The uniform polyhedra arepolyhedra consisting of regular (possibly polygrammic) faces of equal edge length whosepolyhedron vertices are all symmetrically equivalent. The uniform polyhedra include thePlatonic 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, theKepler-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).

Coxeteret al.(1954) conjectured that there are 75 uniform polyhedra in which only two faces are allowed to meet at anpolyhedron edge, and this surmise was subsequently proven. The five pentagonalprisms 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, thegreat complex icosidodecahedron andsmall complex icosidodecahedron (both of which are compounds of two uniform polyhedra).

Thepolyhedron vertices of a uniform polyhedron all lie on acircumsphere whose center is theirgeometric centroid (Coxeteret al.1954, Coxeter 1973, p. 44). Thepolyhedron vertices joined to anotherpolyhedron vertex lie on acircle (Coxeteret al.1954).

Not-necessarily circumscriptable versions of uniform polyhedra with exactified numeric vertices and polygrammic faces sometimes split into separate polygons are implemented in theWolfram Language asUniformPolyhedron["name"] orUniformPolyhedron["Uniform",n] (cf. Garcia 2019). The full exact, equilateral, circumscriptable uniform polyhedra are implemented in theWolfram Language asPolyhedronData["name"] orPolyhedronData["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 specialspherical triangle is mapped to all the other vertices by repeated reflections across the three planar sides of this triangle. Similarly, and its kaleidoscopic images must cover the sphere an integral number of times which is referred to as the density of. The density is dependent on the choice of angles,, at,, respectively, where,, are reduced rational numbers greater than one. Such a spherical triangle is called aSchwarz triangle, conveniently denoted. Except for the infinite dihedral family of for, 3, 4, ..., there are only 44 kinds of Schwarz triangles (Coxeteret al.1954, Coxeter 1973). It has been shown that the numerators of,, 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), Coxeteret al.(1954), and Har'El (1993). Coxeteret al.(1954) give many properties of the uniform solids, and Coxeteret 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 thegreat dirhombicosidodecahedron with Maeder index 75 which has pseudo-Wythoff symbol.

Maeder index	Wenninger index	Coxeter index	Har'El index	Wythoff symbol	name	dual polyhedron
1	1	15	6		regular tetrahedron	tetrahedron
2	6	16	7		truncated tetrahedron	triakis tetrahedron
3	68	37	8		octahemioctahedron	octahemioctacron
4	67	36	9		tetrahemihexahedron	tetrahemihexacron
5	2	17	10		regular octahedron	cube
6	3	18	11		cube	octahedron
7	11	19	12		cuboctahedron	rhombic dodecahedron
8	7	20	13		truncated octahedron	tetrakis hexahedron
9	8	21	14		truncated cube	small triakis octahedron
10	13	22	15		small rhombicuboctahedron (rhombicuboctahedron)	deltoidal icositetrahedron
11	15	23	16		great rhombicuboctahedron (truncated cuboctahedron)	disdyakis dodecahedron
12	17	24	17		snub cube	pentagonal icositetrahedron
13	69	38	18		small cubicuboctahedron	small hexacronic icositetrahedron
14	77	50	19		great cubicuboctahedron	great hexacronic icositetrahedron
15	78	51	20		cubohemioctahedron	hexahemioctacron
16	79	52	21		cubitruncated cuboctahedron	tetradyakis hexahedron
17	85	59	22		quasirhombicuboctahedron (great rhombicuboctahedron)	great deltoidal icositetrahedron
18	86	60	23		small rhombihexahedron	small rhombihexacron
19	92	66	24		stellated truncated hexahedron (quasitruncated hexahedron)	great triakis octahedron
20	93	67	25		great truncated cuboctahedron (quasitruncated cuboctahedron)	great disdyakis dodecahedron
21	103	82	26		great rhombihexahedron	great rhombihexacron
22	4	25	27		regular icosahedron	dodecahedron
23	5	26	28		regular dodecahedron	icosahedron
24	12	28	29		icosidodecahedron	rhombic triacontahedron
25	9	27	30		truncated icosahedron	pentakis dodecahedron
26	10	29	31		truncated dodecahedron	triakis icosahedron
27	14	30	32		small rhombicosidodecahedron (rhombicosidodecahedron)	deltoidal hexecontahedron
28	16	31	33		great rhombicosidodecahedron (truncated icosidodechedon)	disdyakis triacontahedron
29	18	32	34		snub dodecahedron	pentagonal hexecontahedron
30	70	39	35		small ditrigonal icosidodecahedron	small triambic icosahedron
31	71	40	36		small icosicosidodecahedron	small icosacronic hexecontahedron
32	110	41	37		small snub icosicosidodecahedron	small hexagonal hexecontahedron
33	72	42	38		small dodecicosidodecahedron	small dodecacronic hexecontahedron
34	20	43	39		small stellated dodecahedron	great dodecahedron
35	21	44	40		great dodecahedron	small stellated dodecahedron
36	73	45	41		dodecadodecahedron	medial rhombic triacontahedron
37	75	47	42		truncated great dodecahedron	small stellapentakis dodecahedron
38	76	48	42		rhombidodecadodecahedron	medial deltoidal hexecontahedron
39	74	46	44		small rhombidodecahedron	small rhombidodecacron
40	111	49	45		snub dodecadodecahedron	medial pentagonal hexecontahedron
41	80	53	46		ditrigonal dodecadodecahedron	medial triambic icosahedron
42	81	54	47		great ditrigonal dodecicosidodecahedron	great ditrigonal dodecacronic hexecontahedron
43	82	55	48		small ditrigonal dodecicosidodecahedron	small ditrigonal dodecacronic hexecontahedron
44	83	56	49		icosidodecadodecahedron	medial icosacronic hexecontahedron
45	84	57	50		icositruncated dodecadodecahedron	tridyakis icosahedron
46	112	58	51		snub icosidodecadodecahedron	medial hexagonal hexecontahedron
47	87	61	52		great ditrigonal icosidodecahedron	great triambic icosahedron
48	88	62	53		great icosicosidodecahedron	great icosacronic hexecontahedron
49	89	63	54		small icosihemidodecahedron	small icosihemidodecacron
50	90	64	55		small dodecicosahedron	small dodecicosacron
51	91	65	56		small dodecahemidodecahedron	small dodecahemidodecacron
52	22	68	57		great stellated dodecahedron	great icosahedron
53	41	69	58		great icosahedron	great stellated dodecahedron
54	94	70	59		great icosidodecahedron	great rhombic triacontahedron
55	95	71	60		great truncated icosahedron	great stellapentakis dodecahedron
56	96	72	61		rhombicosahedron	rhombicosacron
57	116	88	62		great snub icosidodecahedron	great pentagonal hexecontahedron
58	97	74	63		small stellated truncated dodecahedron	great pentakis dodecahedron
59	98	75	64		truncated dodecadodecahedron	medial disdyakis triacontahedron
60	114	76	65		inverted snub dodecadodecahedron	medial inverted pentagonal hexecontahedron
61	99	77	66		great dodecicosidodecahedron	great dodecacronic hexecontahedron
62	100	78	67		small dodecahemicosahedron	small dodecahemicosacron
63	101	79	68		great dodecicosahedron	great dodecicosacron
64	115	80	69		great snub dodecicosidodecahedron	great hexagonal hexecontahedron
65	102	81	70		great dodecahemicosahedron	great dodecahemicosacron
66	104	83	71		great stellated truncated dodecahedron	great triakis icosahedron
67	105	84	72		quasirhombicosidodecahedron (great rhombicosidodecahedron)	great deltoidal hexecontahedron
68	108	87	73		great truncated icosidodecahedron	great disdyakis triacontahedron
69	113	73	74		great inverted snub icosidodecahedron	great inverted pentagonal hexecontahedron
70	107	86	75		great dodecahemidodecahedron	great dodecahemidodecacron
71	106	85	76		great icosihemidodecahedron	great icosihemidodecacron
72	118	91	77		small retrosnub icosicosidodecahedron	small hexagrammic hexecontahedron
73	109	89	78		great rhombidodecahedron	great rhombidodecacron
74	117	90	79		great retrosnub icosidodecahedron	great pentagrammic hexecontahedron
75	119	92	80		great dirhombicosidodecahedron	great dirhombicosidodecacron
76			1		pentagonal prism	pentagonal dipyramid
77			2		pentagonal antiprism	pentagonal trapezohedron
78		33	3		pentagrammic prism	pentagrammic dipyramid
79		34	4		pentagrammic antiprism	pentagrammic deltohedron
80		35	5		pentagrammic crossed antiprism	pentagrammic 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 modifiedSchläfli 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 by pentagrams. 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, CO 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 theSchläfli 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:

1		T	tetrahedron	tetrahedron
2		O	octahedron	cube
3		C	cube	octahedron
4		I	icosahedron	dodecahedron
5		D	dodecahedron	icosahedron
20		D*	small stellated dodecahedron	great dodecahedron
21		E	great dodecahedron	small stellated dodecahedron
22		E*	great stellated dodecahedron	great icosahedron
41		J	great icosahedron	great stellated dodecahedron

Quasi-regular polyhedra:

11	r	CO	cuboctahedron	rhombic dodecahedron
12	r	ID	icosidodecahedron	rhombic triacontahedron
73	r	ED*	dodecadodecahedron	middle rhombic triacontahedron
94	r	JE*	great icosidodecahedron	great rhombic triacontahedron
70	a	ID*	small ditrigonary icosidodecahedron	small triambic icosahedron
80	b	DE*	ditrigonary dodecadodecahedron	middle triambic icosahedron
87	c	JE	great ditrigonary icosidodecahedron	great triambic icosahedron

Versi-regular polyhedra:

67	o	TT	tetrahemihexahedron	no dual
78	o	CO	cubohemioctahedron	no dual
68	o	OC	octahemioctahedron	no dual
91	o	DI	small dodecahemidodecahedron	no dual
89	o	ID	small icosahemidodecahedron	no dual
102	o	ED*	small dodecahemiicosahedron	no dual
100	o	D*E	great dodecahemiicosahedron	no dual
106	o	JE*	great icosahemidodecahedron	no dual
107	o	E*J	great dodecahemidodecahedron	no dual

Truncated regular polyhedra:

6	t	tT	truncated tetrahedron	triakis tetrahedron
7	t	tO	truncated octahedron	tetrakis hexahedron
8	t	tC	truncated cube	triakis octahedron
92	t'	t'C	stellatruncated cube	great triakis octahedron
9	t	tI	truncated icosahedron	pentakis dodecahedron
10	t	tD	truncated dodecahedron	triakis icosahedron
97	t'	t'D*	small stellatruncated dodecahedron	great pentakis dodecahedron
75	t	tE	great truncated dodecahedron	small stellapentakis dodecahedron
104	t'	t'E*	great stellatruncated dodecahedron	great triakis icosahedron
95	t	tJ	great truncated icosahedron	great stellapentakis dodecahedron

Quasi-quasi-regular polyhedra: and

13	rr	rCO	rhombicuboctahedron	strombic disdodecahedron
69	R'r	R'CO	small cubicuboctahedron	small sagittal disdodecahedron
77	Rr	RCO	great cubicuboctahedron	great strombic disdodecahedron
85	r'r	r'CO	great rhombicuboctahedron	great sagittal disdodecahedron
14	rr	rID	rhombicosidodecahedron	strombic hexecontahedron
72	R'r	R'ID	small dodekicosidodecahedron	small sagittal hexecontahedron
71	ra	rID*	small icosified icosidodecahedron	small strombic trisicosahedron
82	R'a	R'ID*	small dodekified icosidodecahedron	small sagittal trisicosahedron
76	rr	rED*	rhombidodecadodecahedron	middle strombic trisicosahedron
83	R'r	R'ED*	icosified dodecadodecahedron	middle sagittal trisicosahedron
81	Rc	RJE	great dodekified icosidodecahedron	great strombic trisicosahedron
88	r'c	r'JE	great icosified icosidodecahedron	great sagittal trisicosahedron
99	Rr	RJE*	great dodekicosidodecahedron	great strombic hexecontahedron
105	r'r	r'JE*	great rhombicosidodecahedron	great sagittal hexecontahedron

Versi-quasi-regular polyhedra:

86	or	rR'CO	small rhombicube	small dipteral disdodecahedron
103	Or	Rr'CO	great rhombicube	great dipteral disdodecahedron
74	or	rR'ID	small rhombidodecahedron	small dipteral hexecontahedron
90	oa	rR'ID*	small dodekicosahedron	small dipteral trisicosahedron
96	or	rR'ED*	rhombicosahedron	middle dipteral trisicosahedron
101	Oc	Rr'JE	great dodekicosahedron	great dipteral trisicosahedron
109	Or	Rr'JE*	great rhombidodecahedron	great dipteral hexecontahedron

Truncated quasi-regular polyhedra:

15	tr	tCO	truncated cuboctahedron	disdyakis dodecahedron
93	t'r	t'CO	stellatruncated cuboctahedron	great disdyakis dodecahedron
79	Tr	TCO	cubitruncated cuboctahedron	trisdyakis octahedron
16	tr	tID	truncated icosidodecahedron	disdyakis triacontahedron
98	t'r	t'ED*	stellatruncated dodecadodecahedron	middle disdyakis triacontahedron
84	T'r	T'ED*	icositruncated dodecadodecahedron	trisdyakis icosahedron
108	t'r	t'JE*	stellatruncated icosidodecahedron	great disdyakis triacontahedron

Snub quasi-regular polyhedra: or

17	sr	sCO	snub cuboctahedron	petaloidal disdodecahedron
18	sr	sID	snub icosidodecahedron	petaloidal hexecontahedron
110	sa	sID*	snub disicosidodecahedron	no dual
118	s*a	s*ID*	retrosnub disicosidodecahedron	no dual
111	sr	sED*	snub dodecadodecahedron	petaloidal trisicosahedron
114	s'r	s'ED*	vertisnub dodecadodecahedron	vertipetaloidal trisicosahedron
112	S'r	S'ED*	snub icosidodecadodecahedron	hexaloidal trisicosahedron
113	sr	sJE*	great snub icosidodecahedron	great petaloidal hexecontahedron
116	s'r	s'JE*	great vertisnub icosidodecahedron	great vertipetaloidal hexecontahedron
117	s*r	s*JE*	great retrosnub icosidodecahedron	great retropetaloidal hexecontahedron

Snub quasi-regular polyhedron:

119

SSr

SSJE*

great disnub disicosidisdodecahedron

no dual

Prisms:

	P(p)	-gonal prism,, 5, 6, ...	-gonal bipyramid
	P(p/d)	-fold-gonal prism,	-fold-gonal bipyramid

Antiprisms and crossed antiprisms:

sh	Q(p)	-gonal antiprism,, 5, 6, ...	-gonal antibipyramid
sh	Q(p/d)	-fold-gonal antiprism,	-fold-gonal antibipyramid
s'h	Q'(p/d)	-fold-gonal crossed antiprism,	-fold-gonal crossed antibipyramid

See also

Explore with Wolfram|Alpha

More things to try:

• uniform polyhedron
• apply bilateral filter to dog image
• derivative of x^2 y+ x y^2 in the direction (1,1)

References

Maeder, R. E. "Uniform Polyhedra."Mathematica J.3, 48-57, 1993.http://library.wolfram.com/infocenter/Articles/2254/.

Maeder, R. E.Polyhedra.m andPolyhedraExamplesMathematica notebooks.http://www.inf.ethz.ch/department/TI/rm/programs.html.

Smith, A. "Uniform Compounds and the Group."Proc. Cambridge Philos. Soc.75, 115-117, 1974.

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Uniform Polyhedron."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/UniformPolyhedron.html

Subject classifications