View source for Snub icosidodecadodecahedron

{{Short description|Polyhedron with 104 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Sided}}
[[File:Snub icosidodecadodecahedron.stl|thumb|3D model of a snub icosidodecadodecahedron]]
In [[geometry]], the '''snub [[icosidodecadodecahedron]]''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>46</sub>. It has 104 faces (80 [[triangle]]s, 12 [[pentagon]]s, and 12 [[pentagram]]s), 180 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/static/unipoly/46.html|title=46: snub icosidodecadodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> As the name indicates, it belongs to the family of [[snub polyhedra]].

==Cartesian coordinates==

Let <math>\rho\approx 1.3247179572447454</math> be the real zero of the polynomial <math>x^3-x-1</math>. The number <math>\rho</math> is known as the [[plastic ratio]]. Denote by <math>\phi</math> the [[golden ratio]]. Let the point <math>p</math> be given by
:<math>p=
\begin{pmatrix}
        \rho \\
        \phi^2\rho^2-\phi^2\rho-1\\
        -\phi\rho^2+\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 snub icosidodecadodecahedron. The edge length equals <math>2\sqrt{\phi^2\rho^2-2\phi-1}</math>, the circumradius equals <math>\sqrt{(\phi+2)\rho^2+\rho-3\phi-1}</math>, and the midradius equals <math>\sqrt{\rho^2+\rho-\phi}</math>.

For a snub icosidodecadodecahedron whose edge length is 1,
the circumradius is
:<math>R = \frac12\sqrt{\rho^2+\rho+2} \approx 1.126897912799939</math>
Its midradius is
:<math>r = \frac12\sqrt{\rho^2+\rho+1} \approx 1.0099004435452335</math>

== Related polyhedra ==

=== Medial hexagonal hexecontahedron===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|Sided}}
[[File:Medial hexagonal hexecontahedron.stl|thumb|3D model of a medial hexagonal hexecontahedron]]
The [[medial hexagonal hexecontahedron]] is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]]. It is the [[Dual polyhedron|dual]] of the [[uniform star polyhedron|uniform]] snub icosidodecadodecahedron.

== See also ==
* [[List of uniform polyhedra]]

==References==
{{Reflist}}
*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 |mr=730208 | year=1983}}

== External links ==
* {{mathworld | urlname = MedialHexagonalHexecontahedron| title =Medial hexagonal hexecontahedron}}

{{Nonconvex polyhedron navigator}}

[[Category:Uniform polyhedra]]


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