Differential stress

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[[File:Hexagonal hosohedron.png|thumb|The hexagonal [[hosohedron]], a regular map on the sphere with two vertices, six edges, six faces, and 24 flags.]]
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 (geometry)|flag]] (an incident vertex-edge-face triple) can be transformed into any other flag by a [[automorphism group|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 surface]]s, [[hyperbolic geometry]], and [[Galois theory]]. Regular maps are classified according to either: the [[genus (mathematics)|genus]] and [[orientability]] of the supporting surface, the [[Graph embedding |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 [[CW complex |2-cell]] decomposition of a closed compact 2-manifold.

The genus g, of a map M is given by [[Euler characteristic|Euler's relation ]] <math> \chi (M) = |V| - |E| +|F| </math> which is equal to <math> 2 -2g </math> if the map is orientable, and <math> 2 - g </math> 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 <math>\Omega</math> of [[Flag (geometry)|flags]], generated by a fixed-point free involutions ''r''0, ''r''1, ''r''2 satisfying (r0r2)2= I. In this definition the faces are the orbit of ''F'' = ''<''r0, ''r''1>, edges are the orbit of ''E'' = <''r''0, ''r''2>, and vertices are the orbit of ''V'' = <''r''1, ''r''2>. 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 <math>\Gamma</math> with edges coloured blue, yellow, red such that: <math>\Gamma</math> is connected, every vertex is incident to one edge of each colour, and cycles of edges not coloured blue, have length 4. Note that <math>\Gamma</math> is the ''flag graph'' or ''graph encoded map (GEM)'' of the map, defined on the vertex set of flags <math>\Omega</math> and is not the skeleton G = (V,E) of the map. In general, |<math>\Omega</math>| = 4|E|.

A map M is regular iff Aut(M) [[Group action|acts]] [[Group action#Types_of_actions|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 <math>\phi</math> 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 [[Chirality (mathematics)|chiral]].

==Examples==

* The [[great dodecahedron]] is a regular map with pentagonal faces in the orientable surface of genus 4.
* The [[Hemicube (geometry)|hemicube]] is a regular map of type {4,3} [[File:Hemicube2.PNG|thumb|The hemicube, a regular map.]]
* 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}. Note that the hosohedron is non-polyhedral in the sense that it is not an [[abstract polytope]]. In particular, it doesn't satisfy the diamond property.
* 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]]: the sphere and the projective plane (Coxeter 80).
{| class="wikitable"
|-
|Characteristic|| Genus|| [[Schläfli symbol]] || Group || Graph || Notes
|-
|2 || 0 || {p,2} || C2 × Dih''p'' || [[Cycle graph|C''p'']] || Dihedron
|-
|2 || 0 || {2,p} || C2 × Dih''p'' || ''p''-fold [[Complete graph|K2]] || Hosohedron
|-
|2 || 0 || {3,3} || Sym4 || [[Complete graph|K4]] || Tetrahedron
|-
|2 || 0 || {4,3} || C2 × Sym4 || [[Complete graph|K4]] [[Tensor product of graphs|×]] [[Complete graph|K2]] || Cube
|-
|2 || 0 || {3,4} || C2 × Sym4 || K2,2,2 || Octahedron
|-
|2 || 0 || {5,3} || C2 × Alt5 || || Dodecahedron
|-
|2 || 0 || {3,5} || C2 × Alt5 || [[Complete graph|K6]] [[Tensor product of graphs|×]] [[Complete graph|K2]] || Icosahedron
|-
|1 || - || {2p,2}/2 || Dih2''p'' || [[Cycle graph|C''p'']] || Hemidihedron
|-
|1 || - || {2,2p}/2 || Dih2''p'' || ''p''-fold [[Complete graph|K2]] || Hemihosohedron
|-
|1 || - || {4,3} || Sym4 || [[Complete graph|K4]] || Hemicube
|-
|1 || - || {3,4} || Sym4 || 2-fold [[Complete graph|K3]]|| Hemioctahedron
|-
|1 || - || {5,3} || Alt5 || [[Petersen graph]] || Hemidodecahedron
|-
|1 || - || {3,5} || Alt5 || [[Complete graph|K6]] || Hemi-icosahedron
|-
|}

== See also ==
*[[Topological graph theory]]
*[[Abstract polytope]]
*[[Planar graph]]
*[[Toroidal graph]]
*[[Graph embedding]]
*[[Regular tiling]]
*[[Platonic solid]]

== References ==
* {{citation
| last1 = Coxeter | first1 = H. S. M. | author1-link = Harold Scott MacDonald Coxeter
| last2 = Moser | first2 = W. O. J.
| edition = 4th
| isbn = 978-0-387-09212-6
| publisher = Springer Verlag
| series = Ergebnisse der Mathematik und ihrer Grenzgebiete
| title = Generators and Relations for Discrete Groups
| volume = 14
| year = 1980}}.
*{{citation
| last = van Wijk | first = Jarke J. | authorlink = Jack van Wijk
| doi = 10.1145/1531326.1531355
| journal = Proc. SIGGRAPH (ACM Transactions on Graphics)
| page = 12
| title = Symmetric tiling of closed surfaces: visualization of regular maps
| issue = 3
| url = http://www.win.tue.nl/~vanwijk/regularmaps_siggraph09.pdf
| volume = 28
| year = 2009}}.
*{{citation
| last1 = Conder | first1 = Marston | author1-link = Marston Conder
| last2 = Dobcsányi | first2 = Peter
| doi = 10.1006/jctb.2000.2008
| issue = 2
| journal = Journal of Combinatorial Theory, Series B
| pages = 224–242
| title = Determination of all regular maps of small genus
| volume = 81
| year = 2001}}.
*{{citation
| last = Nedela | first = Roman
| title = Maps, Hypermaps, and Related Topics
| url = http://www.savbb.sk/~nedela/CMbook.pdf
| year = 2007}}.
*{{citation
| last = Vince | first = Andrew
| contribution = Maps
| title = Handbook of Graph Theory
| year = 2004}}.
*{{citation
| last1 = Brehm | first1 = Ulrich
| last2 = Schulte | first2 = Egon
| contribution = Polyhedral Maps
| title = Handbook of Discrete and Computational Geometry
| year = 2004}}.

[[Category:Topological graph theory]]
[[Category:Discrete geometry]]

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Differential stress