View source for Small retrosnub icosicosidodecahedron

{{short description|Uniform star polyhedron with 112 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Sirsid}}
[[File:Small retrosnub icosicosidodecahedron.stl|thumb|3D model of a small retrosnub icosicosidodecahedron]]

In [[geometry]], the '''small retrosnub icosicosidodecahedron''' (also known as a '''retrosnub disicosidodecahedron''', '''small inverted retrosnub icosicosidodecahedron''', or '''retroholosnub icosahedron''') is a [[nonconvex uniform polyhedron]], indexed as {{math|U{{sub|72}}}}. It has 112 faces (100 [[triangle]]s and 12 [[pentagram]]s), 180 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/72.html|title=72: small retrosnub icosicosidodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> It is given a [[Schläfli symbol]] sr{⁵/₃,³/₂}.

The 40 non-snub triangular faces form 20 coplanar pairs, forming star [[hexagon]]s that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.

George Olshevsky nicknamed it the '''yog-sothoth''' (after [[Yog-Sothoth|the Cthulhu Mythos deity]]).<ref>{{cite thesis |type=M.S. |last=Birrell |first=Robert J. |date=May 1992 |title=The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron |publisher=California State University}}</ref><ref>{{cite conference |url=https://archive.bridgesmathart.org/2000/bridges2000-239.pdf |title=Uniform Polychora |last1=Bowers |first1=Jonathan |author-link1= |last2= |first2= |author-link2= |date=2000 |publisher= |editor=Reza Sarhagi |book-title=Bridges 2000 |pages=239–246 |location= |conference=Bridges Conference |id=}}</ref>

== Convex hull ==

Its [[convex hull]] is a nonuniform [[truncated dodecahedron]].

{| class=wikitable width=200
|[[File:Truncated dodecahedron.png|100px]]<BR>[[Truncated dodecahedron]]
|[[File:Small retrosnub icosicosidodecahedron convex hull.png|100px]]<BR>[[Convex hull]]
|[[File:Small retrosnub icosicosidodecahedron.png|100px]]<BR>'''Small retrosnub icosicosidodecahedron'''
|}

==Cartesian coordinates==

Let <math>\xi=-\frac32-\frac12\sqrt{1+4\phi}\approx -2.866760399173862</math> be the smallest (most negative) zero of the polynomial <math>P=x^2+3x+\phi^{-2}</math>, where <math>\phi</math> is the [[golden ratio]]. Let the point <math>p</math> be given by
:<math>p=
\begin{pmatrix}
        \phi^{-1}\xi+\phi^{-3} \\
        \xi \\
        \phi^{-2}\xi+\phi^{-2}
\end{pmatrix}
</math>.
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 small snub icosicosidodecahedron. The edge length equals <math>-2\xi</math>, the circumradius equals <math>\sqrt{-4\xi-\phi^{-2}}</math>, and the midradius equals <math>\sqrt{-\xi}</math>.

For a small snub icosicosidodecahedron whose edge length is 1,
the circumradius is
:<math>R = \frac12\sqrt{\frac{\xi-1}{\xi}} \approx 0.5806948001339209</math>
Its midradius is
:<math>r = \frac12\sqrt{\frac{-1}{\xi}} \approx 0.2953073837589815</math>

The other zero of <math>P</math> plays a similar role in the description of the [[small snub icosicosidodecahedron]].

== See also ==
* [[List of uniform polyhedra]]
* [[Small snub icosicosidodecahedron]]

== References ==
{{Reflist}}

== External links ==
* {{mathworld | urlname = SmallRetrosnubIcosicosidodecahedron| title =Small retrosnub icosicosidodecahedron}}
* {{KlitzingPolytopes|../incmats/sirsid.htm|3D star|small retrosnub icosicosidodecahedron}}

[[Category:Uniform polyhedra]]


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