|---|---|
0 | 1 | {4,4}b,0 n=b2 | n | 2n | n | [4,4](b,0) | 8n | Flat toroidal polyhedra Same as {4,4 |b} |
0 | 1 | {4,4}b,b n=2b2 | n | 2n | n | [4,4](b,b) | 8n | Flat toroidal polyhedra Same as rectified {4,4 |b} |
0 | 1 | {4,4}b,c n=b2+c2 | n | 2n | n | [4,4]+ (b,c) | 4n | Flat chiral toroidal polyhedra |
0 | 1 | {3,6}b,0 t=b2 | t | 3t | 2t | [3,6](b,0) | 12t | Flat toroidal polyhedra |
0 | 1 | {3,6}b,b t=3b2 | t | 3t | 2t | [3,6](b,b) | 12t | Flat toroidal polyhedra |
0 | 1 | {3,6}b,c t=b2+bc+c2 | t | 3t | 2t | [3,6]+ (b,c) | 6t | Flat chiral toroidal polyhedra |
0 | 1 | {6,3}b,0 t=b2 | 2t | 3t | t | [3,6](b,0) | 12t | Flat toroidal polyhedra |
0 | 1 | {6,3}b,b t=3b2 | 2t | 3t | t | [3,6](b,b) | 12t | Flat toroidal polyhedra |
0 | 1 | {6,3}b,c t=b2+bc+c2 | 2t | 3t | t | [3,6]+ (b,c) | 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.
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}π with 6 vertices, 12 edges and 4 skew hexagon faces. |
In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.
In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
In geometry and topology, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
In topological graph theory, an embedding of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs are associated with edges in such a way that:
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it.
In the mathematical field of graph theory, the Shrikhande graph is a graph discovered by S. S. Shrikhande in 1959. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having degree 6. Every pair of nodes has exactly two other neighbors in common, whether or not the pair of nodes is connected.
In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
In the mathematical field of graph theory, the Nauru graph is a symmetric, bipartite, cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru.
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid, having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
In the mathematical field of graph theory, the Klein graphs are two different but related regular graphs, each with 84 edges. Each can be embedded in the orientable surface of genus 3, in which they form dual graphs.
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
In topological graph theory, the Petrie dual of an embedded graph is another embedded graph that has the Petrie polygons of the first embedding as its faces.