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|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|triangle]]s and 12 [[Pentagram|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|hexagon]]s that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.

George Olshevsky nicknamed it the '''yog-sothoth''' (after 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 |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|convex hull]] is a nonuniform [[Truncated dodecahedron|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 ==
Cartesian coordinates for the vertices of a small retrosnub icosicosidodecahedron are all the even permutations of
<math display=block>\begin{array}{crrrc}
   \Bigl(& \pm  \bigl[1-\varphi-\alpha\bigr],& 0\,,& \pm \bigl[3-\varphi\alpha\bigr] &\Bigr), \\
   \Bigl(& \pm  \bigl[\varphi-1-\alpha\bigr],& \pm\,2\,,& \pm  \bigl[2\varphi-1-\varphi\alpha\bigr] &\Bigr), \\
   \Bigl(& \pm  \bigl[\varphi+1-\alpha\bigr],& \pm\,2\bigl[\varphi-1\bigr],& \pm  \bigl[1-\varphi\alpha\bigr] &\Bigr),
\end{array}</math>
where <math>\varphi = \tfrac{1+\sqrt 5}{2}</math> is the [[Golden ratio|golden ratio]] and <math>\alpha = \sqrt{3\varphi-2}.</math>

== 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]]



{{Sourceattribution|Small retrosnub icosicosidodecahedron}}