Great dirhombicosidodecahedron

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

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U₇₅.

This is the only 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.

This is also the only uniform polyhedron that cannot be made by Wythoff construction. It has a special Wythoff symbol | ³/₂ ⁵/₃ 3 ⁵/₂.

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

(0, ±2/τ, ±2/√τ)

(±(−1+1/√τ³), ±(1/τ²−1/√τ), ±(1/τ+√τ))

(±(−1/τ+√τ), ±(−1−1/√τ³, ±(1/τ²+1/√τ))

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

Filling

There is some controversy on how to colour the faces of this polyhedron. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out. Usage of the "neo filling" makes the great dirhombicosidodecahedron a hollow polyhedron.^[1]

Traditional filling	"Neo filling"	Interior view, "neo 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.
Richard Klitzing, 3D, uniform polyhedra

↑ http://polytope.net/hedrondude/polychora.htm

External links

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