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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.]]
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| 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]].
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| ==Overview==
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| Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically.
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| ===Topological approach===
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| Topologically, a map is a [[CW complex |2-cell]] decomposition of a closed compact 2-manifold.
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| 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.
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| ===Group-theoretical approach===
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| 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''<sub>0</sub>, ''r''<sub>1</sub>, ''r''<sub>2</sub> satisfying (r<sub>0</sub>r<sub>2</sub>)<sup>2</sup>= I. In this definition the faces are the orbit of ''F'' = ''<''r<sub>0</sub>, ''r''<sub>1</sub>>, edges are the orbit of ''E'' = <''r''<sub>0</sub>, ''r''<sub>2</sub>>, and vertices are the orbit of ''V'' = <''r''<sub>1</sub>, ''r''<sub>2</sub>>. More abstractly, the automorphism group of any regular map is the non-degenerate, homomorphic image of a <2,m,n>-[[triangle group]].
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| ===Graph-theoretical approach===
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| 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|.
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| 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]].
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| ==Examples==
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| * The [[great dodecahedron]] is a regular map with pentagonal faces in the orientable surface of genus 4.
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| * The [[Hemicube (geometry)|hemicube]] is a regular map of type {4,3} [[File:Hemicube2.PNG|thumb|The hemicube, a regular map.]]
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| * The [[hemi-dodecahedron]] is a regular map produced by pentagonal embedding of the Petersen graph in the projective plane.
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| * 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.
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| * 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.
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| The following is a complete list of regular maps in surfaces of positive [[Euler characteristic]]: the sphere and the projective plane (Coxeter 80).
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| {| class="wikitable"
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| |-
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| |Characteristic|| Genus|| [[Schläfli symbol]] || Group || Graph || Notes
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| |-
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| |2 || 0 || {p,2} || C<sub>2</sub> × Dih<sub>''p''</sub> || [[Cycle graph|C<sub>''p''</sub>]] || Dihedron
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| |-
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| |2 || 0 || {2,p} || C<sub>2</sub> × Dih<sub>''p''</sub> || ''p''-fold [[Complete graph|K<sub>2</sub>]] || Hosohedron
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| |-
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| |2 || 0 || {3,3} || Sym<sub>4</sub> || [[Complete graph|K<sub>4</sub>]] || Tetrahedron
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| |-
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| |2 || 0 || {4,3} || C<sub>2</sub> × Sym<sub>4</sub> || [[Complete graph|K<sub>4</sub>]] [[Tensor product of graphs|×]] [[Complete graph|K<sub>2</sub>]] || Cube
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| |-
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| |2 || 0 || {3,4} || C<sub>2</sub> × Sym<sub>4</sub> || K<sub>2,2,2</sub> || Octahedron
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| |-
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| |2 || 0 || {5,3} || C<sub>2</sub> × Alt<sub>5</sub> || || Dodecahedron
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| |-
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| |2 || 0 || {3,5} || C<sub>2</sub> × Alt<sub>5</sub> || [[Complete graph|K<sub>6</sub>]] [[Tensor product of graphs|×]] [[Complete graph|K<sub>2</sub>]] || Icosahedron
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| |-
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| |1 || - || {2p,2}/2 || Dih<sub>2''p''</sub> || [[Cycle graph|C<sub>''p''</sub>]] || Hemidihedron
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| |-
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| |1 || - || {2,2p}/2 || Dih<sub>2''p''</sub> || ''p''-fold [[Complete graph|K<sub>2</sub>]] || Hemihosohedron
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| |-
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| |1 || - || {4,3} || Sym<sub>4</sub> || [[Complete graph|K<sub>4</sub>]] || Hemicube
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| |-
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| |1 || - || {3,4} || Sym<sub>4</sub> || 2-fold [[Complete graph|K<sub>3</sub>]]|| Hemioctahedron
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| |-
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| |1 || - || {5,3} || Alt<sub>5</sub> || [[Petersen graph]] || Hemidodecahedron
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| |-
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| |1 || - || {3,5} || Alt<sub>5</sub> || [[Complete graph|K<sub>6</sub>]] || Hemi-icosahedron
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| |-
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| |}
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| == See also ==
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| *[[Topological graph theory]]
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| *[[Abstract polytope]]
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| *[[Planar graph]]
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| *[[Toroidal graph]]
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| *[[Graph embedding]]
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| *[[Regular tiling]]
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| *[[Platonic solid]]
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| == References ==
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| * {{citation
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| | last1 = Coxeter | first1 = H. S. M. | author1-link = Harold Scott MacDonald Coxeter
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| | last2 = Moser | first2 = W. O. J.
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| | edition = 4th
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| | isbn = 978-0-387-09212-6
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| | publisher = Springer Verlag
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| | series = Ergebnisse der Mathematik und ihrer Grenzgebiete
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| | title = Generators and Relations for Discrete Groups
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| | volume = 14
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| | year = 1980}}.
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| *{{citation
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| | last = van Wijk | first = Jarke J. | authorlink = Jack van Wijk
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| | doi = 10.1145/1531326.1531355
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| | journal = Proc. SIGGRAPH (ACM Transactions on Graphics)
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| | page = 12
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| | title = Symmetric tiling of closed surfaces: visualization of regular maps
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| | issue = 3
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| | url = http://www.win.tue.nl/~vanwijk/regularmaps_siggraph09.pdf
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| | volume = 28
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| | year = 2009}}.
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| *{{citation
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| | last1 = Conder | first1 = Marston | author1-link = Marston Conder
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| | last2 = Dobcsányi | first2 = Peter
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| | doi = 10.1006/jctb.2000.2008
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| | issue = 2
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| | journal = Journal of Combinatorial Theory, Series B
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| | pages = 224–242
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| | title = Determination of all regular maps of small genus
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| | volume = 81
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| | year = 2001}}.
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| *{{citation
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| | last = Nedela | first = Roman
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| | title = Maps, Hypermaps, and Related Topics
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| | url = http://www.savbb.sk/~nedela/CMbook.pdf
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| | year = 2007}}.
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| *{{citation
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| | last = Vince | first = Andrew
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| | contribution = Maps
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| | title = Handbook of Graph Theory
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| | year = 2004}}.
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| *{{citation
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| | last1 = Brehm | first1 = Ulrich
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| | last2 = Schulte | first2 = Egon
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| | contribution = Polyhedral Maps
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| | title = Handbook of Discrete and Computational Geometry
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| | year = 2004}}.
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| [[Category:Topological graph theory]]
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| [[Category:Discrete geometry]]
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