Great dirhombicosidodecahedron

Short description: Uniform star polyhedron with 124 faces

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	Ih, [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

File:Great dirhombicosidodecahedron.stl

3D model of a 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.

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. Since all its 60 double-square facial planes pass through its center, the great dirhombicosidodecahedron classifies as a hemipolyhedron; however, the great dirhombicosidedecahedron does not form a hemi-face like the simpler hemipolyhedra that are based on the two quasiregular polyhedra. 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.

100px Convex hull	100px Great snub dodecicosidodecahedron	100px Great dirhombicosidodecahedron
100px Great disnub dirhombidodecahedron	100px Compound of twenty octahedra	100px Compound of twenty tetrahemihexahedra

This polyhedron is related to the nonconvex great rhombicosidodecahedron (quasirhombicosidodecahedron) by a branched cover: there is a function from the great dirhombicosidodecahedron to the quasirhombicosidodecahedron that is 2-to-1 everywhere, except for the vertices.

Cartesian coordinates

Let the point $p$ be given by

$p=\left(\begin{array}{c}{\varphi }^{-\frac{1}{2}}\\ 0\\ {\varphi }^{-1}\end{array}\right)$,

where $\varphi$ is the golden ratio. Let the matrix $M$ be given by

$M=\left(\begin{array}{ccc}1/2& -\varphi /2& 1/(2\varphi )\\ \varphi /2& 1/(2\varphi )& -1/2\\ 1/(2\varphi )& 1/2& \varphi /2\end{array}\right)$.

$M$ is the rotation around the axis $(1,0,\varphi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_0,\dots ,T_{11}$ be the transformations which send a point $(x,y,z)$ to the even permutations of $(\pm x,\pm y,\pm z)$ with an even number of minus signs. The transformations $T_i$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations $T_i{M}^j$ $(i=0,\dots ,11$, $j=0,\dots ,4)$ constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points $T_i{M}^{j}p$ are the vertices of a great dirhombicosidodecahedron. The edge length equals $\sqrt{2}$, the circumradius equals $1$, and the midradius equals $\frac{1}{2}\sqrt{2}$.

For a great dirhombicosidodecahedron whose edge length is 1, the circumradius is

$R=\frac{1}{2}\sqrt{2}$.

Its midradius is

$r=\frac{1}{2}$.

Gallery

150px Traditional filling

150px Modulo-2 filling

300px 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 (916): 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, Bibcode: 1954RSPTA.246..401C 
• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. OCLC 1738087. 
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. pp. 1–156. ISBN 0521098599. https://www.cambridge.org/core/books/dual-models/7F1B1D1AEF65D83BB77F516AEBA22919. 
• 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". https://bendwavy.org/klitzing/dimensions/polyhedra-neu.htm. 
• Webb, Robert (2001). "Great Dirhombicosidodecahedron ("Miller's Monster")". https://www.software3d.com/MillersMonster.php. 

External links

• Weisstein, Eric W.. "Great dirhombicosidodecahedron". http://mathworld.wolfram.com/GreatDirhombicosidodecahedron.html.