Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as U₇₅. It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.[1]

Great dirhombicosidodecahedron

Type	Uniform star polyhedron
Elements	F = 124, E = 240 V = 60 (χ = −56)
Faces by sides	40{3}+60{4}+24{5/2}
Wythoff symbol	\| 3/2 5/3 3 5/2
Symmetry group	I_h, [5,3], *532
Index references	U₇₅, C₉₂, W₁₁₉
Dual polyhedron	Great dirhombicosidodecacron
Vertex figure	4.5/3.4.3.4.5/2.4.3/2
Bowers acronym	Gidrid

3D model of a great dirhombicosidodecahedron.

This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams. Another unusual feature is that the faces all occur in coplanar pairs.

This is also the only uniform polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol | 3/2 5/3 3 5/2 relating it to a spherical quadrilateral. This symbol suggests that it is a sort of snub polyhedron, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares.

It has been nicknamed "Miller's monster" (after J. C. P. Miller, who with H. S. M. Coxeter and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954).

Related polyhedra

If the definition of a uniform polyhedron is relaxed to allow any even number of faces adjacent to an edge, then this definition gives rise to one further polyhedron: the great disnub dirhombidodecahedron which has the same vertices and edges but with a different arrangement of triangular faces.

The vertices and edges are also shared with the uniform compounds of 20 octahedra or 20 tetrahemihexahedra. 180 of the 240 edges are shared with the great snub dodecicosidodecahedron.

Convex hull	Great snub dodecicosidodecahedron	Great dirhombicosidodecahedron
Great disnub dirhombidodecahedron	Compound of twenty octahedra	Compound of twenty tetrahemihexahedra

Cartesian coordinates

Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of

\left(0,\pm {\frac {2}{\tau }},\pm {\frac {2}{\sqrt {\tau }}}\right),\left(\pm \left(-1+{\frac {1}{\sqrt {\tau ^{3}}}}\right),\pm \left({\frac {1}{\tau ^{2}}}-{\frac {1}{\sqrt {\tau }}}\right),\pm \left({\frac {1}{\tau }}+{\sqrt {\tau }}\right)\right),

\left(\pm \left({\frac {-1}{\tau }}+{\sqrt {\tau }}\right),\pm \left(-1-{\frac {1}{\sqrt {\tau ^{3}}}}\right),\pm \left({\frac {1}{\tau ^{2}}}+{\frac {1}{\sqrt {\tau }}}\right)\right),

where τ = (1+√5)/2 is the golden ratio (sometimes written φ). These vertices result in an edge length of 2√2.

Gallery

Traditional filling	Modulo-2 filling	Interior view, modulo-2 filling

References

Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087.
Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
Klitzing, Richard. "3D uniform polyhedra".

Maeder, Roman. "75: great dirhombicosidodecahedron". MathConsult.

External links

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[1] Maeder, Roman. "75: great dirhombicosidodecahedron". MathConsult.