Singular regular maps
We consider a regular map as "singular" if no two vertices have more than one common edge, and no
two faces have more than one common edge. This page lists singular regular maps of genus up to 11,
omitting those of genus 1.
| Genus | Name | Schläfli | V / F / E | mV, mF | notes | C&D no. | images |
|---|
| 0 | the tetrahedron | {3,3}4 | 4 / 4 / 6 | 1,1 |     | R0.1 | 1 |
| 0 | the cube | {4,3}6 | 8 / 6 / 12 | 1,1 | | R0.2′ | 1 |
| 0 | the octahedron | {3,4}6 | 6 / 8 / 12 | 1,1 |  | R0.2 | 1 |
| 0 | the dodecahedron | {5,3}10 | 20 / 12 / 30 | 1,1 | | R0.3′ | 1 |
| 0 | the icosahedron | {3,5}10 | 12 / 20 / 30 | 1,1 | | R0.3 | 1 |
| 5N | C5:{5,5} | {5,5}3 | 6 / 6 / 15 | 1,1 |  | N5.3 | 2 |
| 5N | C5:{5,4} | {5,4}6 | 15 / 12 / 30 | 1,1 | | N5.1′ | 1 |
| 5N | C5:{4,5} | {4,5}6 | 12 / 15 / 30 | 1,1 | | N5.1 | 1 |
| 3 | the Dyck map | {8,3}6 | 32 / 12 / 48 | 1,1 | | R3.2′ | 3 |
| 3 | the dual Dyck map | {3,8}6 | 12 / 32 / 48 | 1,1 | | R3.2 | 2 |
| 3 | the Klein map | {7,3}8 | 56 / 24 / 84 | 1,1 | | R3.1′ | 3 |
| 3 | the dual Klein map | {3,7}8 | 24 / 56 / 84 | 1,1 | | R3.1 | 2 |
| 6N | C6:{5,4} | {5,4}5 | 20 / 16 / 40 | 1,1 | | N6.3′ | 1 |
| 6N | C6:{4,5} | {4,5}5 | 16 / 20 / 40 | 1,1 | | N6.3 | 1 |
| 7N | C7:{6,4} | {6,4}5 | 15 / 10 / 30 | 1,1 | | N7.1′ | 1 |
| 7N | C7:{4,6} | {4,6}5 | 10 / 15 / 30 | 1,1 | | N7.1 | 1 |
| 4 | S4:{5,5} | {5,5}6 | 12 / 12 / 30 | 1,1 | | R4.6 | 2 |
| 4 | S4:{6,4} | {6,4}4 | 18 / 12 / 36 | 1,1 | | R4.3′ | 1 |
| 4 | S4:{4,6} | {4,6}4 | 12 / 18 / 36 | 1,1 | | R4.3 | 1 |
| 4 | S4:{5,4} | {5,4}6 | 30 / 24 / 60 | 1,1 | | R4.2′ | 1 |
| 4 | S4:{4,5} | {4,5}6 | 24 / 30 / 60 | 1,1 | | R4.2 | 1 |
| 8N | N8.1′ | {7,3}9 | 84 / 36 / 126 | 1,1 | | N8.1′ | 0 |
| 8N | N8.1 | {3,7}9 | 36 / 84 / 126 | 1,1 | | N8.1 | 0 |
| 9N | N9.1′ | {8,3}7 | 56 / 21 / 84 | 1,1 | | N9.1′ | 0 |
| 9N | N9.2′ | {8,3}8 | 56 / 21 / 84 | 1,1 | | N9.2′ | 0 |
| 9N | N9.1 | {3,8}7 | 21 / 56 / 84 | 1,1 | | N9.1 | 0 |
| 9N | N9.2 | {3,8}8 | 21 / 56 / 84 | 1,1 | | N9.2 | 0 |
| 5 | S5:{5,5} | {5,5}4 | 16 / 16 / 40 | 1,1 | | R5.9 | 0 |
| 5 | S5:{6,4} | {6,4}12 | 24 / 16 / 48 | 1,1 | | R5.4′ | 0 |
| 5 | S5:{4,6} | {4,6}12 | 16 / 24 / 48 | 1,1 | | R5.4 | 0 |
| 5 | S5:{5,4} | {5,4}10 | 40 / 32 / 80 | 1,1 | | R5.3′ | 0 |
| 5 | S5:{4,5} | {4,5}10 | 32 / 40 / 80 | 1,1 | | R5.3 | 0 |
| 5 | S5:{8,3} | {8,3}12 | 64 / 24 / 96 | 1,1 | | R5.1′ | 0 |
| 5 | The Fricke-Klein map | {3,8}12 | 24 / 64 / 96 | 1,1 | | R5.1 | 0 |
| 10N | N10.6′ | {6,5}4 | 12 / 10 / 30 | 1,1 | | N10.6′ | 0 |
| 10N | N10.6 | {5,6}4 | 10 / 12 / 30 | 1,1 | | N10.6 | 0 |
| 10N | N10.1′ | {6,4}6 | 24 / 16 / 48 | 1,1 | | N10.1′ | 0 |
| 10N | N10.1 | {4,6}6 | 16 / 24 / 48 | 1,1 | | N10.1 | 0 |
| 11N | C11:{6,6} | {6,6}3 | 9 / 9 / 27 | 1,1 | | N11.2 | 1 |
| 11N | C11:{6,4} | {6,4}12 | 27 / 18 / 54 | 1,1 | | N11.1′ | 1 |
| 11N | C11:{4,6} | {4,6}12 | 18 / 27 / 54 | 1,1 | | N11.1 | 1 |
| 6 | S6:{6,4} | {6,4}10 | 30 / 20 / 60 | 1,1 | | R6.2′ | 1 |
| 6 | S6:{4,6} | {4,6}10 | 20 / 30 / 60 | 1,1 | | R6.2 | 1 |
| 6 | S6:{10,3} | {10,3}6 | 50 / 15 / 75 | 1,1 | | R6.1′ | 0 |
| 6 | S6:{3,10} | {3,10}6 | 15 / 50 / 75 | 1,1 | | R6.1 | 0 |
| 12N | C12{6,4}5 | {6,4}5 | 30 / 20 / 60 | 1,1 | | N12.1′ | 1 |
| 12N | N12.2′ | {6,4}10 | 30 / 20 / 60 | 1,1 | | N12.2′ | 0 |
| 12N | C12{4,6}5 | {4,6}5 | 20 / 30 / 60 | 1,1 | | N12.1 | 1 |
| 12N | N12.2 | {4,6}10 | 20 / 30 / 60 | 1,1 | | N12.2 | 0 |
| 7 | C7.2 | {7,7}4 | 8 / 8 / 28 | 1,1 |  | C7.2 | 1 |
| 7 | C7.2′ | {7,7}4 | 8 / 8 / 28 | 1,1 | | C7.2′ | 1 |
| 7 | S7:{7,3} | {7,3}18 | 168 / 72 / 252 | 1,1 | | R7.1′ | 0 |
| 7 | S7:{3,7} | {3,7}18 | 72 / 168 / 252 | 1,1 | | R7.1 | 0 |
| 14N | N14.1′ | {10,3}15 | 60 / 18 / 90 | 1,1 | | N14.1′ | 0 |
| 14N | N14.1 | {3,10}15 | 18 / 60 / 90 | 1,1 | | N14.1 | 0 |
| 15N | N15.1′ | {7,3}13 | 182 / 78 / 273 | 1,1 | | N15.1′ | 0 |
| 15N | N15.1 | {3,7}13 | 78 / 182 / 273 | 1,1 | | N15.1 | 0 |
| 8 | R8.1′ | {8,3}8 | 112 / 42 / 168 | 1,1 | | R8.1′ | 0 |
| 8 | R8.2′ | {8,3}14 | 112 / 42 / 168 | 1,1 | | R8.2′ | 0 |
| 8 | R8.1 | {3,8}8 | 42 / 112 / 168 | 1,1 | | R8.1 | 0 |
| 8 | R8.2 | {3,8}14 | 42 / 112 / 168 | 1,1 | | R8.2 | 0 |
| 16N | N16.2′ | {6,4}8 | 42 / 28 / 84 | 1,1 | | N16.2′ | 0 |
| 16N | N16.2 | {4,6}8 | 28 / 42 / 84 | 1,1 | | N16.2 | 0 |
| 16N | N16.1′ | {9,3}7 | 84 / 28 / 126 | 1,1 | | N16.1′ | 0 |
| 16N | N16.1 | {3,9}7 | 28 / 84 / 126 | 1,1 | | N16.1 | 0 |
| 17N | N17.1′ | {8,3}10 | 120 / 45 / 180 | 1,1 | | N17.1′ | 0 |
| 17N | N17.1 | {3,8}10 | 45 / 120 / 180 | 1,1 | | N17.1 | 0 |
| 9 | R9.17 | {6,6}8 | 16 / 16 / 48 | 1,1 | | R9.17 | 0 |
| 9 | R9.18 | {6,6}4 | 16 / 16 / 48 | 1,1 | | R9.18 | 0 |
| 9 | R9.16′ | {6,5}4 | 24 / 20 / 60 | 1,1 | | R9.16′ | 0 |
| 9 | R9.16 | {5,6}4 | 20 / 24 / 60 | 1,1 | | R9.16 | 0 |
| 9 | R9.5′ | {8,4}8 | 32 / 16 / 64 | 1,1 | | R9.5′ | 0 |
| 9 | R9.6′ | {8,4}4 | 32 / 16 / 64 | 1,1 | | R9.6′ | 0 |
| 9 | R9.5 | {4,8}8 | 16 / 32 / 64 | 1,1 | | R9.5 | 0 |
| 9 | R9.6 | {4,8}4 | 16 / 32 / 64 | 1,1 | | R9.6 | 0 |
| 9 | R9.14 | {5,5}8 | 32 / 32 / 80 | 1,1 | | R9.14 | 0 |
| 9 | R9.3′ | {6,4}24 | 48 / 32 / 96 | 1,1 | | R9.3′ | 0 |
| 9 | R9.4′ | {6,4}6 | 48 / 32 / 96 | 1,1 | | R9.4′ | 0 |
| 9 | R9.3 | {4,6}24 | 32 / 48 / 96 | 1,1 | | R9.3 | 0 |
| 9 | R9.4 | {4,6}6 | 32 / 48 / 96 | 1,1 | | R9.4 | 0 |
| 9 | R9.2′ | {5,4}20 | 80 / 64 / 160 | 1,1 | | R9.2′ | 0 |
| 9 | R9.2 | {4,5}20 | 64 / 80 / 160 | 1,1 | | R9.2 | 0 |
| 10 | C10.3 | {8,8}6 | 9 / 9 / 36 | 1,1 | | C10.3 | 0 |
| 10 | R10.13 | {6,6}6 | 18 / 18 / 54 | 1,1 | | R10.13 | 0 |
| 10 | R10.14 | {6,6}6 | 18 / 18 / 54 | 1,1 | | R10.14 | 0 |
| 10 | C10.2′ | {8,4}8 | 36 / 18 / 72 | 1,1 | | C10.2′ | 0 |
| 10 | C10.2 | {4,8}8 | 18 / 36 / 72 | 1,1 | | C10.2 | 0 |
| 10 | R10.9′ | {7,4}8 | 42 / 24 / 84 | 1,1 | | R10.9′ | 0 |
| 10 | R10.9 | {4,7}8 | 24 / 42 / 84 | 1,1 | | R10.9 | 0 |
| 10 | R10.2′ | {12,3}6 | 72 / 18 / 108 | 1,1 | | R10.2′ | 0 |
| 10 | R10.7′ | {6,4}12 | 54 / 36 / 108 | 1,1 | | R10.7′ | 0 |
| 10 | R10.8′ | {6,4}12 | 54 / 36 / 108 | 1,1 | | R10.8′ | 0 |
| 10 | R10.7 | {4,6}12 | 36 / 54 / 108 | 1,1 | | R10.7 | 0 |
| 10 | R10.8 | {4,6}12 | 36 / 54 / 108 | 1,1 | | R10.8 | 0 |
| 10 | R10.2 | {3,12}6 | 18 / 72 / 108 | 1,1 | | R10.2 | 0 |
| 10 | R10.1′ | {9,3}12 | 108 / 36 / 162 | 1,1 | | R10.1′ | 0 |
| 10 | R10.1 | {3,9}12 | 36 / 108 / 162 | 1,1 | | R10.1 | 0 |
| 10 | R10.6′ | {5,4}8 | 90 / 72 / 180 | 1,1 | | R10.6′ | 0 |
| 10 | R10.6 | {4,5}8 | 72 / 90 / 180 | 1,1 | | R10.6 | 0 |
| 10 | C10.1′ | {8,3}12 | 144 / 54 / 216 | 1,1 | | C10.1′ | 0 |
| 10 | C10.1 | {3,8}12 | 54 / 144 / 216 | 1,1 | | C10.1 | 0 |
| 11 | R11.5 | {6,6}6 | 20 / 20 / 60 | 1,1 | | R11.5 | 0 |
| 11 | R11.1′ | {6,4}10 | 60 / 40 / 120 | 1,1 | | R11.1′ | 0 |
| 11 | R11.1 | {4,6}10 | 40 / 60 / 120 | 1,1 | | R11.1 | 0 |
| 22N | N22.4′ | {8,6}3 | 16 / 12 / 48 | 1,1 | | N22.4′ | 0 |
| 22N | N22.4 | {6,8}3 | 12 / 16 / 48 | 1,1 | | N22.4 | 0 |
Other Regular Maps
General Index