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Lithic polyhedra

Lithic polyhedra

The Lithic polyhedra are abstract regular polyhedra with the requirements that it must be finite, of the form {p,q}r and not part of an infinite family (specifically, the square toroids {4,4}2r and the trihexagonal toroids {6,3}2r and their duals, but including the {4,3}6 cubic family). There are 112 Lithic polyhedra currently known, and if additional Lithic polyhedra are ever discovered they would (probably) have an order of at least several million.

The Lithic polyhedra can be grouped into 22 families. Families can be described with three integers p, q, r, and the polyhedra within a family can be generated from the permutations of that family's integers as the Schläfli symbol {p,q}r. Families can have either one, three or six polyhedra, corresponding to repeat integers: 16 families have six polyhedra, 5 have three and 1 has one (as {5,5}5).

This page's contents were based on Plasmath's Finite abstract regular polyhedra of the form p,q:r, but with the inclusion of the non-base polyhedra (without the r≥p≥q requirement).

Table (by order)

Schläfli symbol Order F E V Euler characteristic Systematic name Polytope Wiki Atlas
{3,3}4 24 4 6 4 2, o-genus 0 spherical tetrahedron tetrahedron {3,3}*24
{3,4}3 24 4 6 3 1, n-genus 1 projective tetrahedron hemioctahedron {3,4}*24
{4,3}3 24 3 6 4 1, n-genus 1 projective trihedron hemicube {4,3}*24
{3,4}6 48 8 12 6 2, o-genus 0 spherical octahedron octahedron {3,4}*48
{3,6}4 48 8 12 4 0, o-genus 1 Heawood octahedron {3,6}*48
{4,3}6 48 6 12 8 2, o-genus 0 spherical hexahedron cube {4,3}*48
{4,6}3 48 6 12 4 -2, n-genus 4 quadriprojective hexahedron {4,6}*48
{6,3}4 48 4 12 8 0, o-genus 1 Heawood tetrahedron Petrial cube {6,3}*48
{6,4}3 48 4 12 6 -2, n-genus 4 quadriprojective tetrahedron Petrial octahedron {6,4}*48
{3,5}5 60 10 15 6 1, n-genus 1 projective decahedron hemiicosahedron {3,5}*60
{5,3}5 60 6 15 10 1, n-genus 1 projective hexahedron hemidodecahedron {5,3}*60
{5,5}3 60 6 15 6 -3, n-genus 5 quinqueprojective hexahedron Petrial hemiicosahedron {5,5}*60
{3,5}10 120 20 30 12 2, o-genus 0 spherical icosahedron icosahedron {3,5}*120
{3,10}5 120 20 30 6 -4, n-genus 6 sexaprojective icosahedron {3,10}*120a
{5,3}10 120 12 30 20 2, o-genus 0 spherical dodecahedron dodecahedron {5,3}*120
{5,10}3 120 12 30 6 -12, n-genus 14 quattuordecaprojective dodecahedron {5,10}*120a
{10,3}5 120 6 30 20 -4, n-genus 6 sexaprojective hexahedron Petrial dodecahedron {10,3}*120b
{10,5}3 120 6 30 12 -12, n-genus 14 quattuordecaprojective hexahedron Petrial icosahedron {10,5}*120
{4,5}5 160 20 40 16 -4, n-genus 6 sexaprojective icosahedron {4,5}*160
{5,4}5 160 16 40 20 -4, n-genus 6 sexaprojective hexadecahedron {5,4}*160
{5,5}4 160 16 40 16 -8, o-genus 5 Sherk hexadecahedron {5,5}*160
{4,5}6 240 30 60 24 -6, o-genus 4 Bring triacontahedron medial rhombic triacontahedron {4,5}*240
{4,6}5 240 30 60 20 -10, n-genus 12 duodecaprojective triacontahedron {4,6}*240c
{5,4}6 240 24 60 30 -6, o-genus 4 Bring icositetrahedron dodecadodecahedron {5,4}*240
{5,6}4 240 24 60 20 -16, o-genus 9 noventoral icositetrahedron ditrigonary dodecadodecahedron {5,6}*240a
{6,4}5 240 20 60 30 -10, n-genus 12 duodecaprojective icosahedron {6,4}*240c
{6,5}4 240 20 60 24 -16, o-genus 9 noventoral icosahedron medial triambic icosahedron {6,5}*240a
{3,7}8 336 56 84 24 -4, o-genus 3 Klein pentecontahexahedron dual Klein map {3,7}*336
{3,8}7 336 56 84 21 -7, n-genus 9 novemprojective pentecontahexahedron {3,8}*336a
{7,3}8 336 24 84 56 -4, o-genus 3 Klein icositetrahedron Klein map {7,3}*336
{7,8}3 336 24 84 21 -39, n-genus 41 quadragintuniprojective icositetrahedron {7,8}*336b
{8,3}7 336 21 84 56 -7, n-genus 9 novemprojective icosihenahedron Petrial Klein map {8,3}*336b
{8,7}3 336 21 84 24 -39, n-genus 41 quadragintuniprojective icosihenahedron {8,7}*336a
{3,7}9 504 84 126 36 -6, n-genus 8 octoprojective ogdoëcontatetrahedron {3,7}*504
{3,9}7 504 84 126 28 -14, n-genus 16 sedecaprojective ogdoëcontatetrahedron {3,9}*504
{7,3}9 504 36 126 84 -6, n-genus 8 octoprojective triacontahexahedron {7,3}*504
{7,9}3 504 36 126 28 -62, n-genus 64 sexagintaquadriprojective triacontahexahedron {7,9}*504a
{9,3}7 504 28 126 84 -14, n-genus 16 sedecaprojective icosoctahedron {9,3}*504
{9,7}3 504 28 126 36 -62, n-genus 64 sexagintaquadriprojective icosoctahedron {9,7}*504c
{5,5}5 660 66 165 66 -33, n-genus 35 trigintaquinqueprojective hexecontahexahedron {5,5}*660
{3,8}8 672 112 168 42 -14, o-genus 8 octotoral hecatondodecahedron {3,8}*672b
{8,3}8 672 42 168 112 -14, o-genus 8 octotoral tessaracontadihedron {8,3}*672a
{8,8}3 672 42 168 42 -84, n-genus 86 octogintasexaprojective tessaracontadihedron {8,8}*672c
{4,5}8 1440 180 360 144 -36, o-genus 19 novendecatoral hecatonogdoëcontahedron {4,5}*1440
{4,8}5 1440 180 360 90 -90, n-genus 92 nonagintaduoprojective hecatonogdoëcontahedron {4,8}*1440f
{5,4}8 1440 144 360 180 -36, o-genus 19 novendecatoral hecatontessaracontatetrahedron {5,4}*1440
{5,8}4 1440 144 360 90 -126, o-genus 64 sexagintaquadritoral hecatontessaracontatetrahedron {5,8}*1440b
{8,4}5 1440 90 360 180 -90, n-genus 92 nonagintaduoprojective enenecontahedron {8,4}*1440f
{8,5}4 1440 90 360 144 -126, o-genus 64 sexagintaquadritoral enenecontahedron {8,5}*1440a
{3,7}13 1092 182 273 78 -13, n-genus 15 quindecaprojective hecatonogdoëcontadihedron {3,7}*1092
{3,13}7 1092 182 273 42 -49, n-genus 51 quinquagintuniprojective hecatonogdoëcontadihedron {3,13}*1092
{7,3}13 1092 78 273 182 -13, n-genus 15 quindecaprojective hebdomecontoctahedron {7,3}*1092
{7,13}3 1092 78 273 42 -153, n-genus 155 centiquinquagintaquinqueprojective hebdomecontoctahedron {7,13}*1092a
{13,3}7 1092 42 273 182 -49, n-genus 51 quinquagintuniprojective tessaracontadihedron {13,3}*1092
{13,7}3 1092 42 273 78 -153, n-genus 155 centiquinquagintaquinqueprojective tessaracontadihedron {13,7}*1092c
{3,7}12 2184 364 546 156 -26, o-genus 14 quattuordecatoral triacosihexecontatetrahedron
{3,12}7 2184 364 546 91 -91, n-genus 93 nonagintatriprojective triacosihexecontatetrahedron
{7,3}12 2184 156 546 364 -26, o-genus 14 quattuordecatoral hecatonpentecontahexahedron
{7,12}3 2184 156 546 91 -299, n-genus 301 trecentuniprojective hecatonpentecontahexahedron
{12,3}7 2184 91 546 364 -91, n-genus 93 nonagintatriprojective enenecontahenahedron
{12,7}3 2184 91 546 156 -299, n-genus 301 trecentuniprojective enenecontahenahedron
{3,7}14 2184 364 546 156 -26, o-genus 14 quattuordecatoral triacosihexecontatetrahedron
{3,14}7 2184 364 546 78 -104, n-genus 106 centisexaprojective triacosihexecontatetrahedron
{7,3}14 2184 156 546 364 -26, o-genus 14 quattuordecatoral hecatonpentecontahexahedron
{7,14}3 2184 156 546 78 -312, n-genus 314 trecentiquattuordecaprojective hecatonpentecontahexahedron
{14,3}7 2184 78 546 364 -104, n-genus 106 centisexaprojective hebdomecontoctahedron
{14,7}3 2184 78 546 156 -312, n-genus 314 trecentiquattuordecaprojective hebdomecontoctahedron
{3,9}9 3420 570 855 190 -95, n-genus 97 nonagintaseptemprojective pentacosihebdomecontahedron
{9,3}9 3420 190 855 570 -95, n-genus 97 nonagintaseptemprojective hecatonenenecontahedron
{9,9}3 3420 190 855 190 -475, n-genus 477 quadringentiseptuagintaseptemprojective hecatonenenecontahedron
{3,8}10 4320 720 1080 270 -90, o-genus 46 quadragintasexatoral heptacosicosahedron
{3,10}8 4320 720 1080 216 -144, o-genus 73 septuagintatritoral heptacosicosahedron
{8,3}10 4320 270 1080 720 -90, o-genus 46 quadragintasexatoral diacosihebdomecontahedron
{8,10}3 4320 270 1080 216 -594, n-genus 596 quingentinonagintasexaprojective diacosihebdomecontahedron
{10,3}8 4320 216 1080 720 -144, o-genus 73 septuagintatritoral diacosihexadecahedron
{10,8}3 4320 216 1080 270 -594, n-genus 596 quingentinonagintasexaprojective diacosihexadecahedron
{4,6}7 4368 546 1092 364 -182, n-genus 184 centoctogintaquadriprojective pentacositessaracontahexahedron
{4,7}6 4368 546 1092 312 -234, o-genus 118 centoctodecatoral pentacositessaracontahexahedron
{6,4}7 4368 364 1092 546 -182, n-genus 184 centoctogintaquadriprojective triacosihexecontatetrahedron
{6,7}4 4368 364 1092 312 -416, o-genus 209 ducentinoventoral triacosihexecontatetrahedron
{7,4}6 4368 312 1092 546 -234, o-genus 118 centoctodecatoral triacosidodecahedron
{7,6}4 4368 312 1092 364 -416, o-genus 209 ducentinoventoral triacosidodecahedron
{4,5}9 6840 855 1710 684 -171, n-genus 173 centiseptuagintatriprojective octacosipentecontapentahedron
{4,9}5 6840 855 1710 380 -475, n-genus 477 quadringentiseptuagintaseptemprojective octacosipentecontapentahedron
{5,4}9 6840 684 1710 855 -171, n-genus 173 centiseptuagintatriprojective hexacosogdoëcontatetrahedron
{5,9}4 6840 684 1710 380 -646, o-genus 324 trecentivigintiquadritoral hexacosogdoëcontatetrahedron
{9,4}5 6840 380 1710 855 -475, n-genus 477 quadringentiseptuagintaseptemprojective triacosogdoëcontahedron
{9,5}4 6840 380 1710 684 -646, o-genus 324 trecentivigintiquadritoral triacosogdoëcontahedron
{3,8}11 12144 2024 3036 759 -253, n-genus 255 ducentiquinquagintaquinqueprojective dischiliicositetrahedron
{3,11}8 12144 2024 3036 552 -460, o-genus 231 ducentitrigintunitoral dischiliicositetrahedron
{8,3}11 12144 759 3036 2024 -253, n-genus 255 ducentiquinquagintaquinqueprojective heptacosipentecontenneahedron
{8,11}3 12144 759 3036 552 -1725, n-genus 1727 milliseptingentivigintiseptemprojective heptacosipentecontenneahedron
{11,3}8 12144 552 3036 2024 -460, o-genus 231 ducentitrigintunitoral pentacosipentecontadihedron
{11,8}3 12144 552 3036 759 -1725, n-genus 1727 milliseptingentivigintiseptemprojective pentacosipentecontadihedron
{3,7}15 12180 2030 3045 870 -145, n-genus 147 centiquadragintaseptemprojective dischiliatriacontahedron
{3,15}7 12180 2030 3045 406 -609, n-genus 611 sescentundecaprojective dischiliatriacontahedron
{7,3}15 12180 870 3045 2030 -145, n-genus 147 centiquadragintaseptemprojective octacosihebdomecontahedron
{7,15}3 12180 870 3045 406 -1769, n-genus 1771 milliseptingentiseptuagintuniprojective octacosihebdomecontahedron
{15,3}7 12180 406 3045 2030 -609, n-genus 611 sescentundecaprojective tetracosihexahedron
{15,7}3 12180 406 3045 870 -1769, n-genus 1771 milliseptingentiseptuagintuniprojective tetracosihexahedron
{3,9}10 20520 3420 5130 1140 -570, o-genus 286 ducentoctogintasexatoral trischiliatetracosicosahedron
{3,10}9 20520 3420 5130 1026 -684, n-genus 686 sescentoctogintasexaprojective trischiliatetracosicosahedron
{9,3}10 20520 1140 5130 3420 -570, o-genus 286 ducentoctogintasexatoral chiliahecatontessaracontahedron
{9,10}3 20520 1140 5130 1026 -2964, n-genus 2966 dumillinongentisexagintasexaprojective chiliahecatontessaracontahedron
{10,3}9 20520 1026 5130 3420 -684, n-genus 686 sescentoctogintasexaprojective chiliicosihexahedron
{10,9}3 20520 1026 5130 1140 -2964, n-genus 2966 dumillinongentisexagintasexaprojective chiliicosihexahedron
{3,7}16 21504 3584 5376 1536 -256, o-genus 129 centivigintinoventoral trischiliapentacosogdoëcontatetrahedron
{3,16}7 21504 3584 5376 672 -1120, n-genus 1122 millicentivigintiduoprojective trischiliapentacosogdoëcontatetrahedron
{7,3}16 21504 1536 5376 3584 -256, o-genus 129 centivigintinoventoral chiliapentacositriacontahexahedron
{7,16}3 21504 1536 5376 672 -3168, n-genus 3170 tremillicentiseptuagintaprojective chiliapentacositriacontahexahedron
{16,3}7 21504 672 5376 3584 -1120, n-genus 1122 millicentivigintiduoprojective hexacosihebdomecontadihedron
{16,7}3 21504 672 5376 1536 -3168, n-genus 3170 tremillicentiseptuagintaprojective hexacosihebdomecontadihedron

Generalization

Lithic polyhedra can be generalized to rank 4 as polychora of the form {{p,q}a,{q,r}b}c. They have not been studied as extensively as the Lithic polyhedra have.