Singular regular maps

We consider a regular map as "singular" if no two vertices have more than one common edge, and no two facces have more than one common edge. This page lists singular regular maps of genus up to 11, omitting those of genus 1.

GenusNameSchläfliV / F / EmV, mFnotesC&D no.images
0the tetrahedron{3,3}44 / 4 / 61,1 replete singular is a polyhedral map permutes its vertices evenly R0.11
0the cube{4,3}68 / 6 / 121,1 replete singular is a polyhedral map permutes its vertices evenly R0.2′1
0the octahedron{3,4}66 / 8 / 121,1 replete singular is a polyhedral map permutes its vertices oddly R0.21
0the dodecahedron{5,3}1020 / 12 / 301,1 replete singular is a polyhedral map permutes its vertices evenly R0.3′1
0the icosahedron{3,5}1012 / 20 / 301,1 replete singular is a polyhedral map permutes its vertices evenly R0.31
5NC5:{5,5}{5,5}36 / 6 / 151,1 replete singular is not a polyhedral map permutes its vertices evenly N5.32
5NC5:{5,4}{5,4}615 / 12 / 301,1 replete singular is a polyhedral map permutes its vertices evenly N5.1′1
5NC5:{4,5}{4,5}612 / 15 / 301,1 replete singular is a polyhedral map permutes its vertices evenly N5.11
3the Dyck Map, S3:{8,3}{8,3}632 / 12 / 481,1 replete singular is a polyhedral map permutes its vertices evenly R3.2′3
3S3:{3,8}{3,8}612 / 32 / 481,1 replete singular is a polyhedral map permutes its vertices evenly R3.22
3the Klein map, S3:{7,3}{7,3}856 / 24 / 841,1 replete singular is a polyhedral map permutes its vertices evenly R3.1′2
3S3:{3,7}{3,7}824 / 56 / 841,1 replete singular is a polyhedral map permutes its vertices evenly R3.11
6NC6:{5,4}{5,4}520 / 16 / 401,1 replete singular is a polyhedral map permutes its vertices oddly N6.3′1
6NC6:{4,5}{4,5}516 / 20 / 401,1 replete singular is a polyhedral map permutes its vertices evenly N6.31
7NC7:{6,4}{6,4}515 / 10 / 301,1 replete singular is not a polyhedral map permutes its vertices evenly N7.1′1
7NC7:{4,6}{4,6}510 / 15 / 301,1 replete singular is not a polyhedral map permutes its vertices oddly N7.11
4S4:{5,5}{5,5}612 / 12 / 301,1 replete singular is not a polyhedral map permutes its vertices evenly R4.62
4S4:{6,4}{6,4}418 / 12 / 361,1 replete singular is not a polyhedral map permutes its vertices oddly R4.3′1
4S4:{4,6}{4,6}412 / 18 / 361,1 replete singular is not a polyhedral map permutes its vertices oddly R4.31
4S4:{5,4}{5,4}630 / 24 / 601,1 replete singular is a polyhedral map permutes its vertices oddly R4.2′1
4S4:{4,5}{4,5}624 / 30 / 601,1 replete singular is a polyhedral map permutes its vertices evenly R4.21
8NN8.1′{7,3}984 / 36 / 1261,1 replete singular N8.1′0
8NN8.1{3,7}936 / 84 / 1261,1 replete singular N8.10
9NN9.2′{8,3}856 / 21 / 841,1 replete singular N9.2′0
9NN9.1′{8,3}756 / 21 / 841,1 replete singular N9.1′0
9NN9.2{3,8}821 / 56 / 841,1 replete singular N9.20
9NN9.1{3,8}721 / 56 / 841,1 replete singular N9.10
5S5:{5,5}{5,5}416 / 16 / 401,1 replete singular R5.90
5S5:{6,4}{6,4}1224 / 16 / 481,1 replete singular R5.4′0
5S5:{4,6}{4,6}1216 / 24 / 481,1 replete singular R5.40
5S5:{5,4}{5,4}1040 / 32 / 801,1 replete singular R5.3′0
5S5:{4,5}{4,5}1032 / 40 / 801,1 replete singular R5.30
5S5:{8,3}{8,3}1264 / 24 / 961,1 replete singular R5.1′0
5S5:{3,8}{3,8}1224 / 64 / 961,1 replete singular R5.10
10NN10.6′{6,5}412 / 10 / 301,1 replete singular is not a polyhedral map N10.6′0
10NN10.6{5,6}410 / 12 / 301,1 replete singular is not a polyhedral map N10.60
10NN10.1′{6,4}624 / 16 / 481,1 replete singular N10.1′0
10NN10.1{4,6}616 / 24 / 481,1 replete singular N10.10
11NC11:{6,6}{6,6}39 / 9 / 271,1 replete singular N11.21
11NC11:{6,4}{6,4}1227 / 18 / 541,1 replete singular N11.1′1
11NC11:{4,6}{4,6}1218 / 27 / 541,1 replete singular N11.11
6S6:{6,4}{6,4}1030 / 20 / 601,1 replete singular is a polyhedral map permutes its vertices oddly R6.2′1
6S6:{4,6}{4,6}1020 / 30 / 601,1 replete singular is a polyhedral map permutes its vertices oddly R6.21
6S6:{10,3}{10,3}650 / 15 / 751,1 replete singular R6.1′0
6S6:{3,10}{3,10}615 / 50 / 751,1 replete singular R6.10
12NC12{6,4}5{6,4}530 / 20 / 601,1 singular replete N12.1′1
12NN12.2′{6,4}1030 / 20 / 601,1 replete singular N12.2′0
12NC12{4,6}5{4,6}520 / 30 / 601,1 replete singular N12.11
12NN12.2{4,6}1020 / 30 / 601,1 replete singular N12.20
7C7.2′{7,7}48 / 8 / 281,1 replete Chiral singular is not a polyhedral map C7.2′1
7C7.2{7,7}48 / 8 / 281,1 replete Chiral singular is not a polyhedral map C7.21
7S7:{7,3}{7,3}18168 / 72 / 2521,1 replete singular R7.1′0
7S7:{3,7}{3,7}1872 / 168 / 2521,1 replete singular R7.10
14NN14.1′{10,3}1560 / 18 / 901,1 replete singular N14.1′0
14NN14.1{3,10}1518 / 60 / 901,1 replete singular N14.10
15NN15.1′{7,3}13182 / 78 / 2731,1 replete singular N15.1′0
15NN15.1{3,7}1378 / 182 / 2731,1 replete singular N15.10
8R8.2′{8,3}14112 / 42 / 1681,1 replete singular R8.2′0
8R8.1′{8,3}8112 / 42 / 1681,1 replete singular R8.1′0
8R8.2{3,8}1442 / 112 / 1681,1 replete singular R8.20
8R8.1{3,8}842 / 112 / 1681,1 replete singular R8.10
16NN16.2′{6,4}842 / 28 / 841,1 replete singular N16.2′0
16NN16.2{4,6}828 / 42 / 841,1 replete singular N16.20
16NN16.1′{9,3}784 / 28 / 1261,1 replete singular N16.1′0
16NN16.1{3,9}728 / 84 / 1261,1 replete singular N16.10
17NN17.1′{8,3}10120 / 45 / 1801,1 replete singular N17.1′0
17NN17.1{3,8}1045 / 120 / 1801,1 replete singular N17.10
9R9.18{6,6}416 / 16 / 481,1 replete singular R9.180
9R9.17{6,6}816 / 16 / 481,1 replete singular R9.170
9R9.16′{6,5}424 / 20 / 601,1 replete singular R9.16′0
9R9.16{5,6}420 / 24 / 601,1 replete singular R9.160
9R9.6′{8,4}432 / 16 / 641,1 replete singular R9.6′0
9R9.5′{8,4}832 / 16 / 641,1 replete singular R9.5′0
9R9.5{4,8}816 / 32 / 641,1 replete singular R9.50
9R9.6{4,8}416 / 32 / 641,1 replete singular R9.60
9R9.14{5,5}832 / 32 / 801,1 replete singular R9.140
9R9.4′{6,4}648 / 32 / 961,1 replete singular R9.4′0
9R9.3′{6,4}2448 / 32 / 961,1 replete singular R9.3′0
9R9.4{4,6}632 / 48 / 961,1 replete singular R9.40
9R9.3{4,6}2432 / 48 / 961,1 replete singular R9.30
9R9.2′{5,4}2080 / 64 / 1601,1 replete singular R9.2′0
9R9.2{4,5}2064 / 80 / 1601,1 replete singular R9.20
10C10.3{8,8}69 / 9 / 361,1 singular replete Chiral is not a polyhedral map C10.30
10R10.14{6,6}618 / 18 / 541,1 replete singular R10.140
10R10.13{6,6}618 / 18 / 541,1 replete singular R10.130
10C10.2′{8,4}836 / 18 / 721,1 singular replete Chiral C10.2′0
10C10.2{4,8}818 / 36 / 721,1 singular replete Chiral C10.20
10R10.9′{7,4}842 / 24 / 841,1 replete singular R10.9′0
10R10.9{4,7}824 / 42 / 841,1 replete singular R10.90
10R10.2′{12,3}672 / 18 / 1081,1 replete singular R10.2′0
10R10.8′{6,4}1254 / 36 / 1081,1 replete singular R10.8′0
10R10.7′{6,4}1254 / 36 / 1081,1 replete singular R10.7′0
10R10.8{4,6}1236 / 54 / 1081,1 replete singular R10.80
10R10.7{4,6}1236 / 54 / 1081,1 replete singular R10.70
10R10.2{3,12}618 / 72 / 1081,1 replete singular R10.20
10R10.1′{9,3}12108 / 36 / 1621,1 replete singular R10.1′0
10R10.1{3,9}1236 / 108 / 1621,1 replete singular R10.10
10R10.6′{5,4}890 / 72 / 1801,1 replete singular R10.6′0
10R10.6{4,5}872 / 90 / 1801,1 replete singular R10.60
10C10.1′{8,3}12144 / 54 / 2161,1 singular replete Chiral C10.1′0
10C10.1{3,8}1254 / 144 / 2161,1 singular replete Chiral C10.10
11R11.5{6,6}620 / 20 / 601,1 replete singular R11.50
11R11.1′{6,4}1060 / 40 / 1201,1 replete singular R11.1′0
11R11.1{4,6}1040 / 60 / 1201,1 replete singular R11.10
22NN22.4′{8,6}316 / 12 / 481,1 replete singular N22.4′0
22NN22.4{6,8}312 / 16 / 481,1 replete singular N22.40

Other Regular Maps

General Index