Uniform Polyhedron

The uniform polyhedra are polyhedra with identical polyhedron vertices. Badoureau discovered 37 nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids.

22 of the 75 uniform polyhedra are equilateral.

Coxeter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this 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 sphere 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).

The uniform polyhedra are implemented in the Wolfram Language as UniformPolyhedron["name"] (cf. Garcia 2019).

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).

Source code and binary programs for generating and viewing the uniform polyhedra are also available at The following depictions of the polyhedra were produced by R. Maeder's UniformPolyhedra.m using the Wolfram Language. In this package, uniform polyhedra are computed to the desired numerical precision by numerically solving the defining fundamental equation, and lengths are normalized to give a midradius of rho=1.

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 numbering of Maeder. 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 U_(75) which has pseudo-Wythoff symbol |3/2 5/3 3 5/2.

nWythoff symbolNamedual polyhedron
13|2 3tetrahedrontetrahedron
22 3|3truncated tetrahedrontriakis tetrahedron
33/2 3|3octahemioctahedronoctahemioctacron
43/2 3|2tetrahemihexahedrontetrahemihexacron
54|2 3octahedroncube
63|2 4cubeoctahedron
72|3 4cuboctahedronrhombic dodecahedron
82 4|3truncated octahedrontetrakis hexahedron
92 3|4truncated cubesmall triakis octahedron
103 4|2small rhombicuboctahedrondeltoidal icositetrahedron
112 3 4|great rhombicuboctahedrondisdyakis dodecahedron
12|2 3 4snub cubepentagonal icositetrahedron
133/2 4|4small cubicuboctahedronsmall hexacronic icositetrahedron
143 4|4/3great cubicuboctahedrongreat hexacronic icositetrahedron
154/3 4|3cubohemioctahedronhexahemioctacron
164/3 3 4|cubitruncated cuboctahedrontetradyakis hexahedron
173/2 4|2uniform great rhombicuboctahedrongreat deltoidal icositetrahedron
183/2 2 4|small rhombihexahedronsmall rhombihexacron
192 3|4/3stellated truncated hexahedrongreat triakis octahedron
204/3 2 3|great truncated cuboctahedrongreat disdyakis dodecahedron
214/3 3/2 2|great rhombihexahedrongreat rhombihexacron
225|2 3icosahedrondodecahedron
233|2 5dodecahedronicosahedron
242|3 5icosidodecahedronrhombic triacontahedron
252 5|3truncated icosahedronpentakis dodecahedron
262 3|5truncated dodecahedrontriakis icosahedron
273 5|2small rhombicosidodecahedrondeltoidal hexecontahedron
282 3 5|great rhombicosidodecahedrondisdyakis triacontahedron
29|2 3 5snub dodecahedronpentagonal hexecontahedron
303|5/2 3small ditrigonal icosidodecahedronsmall triambic icosahedron
315/2 3|3small icosicosidodecahedronsmall icosacronic hexecontahedron
32|5/2 3 3small snub icosicosidodecahedronsmall hexagonal hexecontahedron
333/2 5|5small dodecicosidodecahedronsmall dodecacronic hexecontahedron
345|2 5/2small stellated dodecahedrongreat dodecahedron
355/2|2 5great dodecahedronsmall stellated dodecahedron
362|5/2 5dodecadodecahedronmedial rhombic triacontahedron
372 5/2|5truncated great dodecahedronsmall stellapentakis dodecahedron
385/2 5|2rhombidodecadodecahedronmedial deltoidal hexecontahedron
392 5/2 5|small rhombidodecahedronsmall rhombidodecacron
40|2 5/2 5snub dodecadodecahedronmedial pentagonal hexecontahedron
413|5/3 5ditrigonal dodecadodecahedronmedial triambic icosahedron
423 5|5/3great ditrigonal dodecicosidodecahedrongreat ditrigonal dodecacronic hexecontahedron
435/3 3|5small ditrigonal dodecicosidodecahedronsmall ditrigonal dodecacronic hexecontahedron
445/3 5|3icosidodecadodecahedronmedial icosacronic hexecontahedron
455/3 3 5|icositruncated dodecadodecahedrontridyakis icosahedron
46|5/3 3 5snub icosidodecadodecahedronmedial hexagonal hexecontahedron
473/2|3 5great ditrigonal icosidodecahedrongreat triambic icosahedron
483/2 5|3great icosicosidodecahedrongreat icosacronic hexecontahedron
493/2 3|5small icosihemidodecahedronsmall icosihemidodecacron
503/2 3 5|small dodecicosahedronsmall dodecicosacron
515/4 5|5small dodecahemidodecahedronsmall dodecahemidodecacron
523|2 5/2great stellated dodecahedrongreat icosahedron
535/2|2 3great icosahedrongreat stellated dodecahedron
542|5/2 3great icosidodecahedrongreat rhombic triacontahedron
552 5/2|3great truncated icosahedrongreat stellapentakis dodecahedron
562 5/2 3|rhombicosahedronrhombicosacron
57|2 5/2 3great snub icosidodecahedrongreat pentagonal hexecontahedron
582 5|5/3small stellated truncated dodecahedrongreat pentakis dodecahedron
595/3 2 5|truncated dodecadodecahedronmedial disdyakis triacontahedron
60|5/3 2 5inverted snub dodecadodecahedronmedial inverted pentagonal hexecontahedron
615/2 3|5/3great dodecicosidodecahedrongreat dodecacronic hexecontahedron
625/3 5/2|3small dodecahemicosahedronsmall dodecahemicosacron
635/3 5/2 3|great dodecicosahedrongreat dodecicosacron
64|5/3 5/2 3great snub dodecicosidodecahedrongreat hexagonal hexecontahedron
655/4 5|3great dodecahemicosahedrongreat dodecahemicosacron
662 3|5/3great stellated truncated dodecahedrongreat triakis icosahedron
675/3 3|2uniform great rhombicosidodecahedrongreat deltoidal hexecontahedron
685/3 2 3|great truncated icosidodecahedrongreat disdyakis triacontahedron
69|5/3 2 3great inverted snub icosidodecahedrongreat inverted pentagonal hexecontahedron
705/3 5/2|5/3great dodecahemidodecahedrongreat dodecahemidodecacron
713/2 3|5/3great icosihemidodecahedrongreat icosihemidodecacron
72|3/2 3/2 5/2small retrosnub icosicosidodecahedronsmall hexagrammic hexecontahedron
733/2 5/3 2|great rhombidodecahedrongreat rhombidodecacron
74|3/2 5/3 2great retrosnub icosidodecahedrongreat pentagrammic hexecontahedron
75|3/2 5/3 35/2great dirhombicosidodecahedrongreat dirhombicosidodecacron
762 5|2pentagonal prismpentagonal dipyramid
77|2 2 5pentagonal antiprismpentagonal trapezohedron
782 5/2|2pentagrammic prismpentagrammic dipyramid
79|2 2 5/2pentagrammic antiprismpentagrammic deltohedron
80|2 2 5/3pentagrammic 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 Solid, Möbius Triangles, Platonic Solid, Polyhedron, Schwarz Triangle, Uniform Polychoron, Vertex Figure, Wythoff Symbol

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 136, 1987.Brückner, M. Vielecke under Vielflache. Leipzig, Germany: Teubner, 1900.Bulatov, V. "Compounds of Uniform Polyhedra.", V. "Dual Uniform Polyhedra.", V. "Uniform Polyhedra.", H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Garcia, K. "Building Uniform Polyhedra for Version 12." July 25, 2019.'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Har'El, Z. "Kaleido."'El, Z. "Eighty Dual Polyhedra Generated by Kaleido."'El, Z. "Eighty Uniform Polyhedra Generated by Kaleido."'El, Z. "Uniform Solution for Uniform Polyhedra." Geom. Dedicata 47, 57-110, 1993., A. "Exact Descriptions of Regular and Semi-Regular Polyhedra and Their Duals." Computing Science Tech. Rept. No. 130. Murray Hill, NJ: AT&T Bell Lab., 1986.Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.Johnson, N. W. Uniform Polytopes. Cambridge, England: Cambridge University Press, 2000. Maeder, R. E. "Uniform Polyhedra." Mathematica J. 3, 48-57, 1993. Maeder, R. E. Polyhedra.m and PolyhedraExamples Mathematica notebooks. Maeder, R. E. "The Uniform Polyhedra.", P. W. "Problem 1094." Crux Math. 11, 325, 1985.Messer, P. W. "Solution to Problem 1094." Crux Math. 13, 133, 1987.Messer, P. W. "Closed-Form Expressions for Uniform Polyhedra and Their Duals." Disc. Comput. Geom. 27, 353-375, 2002.Sandia National Laboratories. "Polyhedron Database.", J. "The Complete Set of Uniform Polyhedron." Phil. Trans. Roy. Soc. London, Ser. A 278, 111-136, 1975.Skilling, J. "Uniform Compounds of Uniform Polyhedra." Math. Proc. Cambridge Philos. Soc. 79, 447-457, 1976.Smith, A. "Uniform Compounds and the Group A_4." Proc. Cambridge Philos. Soc. 75, 115-117, 1974.Sopov, S. P. "Proof of the Completeness of the Enumeration of Uniform Polyhedra." Ukrain. Geom. Sbornik 8, 139-156, 1970.Virtual Image. The Uniform Polyhedra CD-ROM. 1997., R. "Uniform/Dual Polyhedra.", R. "Stellated Polyhedra.", M. J. Dual Models. Cambridge, England: Cambridge University Press, 1983.Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 1-10 and 98, 1989.Zalgaller, V. Convex Polyhedra with Regular Faces. New York: Consultants Bureau, 1969.Ziegler, G. M. Lectures on Polytopes. Berlin: Springer-Verlag, 1995.

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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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