Uniform Polyhedron

The uniform polyhedra are polyhedra consisting of regular (possibly polygrammic) faces of equal edge length whose polyhedron vertices are all symmetrically equivalent. The uniform polyhedra include the Platonic solids (consisting of equal convex regular polygon faces), Archimedean soldis (consisting of convex regular faces of more than one type). Unlike these special cases, the uniform polyhedra need not enclose a volume and in general have self-intersections between faces. For example, the Kepler-Poinsot polyhedra (consisting of equal concave regular polygon or polygram faces) are uniform polyhedra whose outer hulls enclose a volume but which contain interior faces corresponding to parts of the faces that are not part of the hull. Badoureau discovered 37 such nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55).

Coxeter et al. (1954) conjectured that there are 75 uniform polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this surmise was subsequently proven. The five pentagonal prisms can also be considered uniform polyhedra, bringing the total to 80. In addition, there are two other polyhedra in which four faces meet at an edge, the great complex icosidodecahedron and small complex icosidodecahedron (both of which are compounds of two uniform polyhedra).

The polyhedron vertices of a uniform polyhedron all lie on a circumsphere whose center is their geometric centroid (Coxeter et al. 1954, Coxeter 1973, p. 44). The polyhedron vertices joined to another polyhedron vertex lie on a circle (Coxeter et al. 1954).

Not-necessarily circumscriptable versions of uniform polyhedra with exactified numeric vertices and polygrammic faces sometimes split into separate polygons are implemented in the Wolfram Language as UniformPolyhedron["name"] or UniformPolyhedron[{"Uniform", n}] (cf. Garcia 2019). The full exact, equilateral, circumscriptable uniform polyhedra will be implemented in a future version of the Wolfram Language as PolyhedronData["name"] or PolyhedronData[{"Uniform", n}].

Except for a single non-Wythoffian case, uniform polyhedra can be generated by Wythoff's kaleidoscopic method of construction. In this construction, an initial vertex inside a special spherical triangle PQR is mapped to all the other vertices by repeated reflections across the three planar sides of this triangle. Similarly, PQR and its kaleidoscopic images must cover the sphere an integral number of times which is referred to as the density d of PQR. The density d>1 is dependent on the choice of angles pi/p, pi/q, pi/r at P, Q, R respectively, where p, q, r are reduced rational numbers greater than one. Such a spherical triangle is called a Schwarz triangle, conveniently denoted (pqr). Except for the infinite dihedral family of (p22) for p=2, 3, 4, ..., there are only 44 kinds of Schwarz triangles (Coxeter et al. 1954, Coxeter 1973). It has been shown that the numerators of p, q, r are limited to 2, 3, 4, 5 (4 and 5 cannot occur together) and so the nine choices for rational numbers are: 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, 5/4 (Messer 2002).

The names of the 75 uniform polyhedra were first formalized in Wenninger (1983, first printed in 1971), based on a list prepared by N. Johnson a few years earlier, as slightly modified by D. Luke. Johnson also suggested a few modifications in the original nomenclature to incorporate some additional thoughts, as well as to undo some of Luke's less felicitous changes. The "List of polyhedra and dual models" in Wenninger (1983) gives revised names for several of the uniform polyhedra. The names of the five pentagonal prisms appeared in Har'El (1993).

The following table gives the names of the uniform polyhedra and their duals as given in Wenninger (1983) and Har'El (1993) and with the numberings of Maeder (1997), Wenninger (1971), Coxeter et al. (1954), and Har'El (1993). Coxeter et al. (1954) give many properties of the uniform solids, and Coxeter et al. (1954), Johnson (2000), and Messer (2002) give the quartic equation for determining the central angle subtending half an edge. The single non-Wythoffian case is the great dirhombicosidodecahedron with Maeder index 75 which has pseudo-Wythoff symbol |3/2 5/3 3 5/2.

Maeder indexWenninger indexCoxeter indexHar'El indexWythoff symbolnamedual polyhedron
111563|23regular tetrahedrontetrahedron
2616723|3truncated tetrahedrontriakis tetrahedron
5217104|23regular octahedroncube
71119122|34cuboctahedronrhombic dodecahedron
87201324|3truncated octahedrontetrakis hexahedron
98211423|4truncated cubesmall triakis octahedron
1013221534|2small rhombicuboctahedron (rhombicuboctahedron)deltoidal icositetrahedron
11152316234|great rhombicuboctahedron (truncated cuboctahedron)disdyakis dodecahedron
12172417|234snub cubepentagonal icositetrahedron
136938183/24|4small cubicuboctahedronsmall hexacronic icositetrahedron
1477501934|4/3great cubicuboctahedrongreat hexacronic icositetrahedron
167952214/334|cubitruncated cuboctahedrontetradyakis hexahedron
178559223/24|2quasirhombicuboctahedron (great rhombicuboctahedron)great deltoidal icositetrahedron
188660233/224|small rhombihexahedronsmall rhombihexacron
1992662423|4/3stellated truncated hexahedron (quasitruncated hexahedron)great triakis octahedron
209367254/323|great truncated cuboctahedron (quasitruncated cuboctahedron)great disdyakis dodecahedron
2110382264/33/22|great rhombihexahedrongreat rhombihexacron
22425275|23regular icosahedrondodecahedron
23526283|25regular dodecahedronicosahedron
241228292|35icosidodecahedronrhombic triacontahedron
259273025|3truncated icosahedronpentakis dodecahedron
2610293123|5truncated dodecahedrontriakis icosahedron
2714303235|2small rhombicosidodecahedron (rhombicosidodecahedron)deltoidal hexecontahedron
28163133235|great rhombicosidodecahedron (truncated icosidodechedon)disdyakis triacontahedron
29183234|235snub dodecahedronpentagonal hexecontahedron
307039353|5/23small ditrigonal icosidodecahedronsmall triambic icosahedron
317140365/23|3small icosicosidodecahedronsmall icosacronic hexecontahedron
321104137|5/233small snub icosicosidodecahedronsmall hexagonal hexecontahedron
337242383/25|5small dodecicosidodecahedronsmall dodecacronic hexecontahedron
342043395|25/2small stellated dodecahedrongreat dodecahedron
352144405/2|25great dodecahedronsmall stellated dodecahedron
367345412|5/25dodecadodecahedronmedial rhombic triacontahedron
3775474225/2|5truncated great dodecahedronsmall stellapentakis dodecahedron
387648425/25|2rhombidodecadodecahedronmedial deltoidal hexecontahedron
3974464425/25|small rhombidodecahedronsmall rhombidodecacron
401114945|25/25snub dodecadodecahedronmedial pentagonal hexecontahedron
418053463|5/35ditrigonal dodecadodecahedronmedial triambic icosahedron
4281544735|5/3great ditrigonal dodecicosidodecahedrongreat ditrigonal dodecacronic hexecontahedron
438255485/33|5small ditrigonal dodecicosidodecahedronsmall ditrigonal dodecacronic hexecontahedron
448356495/35|3icosidodecadodecahedronmedial icosacronic hexecontahedron
458457505/335|icositruncated dodecadodecahedrontridyakis icosahedron
461125851|5/335snub icosidodecadodecahedronmedial hexagonal hexecontahedron
478761523/2|35great ditrigonal icosidodecahedrongreat triambic icosahedron
488862533/25|3great icosicosidodecahedrongreat icosacronic hexecontahedron
498963543/23|5small icosihemidodecahedronsmall icosihemidodecacron
509064553/235|small dodecicosahedronsmall dodecicosacron
519165565/45|5small dodecahemidodecahedronsmall dodecahemidodecacron
522268573|25/2great stellated dodecahedrongreat icosahedron
534169585/2|23great icosahedrongreat stellated dodecahedron
549470592|5/23great icosidodecahedrongreat rhombic triacontahedron
5595716025/2|3great truncated icosahedrongreat stellapentakis dodecahedron
571168862|25/23great snub icosidodecahedrongreat pentagonal hexecontahedron
5897746325|5/3small stellated truncated dodecahedrongreat pentakis dodecahedron
599875645/325|truncated dodecadodecahedronmedial disdyakis triacontahedron
601147665|5/325inverted snub dodecadodecahedronmedial inverted pentagonal hexecontahedron
619977665/23|5/3great dodecicosidodecahedrongreat dodecacronic hexecontahedron
6210078675/35/2|3small dodecahemicosahedronsmall dodecahemicosacron
6310179685/35/23|great dodecicosahedrongreat dodecicosacron
641158069|5/35/23great snub dodecicosidodecahedrongreat hexagonal hexecontahedron
6510281705/45|3great dodecahemicosahedrongreat dodecahemicosacron
66104837123|5/3great stellated truncated dodecahedrongreat triakis icosahedron
6710584725/33|2quasirhombicosidodecahedron (great rhombicosidodecahedron)great deltoidal hexecontahedron
6810887735/323|great truncated icosidodecahedrongreat disdyakis triacontahedron
691137374|5/323great inverted snub icosidodecahedrongreat inverted pentagonal hexecontahedron
7010786755/35/2|5/3great dodecahemidodecahedrongreat dodecahemidodecacron
7110685763/23|5/3great icosihemidodecahedrongreat icosihemidodecacron
721189177|3/23/25/2small retrosnub icosicosidodecahedronsmall hexagrammic hexecontahedron
7310989783/25/32|great rhombidodecahedrongreat rhombidodecacron
741179079|3/25/32great retrosnub icosidodecahedrongreat pentagrammic hexecontahedron
751199280|3/25/335/2great dirhombicosidodecahedrongreat dirhombicosidodecacron
76125|2pentagonal prismpentagonal dipyramid
772|225pentagonal antiprismpentagonal trapezohedron
7833325/2|2pentagrammic prismpentagrammic dipyramid
79344|225/2pentagrammic antiprismpentagrammic deltohedron
80355|225/3pentagrammic crossed antiprismpentagrammic concave deltohedron

Johnson (2000) proposed a further revision of the "official" names of the uniform polyhedra and their duals and, at the same time, devised a literal symbol for each uniform polyhedron. For each uniform polyhedron, Johnson (2000) gives its number in Wenninger (1989), a modified Schläfli symbol (following Coxeter), a literal symbol, and its new designated name. Not every uniform polyhedron has a dual that is free from anomalies like coincident vertices or faces extending to infinity. For those that do, Johnson gives the name of the dual polyhedron. In Johnson's new system, the uniform polyhedra are classified as follows:

1. Regular (regular polygonal vertex figures),

2. Quasi-regular (rectangular or ditrigonal vertex figures),

3. Versi-regular (orthodiagonal vertex figures),

4. Truncated regular (isosceles triangular vertex figures),

5. Quasi-quasi-regular (trapezoidal vertex figures),

6. Versi-quasi-regular (dipteroidal vertex figures),

7. Truncated quasi-regular (scalene triangular vertex figures),

8. Snub quasi-regular (pentagonal, hexagonal, or octagonal vertex figures),

9. Prisms (truncated hosohedra),

10. Antiprisms and crossed antiprisms (snub dihedra)

Here is a brief description of Johnson's symbols for the uniform polyhedra (Johnson). The star operator * appended to "D" or "E" replaces pentagons {5} by pentagrams {5/2}. The bar operator | indicates the removal from a related figure of a set (or sets) of faces, leaving "holes" so that a different set of faces takes their place. Thus, C|O is obtained from the cuboctahedron CO by replacing the eight triangles by four hexagons. In like manner, rR'|CO has the twelve squares of the rhombicuboctahedron rCO and the six octagons of the small cubicuboctahedron R'CO but has holes in place of their six squares and eight triangles. The operator "r" stands for "rectified": a polyhedron is truncated to the midpoints of the edges. Operators "a", "b", and "c" in the Schläfli symbols for the ditrigonary (i.e., having ditrigonal vertex figures) polyhedra stand for "altered," "blended," and "converted." The operator "o" stands for "ossified" (after S. L. van Oss). Operators "s" and "t" stand for "simiated" (snub) and "truncated."

Primes and capital letters are used for certain operators analogous to those just mentioned. For instance, rXY is the "rhombi-XY," with the faces of the quasi-regular XY supplemented by a set of square "rhombical" faces. The isomorphic r'XY has a crossed vertex figure. The operators "R" and "R'" denote a supplementary set of faces of a different kind--hexagons, octagons or octagrams, decagons or decagrams. Likewise, the operators "T" and "S" indicate the presence of faces other than, or in addition to, those produced by the simpler operators "t" and "s." The vertex figure of s'XY, the "vertisnub XY," is a crossed polygon, and that of s*XY, the "retrosnub XY," has density 2 relative to its circumcenter.

Regular polyhedra: p^q

20{5/2,5}D*small stellated dodecahedrongreat dodecahedron
21{5,5/2}Egreat dodecahedronsmall stellated dodecahedron
22{5/2,3}E*great stellated dodecahedrongreat icosahedron
41{3,5/2}Jgreat icosahedrongreat stellated dodecahedron

Quasi-regular polyhedra: (p.q)^r

11r{3,4}COcuboctahedronrhombic dodecahedron
12r{3,5}IDicosidodecahedronrhombic triacontahedron
73r{5/2,5}ED*dodecadodecahedronmiddle rhombic triacontahedron
94r{5/2,3}JE*great icosidodecahedrongreat rhombic triacontahedron
70a{5,3}ID*small ditrigonary icosidodecahedronsmall triambic icosahedron
80b{5,5/2}DE*ditrigonary dodecadodecahedronmiddle triambic icosahedron
87c{3,5/2}JEgreat ditrigonary icosidodecahedrongreat triambic icosahedron

Versi-regular polyhedra: q.h.q.h

67o{3,3}T|Ttetrahemihexahedronno dual
78o{3,4}C|Ocubohemioctahedronno dual
68o{4,3}O|Coctahemioctahedronno dual
91o{3,5}D|Ismall dodecahemidodecahedronno dual
89o{5,3}I|Dsmall icosahemidodecahedronno dual
102o{5/2,5}E|D*small dodecahemiicosahedronno dual
100o{5,5/2}D*|Egreat dodecahemiicosahedronno dual
106o{5/2,3}J|E*great icosahemidodecahedronno dual
107o{3,5/2}E*|Jgreat dodecahemidodecahedronno dual

Truncated regular polyhedra: q.2p.2p

6t{3,3}tTtruncated tetrahedrontriakis tetrahedron
7t{3,4}tOtruncated octahedrontetrakis hexahedron
8t{4,3}tCtruncated cubetriakis octahedron
92t'{4,3}t'Cstellatruncated cubegreat triakis octahedron
9t{3,5}tItruncated icosahedronpentakis dodecahedron
10t{5,3}tDtruncated dodecahedrontriakis icosahedron
97t'{5/2,5}t'D*small stellatruncated dodecahedrongreat pentakis dodecahedron
75t{5,5/2}tEgreat truncated dodecahedronsmall stellapentakis dodecahedron
104t'{5/2,3}t'E*great stellatruncated dodecahedrongreat triakis icosahedron
95t{3,5/2}tJgreat truncated icosahedrongreat stellapentakis dodecahedron

Quasi-quasi-regular polyhedra: p.2r.q.2r and p.2s.q.2s

13rr{3,4}rCOrhombicuboctahedronstrombic disdodecahedron
69R'r{3,4}R'COsmall cubicuboctahedronsmall sagittal disdodecahedron
77Rr{3,4}RCOgreat cubicuboctahedrongreat strombic disdodecahedron
85r'r{3,4}r'COgreat rhombicuboctahedrongreat sagittal disdodecahedron
14rr{3,5}rIDrhombicosidodecahedronstrombic hexecontahedron
72R'r{3,5}R'IDsmall dodekicosidodecahedronsmall sagittal hexecontahedron
71ra{5,3}rID*small icosified icosidodecahedronsmall strombic trisicosahedron
82R'a{5,3}R'ID*small dodekified icosidodecahedronsmall sagittal trisicosahedron
76rr{5/2,5}rED*rhombidodecadodecahedronmiddle strombic trisicosahedron
83R'r{5/2,5}R'ED*icosified dodecadodecahedronmiddle sagittal trisicosahedron
81Rc{3,5/2}RJEgreat dodekified icosidodecahedrongreat strombic trisicosahedron
88r'c{3,5/2}r'JEgreat icosified icosidodecahedrongreat sagittal trisicosahedron
99Rr{5/2,3}RJE*great dodekicosidodecahedrongreat strombic hexecontahedron
105r'r{5/2,3}r'JE*great rhombicosidodecahedrongreat sagittal hexecontahedron

Versi-quasi-regular polyhedra: 2r.2s.2r.2s

86or{3,4}rR'|COsmall rhombicubesmall dipteral disdodecahedron
103Or{3,4}Rr'|COgreat rhombicubegreat dipteral disdodecahedron
74or{3,5}rR'|IDsmall rhombidodecahedronsmall dipteral hexecontahedron
90oa{5,3}rR'|ID*small dodekicosahedronsmall dipteral trisicosahedron
96or{5/2,5}rR'|ED*rhombicosahedronmiddle dipteral trisicosahedron
101Oc{3,5/2}Rr'|JEgreat dodekicosahedrongreat dipteral trisicosahedron
109Or{5/2,3}Rr'|JE*great rhombidodecahedrongreat dipteral hexecontahedron

Truncated quasi-regular polyhedra: 2p.2q.2r

15tr{3,4}tCOtruncated cuboctahedrondisdyakis dodecahedron
93t'r{3,4}t'COstellatruncated cuboctahedrongreat disdyakis dodecahedron
79Tr{3,4}TCOcubitruncated cuboctahedrontrisdyakis octahedron
16tr{3,5}tIDtruncated icosidodecahedrondisdyakis triacontahedron
98t'r{5/2,5}t'ED*stellatruncated dodecadodecahedronmiddle disdyakis triacontahedron
84T'r{5/2,5}T'ED*icositruncated dodecadodecahedrontrisdyakis icosahedron
108t'r{5/2,3}t'JE*stellatruncated icosidodecahedrongreat disdyakis triacontahedron

Snub quasi-regular polyhedra: p.3.q.3.3 or p.3.q.3.r.3

17sr{3,4}sCOsnub cuboctahedronpetaloidal disdodecahedron
18sr{3,5}sIDsnub icosidodecahedronpetaloidal hexecontahedron
110sa{5,3}sID*snub disicosidodecahedronno dual
118s*a{5,3}s*ID*retrosnub disicosidodecahedronno dual
111sr{5/2,5}sED*snub dodecadodecahedronpetaloidal trisicosahedron
114s'r{5/2,5}s'ED*vertisnub dodecadodecahedronvertipetaloidal trisicosahedron
112S'r{5/2,5}S'ED*snub icosidodecadodecahedronhexaloidal trisicosahedron
113sr{5/2,3}sJE*great snub icosidodecahedrongreat petaloidal hexecontahedron
116s'r{5/2,3}s'JE*great vertisnub icosidodecahedrongreat vertipetaloidal hexecontahedron
117s*r{5/2,3}s*JE*great retrosnub icosidodecahedrongreat retropetaloidal hexecontahedron

Snub quasi-regular polyhedron: (p.4.q.4)^2

119SSr{5/2,3}SSJE*great disnub disicosidisdodecahedronno dual

Prisms: p.4.4

{p}x{}P(p)p-gonal prism, p=3, 5, 6, ...p-gonal bipyramid
{p/d}x{}P(p/d)d-fold p-gonal prism, p/d>2d-fold p-gonal bipyramid

Antiprisms and crossed antiprisms: 3.3.3.p

s{p}h{}Q(p)p-gonal antiprism, p=4, 5, 6, ...p-gonal antibipyramid
s{p/d}h{}Q(p/d)d-fold p-gonal antiprism, p/d>2d-fold p-gonal antibipyramid
s'{p/d}h{}Q'(p/d)d-fold p-gonal crossed antiprism, 2<p/d<3d-fold p-gonal crossed antibipyramid

See also

Archimedean Solid, Augmented Polyhedron, Dual Polyhedron, Johnson Solid, Kepler-Poinsot Polyhedron, Möbius Triangles, Platonic Solid, Polyhedron, Schwarz Triangle, Uniform Polychoron, Vertex Figure, Wythoff Symbol

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Uniform Polyhedron

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Weisstein, Eric W. "Uniform Polyhedron." From MathWorld--A Wolfram Web Resource.

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