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Lithic polyhedra

The Lithic polyhedra are the the Lithic polytopes of rank 3. They must be of the form {p,q}_a, or in more mathematical notation, they can be made by taking a base rank 3 Schläfli type $\rho$ and applying a Petrial polygon relation: $\rho =\rho /{⟨{\left({\rho }_1{\rho }_2{\rho }_3\right)}^a⟩}^{\rho }$. In total, there are 112 known Lithic polyhedra with two infinite families. Any additional Lithic polyhedra would likely have a symmetry order of at least several million.

The Lithic polyhedra in their totality are called the Greater Lithic polyhedra, which includes the two infinite families. The subset of these that exclude the two infinite families are the Smaller or exceptional Lithic polyhedra.

The exceptional Lithic polyhedra can be divided into 22 families. Each family has a base polyhedron, {p,q}_a, conventionally a ≥ p ≥ q, from which the whole family can be generated by successive dualing and Petrialing (that is, all permutations of p, q, a). 16 families have six polyhedra, 5 have three and 1 has one (the {5,5}₅ family).

List

The lists below were based on Plasmath's Finite abstract regular polyhedra of the form p,q:r.

Families

Tetrahedral family {3,3}₄:: has symmetry order 24. It is named after the tetrahedron, its main member. Its components are 4 triangles and 3 squares.

Cubic family {4,3}₆:: has order 48, being a double cover of the tetrahedral family. It is named after the cube, its main member. Its components are 8 triangles, 6 squares and 4 hexagons.

Hemidodecahedral family {5,3}₅:: has order 60. It is named after the hemidodecahedron, its main member.. Its components are 10 triangles and 6 pentagons.

Dodecahedral family {5,3}₁₀:: has order 120, being a double cover of the hemidodecahedral family. It is named after the dodecahedron, its main member. Its components are 20 triangles, 12 pentagons and 6 decagons.

{5,4}₅ family: has order 160. Its components are 16 pentagons and 20 squares.

Dodecadodecahedral family {5,4}₆:: has order 240. It is named after the dodecadodecahedron, its main member. Its components are 30 squares, 24 pentagons and 20 hexagons.

Kleinig family {3,7}₈:: has order 336. It is named after the Klein map, its main member. Its components are 56 triangles, 24 heptagons and 21 octagons.

{3,7}₉ family: has order 504. Its components are 84 triangles, 36 heptagons and 28 enneagons.

Triquinary family {5,5}₅:: has order 660. It is named after the fact that it has three fives as its symbol. Its components are 66 pentagons. Only family with only one member.

{8,3}₈ family: has order 672. Its components are 112 triangles and 42 octagons.

{3,7}₁₃ family: has order 1092. Its components are 182 triangles, 78 heptagons and 42 tridecagons.

{4,5}₈ family: has order 1440. Its components are 180 squares, 144 pentagons and 90 octagons.

{3,7}₁₂ family: has order 2184. Its components are 364 triangles, 156 heptagons and 91 dodecagons.

{3,7}₁₄ family: has order 2184. Its components are 364 triangles, 156 heptagons and 78 tetradecagons.

{9,3}₉ family: has order 3420. Its components are 570 triangles and 190 enneagons.

{8,3}₁₀ family: has order 4320. Its components are 720 triangles, 270 octagons and 216 decagons.

{6,4}₇ family: has order 4368. Its components are 546 squares, 364 hexagons and 312 heptagons.

{5,4}₉ family: has order 6840. Its components are 855 squares, 684 pentagons and 380 enneagons.

{8,3}₁₁ family: has order 12144. Its components are 2024 triangles, 759 octagons and 552 hendecagons.

{7,3}₁₅ family: has order 12180. Its components are 2030 triangles, 870 heptagons and 406 pentadecagons.

{9,3}₁₀ family: has order 20520. Its components are 3420 triangles, 1140 enneagons and 1026 decagons.

{7,3}₁₆ family: has order 21504, being a quattuorsexagecuple (64) cover of the Kleinig family. Its components are 3584 triangles, 1536 enneagons and 672 hexadecagons.

Infinite

Square toroidal family {4,4}_2n: has order 16n². Its components are 2n² squares and 4n 2n-gons. All have orientable genus 1 (torus).

Trihexagonal toroidal family {6,3}_2n: has order 12n². Its components are 2n² triangles, n² hexagons and 3n 2n-gons. All have orientable genus 1 (torus).

Exceptional Lithic polyhedra

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