View source for Great dirhombicosidodecahedron

{{Short description|Uniform star polyhedron with 124 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Gidrid}}
[[File:Great dirhombicosidodecahedron.stl|thumb|3D model of a great dirhombicosidodecahedron.]]

In [[geometry]], the '''great dirhombicosidodecahedron''' (or '''great snub disicosidisdodecahedron''') is a [[nonconvex uniform polyhedron]], indexed last as {{math|U{{sub|75}}}}. It has 124 faces (40 [[Triangle|triangles]], 60 [[Square|squares]], and 24 [[Pentagram|pentagrams]]), 240 [[Edge (geometry)|edges]], and 60 [[Vertex (geometry)|vertices]].<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/75.html|title=75: great dirhombicosidodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref>

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]] {{math|{{!}} {{frac|3|2}} {{frac|5|3}} 3 {{frac|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 [[compound of twenty octahedra|20 octahedra]] or [[compound of twenty tetrahemihexahedra|20 tetrahemihexahedra]]. 180 of the 240 edges are shared with the [[great snub dodecicosidodecahedron]].

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
| [[File:Nonuniform2-rhombicosidodecahedron.png|100px]]<br>[[Convex hull]]
| [[File:Great snub dodecicosidodecahedron.png|100px]]<br>[[Great snub dodecicosidodecahedron]]
| [[File:Great dirhombicosidodecahedron.png|100px]]<br>'''Great dirhombicosidodecahedron'''
|-
| [[File:Great disnub dirhombidodecahedron.png|100px]]<br>[[Great disnub dirhombidodecahedron]]
| [[File:UC14-20 octahedra.png|100px]]<br>[[Compound of twenty octahedra]]
| [[File:UC19-20 tetrahemihexahedron.png|100px]]<br>[[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.<ref>{{cite web | url=https://homepages.wmich.edu/~drichter/great_dirhombicosidodecahedron.htm | title=Great Dirhombicosidodecahedron | access-date=2022-07-24 | archive-date=2018-10-18 | archive-url=https://web.archive.org/web/20181018072752/http://homepages.wmich.edu/~drichter/great_dirhombicosidodecahedron.htm | url-status=dead }}</ref>

==Cartesian coordinates==

Let the point <math>p</math> be given by
:<math>p=
\begin{pmatrix}
        \phi^{-\frac12} \\
        0 \\
        \phi^{-1}
\end{pmatrix}
</math>,
where <math>\phi</math> is the [[golden ratio]].
Let the matrix <math>M</math> be given by
:<math>M=
\begin{pmatrix}
         1/2  & -\phi/2 & 1/(2\phi) \\
         \phi/2   & 1/(2\phi)     & -1/2 \\
         1/(2\phi)     & 1/2  & \phi/2
\end{pmatrix}
</math>.
<math>M</math> is the rotation around the axis <math>(1, 0, \phi)</math>  by an angle of <math>2\pi/5</math>, counterclockwise. Let the linear transformations <math>T_0, \ldots, T_{11}</math> 
be the transformations which send a point <math>(x, y, z)</math> to the [[parity of a permutation|even permutations]] of <math>(\pm x, \pm y, \pm z)</math>  with an even number of minus signs. 
The transformations <math>T_i</math> constitute the group of rotational symmetries of a [[regular tetrahedron]].
The transformations <math>T_i M^j</math> <math>(i = 0,\ldots, 11</math>, <math>j = 0,\ldots, 4)</math> constitute the group of rotational symmetries of a [[regular icosahedron]].
Then the 60 points <math>T_i M^j p</math>  are the vertices of a great dirhombicosidodecahedron. The edge length equals <math>\sqrt2</math>, the circumradius equals <math>1</math>, and the midradius equals <math>\frac12\sqrt2</math>.

For a great dirhombicosidodecahedron whose edge length is 1,
the circumradius is
:<math>R = \frac12\sqrt2</math>.
Its midradius is
:<math>r=\frac12</math>.

== Gallery ==
<br />

{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
|align=center|[[File:Great dirhombicosidodecahedron.png|150px]]<br>Traditional filling
|align=center|[[File:Great dirhombicosidodecahedron 2.png|150px]]<br>Modulo-2 filling
|align=center|[[File:Great dirhombicosidodecahedron interior.png|300px]]<br>Interior view, modulo-2 filling
|}

== References==
*{{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Longuet-Higgins | first2=M. S. | last3=Miller | first3=J. C. P. | title=Uniform polyhedra | jstor=91532 | mr=0062446  | year=1954 | journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences | issn=0080-4614 | volume=246 | issue=916 | pages=401–450 | doi=10.1098/rsta.1954.0003| bibcode=1954RSPTA.246..401C | s2cid=202575183 }}
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger  | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 | oclc=1738087 }}
* Har'El, Z. [https://www.math.technion.ac.il/~rl/docs/uniform.pdf ''Uniform Solution for Uniform Polyhedra.''], Geometriae Dedicata 47, 57-110, 1993. [https://www.math.technion.ac.il/~rl Zvi Har’El], [https://www.math.technion.ac.il/~rl/kaleido Kaleido software], [https://www.math.technion.ac.il/~rl/kaleido/poly.html Images], [https://www.math.technion.ac.il/~rl/kaleido/dual.html dual images]
*  [http://www.mathconsult.ch/showroom/unipoly Mäder, R. E.] ''[http://library.wolfram.com/infocenter/Articles/2254 Uniform Polyhedra.]'' Mathematica J. 3, 48-57, 1993. 
* {{KlitzingPolytopes|polyhedra-neu.htm|3D|uniform polyhedra}}
{{reflist}}

== External links ==
* {{mathworld | urlname = GreatDirhombicosidodecahedron| title =Great dirhombicosidodecahedron}}
* http://www.mathconsult.ch/showroom/unipoly/75.html
* http://www.software3d.com/MillersMonster.php

[[Category:Uniform polyhedra]]