Regular Map (graph Theory) - Examples

Examples

  • The great dodecahedron is a regular map with pentagonal faces on the orientable surface of genus 4.
  • The hemicube is a regular map of type {4,3}
  • 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 on a torus.

The following is the complete determination of simple reflexible maps of positive Euler characteristic: the sphere and the projective plane (Coxeter 80).

Characteristic Genus Schläfli symbol Group Graph Notes
2 0 {p,2} C2 × Dihp Cp Dihedron
2 0 {2,p} C2 × Dihp p-fold K2 Hosohedron
2 0 {3,3} Sym4 K4 Tetrahedron
2 0 {4,3} C2 × Sym4 K2,2,2 Octahedron
2 0 {3,4} C2 × Sym4 K4 × K2 Cube
2 0 {5,3} C2 × Alt5 Dodecahedron
2 0 {3,5} C2 × Alt5 K6 × K2 Icosahedron
1 - {2p,2}/2 Dih2p Cp Hemidihedron
1 - {2,2p}/2 Dih2p p-fold K2 Hemihosohedron
1 - {4,3} Sym4 K4 Hemicube
1 - {4,3} Sym4 2-fold K3 Hemioctahedron
1 - {5,3} Alt5 Petersen graph Hemidodecahedron
1 - {3,5} Alt5 K6 Hemi-icosahedron

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