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# 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 are implemented in 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 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 a Schwarz triangle, conveniently denoted . Except for the infinite dihedral family of for , 3, 4, ..., there are only 44 kinds of Schwarz triangles (Coxeter et 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), 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 .

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 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 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:

 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

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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## References

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## Referenced on Wolfram|Alpha

Uniform Polyhedron

## Cite this as:

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