View source for Snub dodecadodecahedron

{{Short description|Uniform star polyhedron with 84 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|Siddid}}
[[File:Snub dodecadodecahedron.stl|thumb|3D model of a snub dodecadodecahedron]]

In [[geometry]], the '''snub dodecadodecahedron''' is a [[nonconvex uniform polyhedron]], indexed as {{math|U{{sub|40}}}}. It has 84 faces (60 [[triangle]]s, 12 [[pentagon]]s, and 12 [[pentagram]]s), 150 edges, and 60 vertices.<ref>{{Cite web|url=https://www.mathconsult.ch/enwiki/static/unipoly/40.html|title=40: snub dodecadodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}}</ref> It is given a [[Schläfli symbol]] {{math|sr{<nowiki/>{{frac|5|2}},5},}} as a [[Snub (geometry)|snub]] [[great dodecahedron]].

==Cartesian coordinates==

Let <math>\xi\approx 1.2223809502469911</math> be the smallest real zero of the polynomial <math>P=2x^4-5x^3+3x+1</math>. Denote by <math>\phi</math> the [[golden ratio]]. Let the point <math>p</math> be given by
:<math>p=
\begin{pmatrix}
        \phi^{-2}\xi^2-\phi^{-2}\xi+\phi^{-1}\\
        -\phi^{2}\xi^2+\phi^{2}\xi+\phi\\
        \xi^2+\xi
\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 dodecadodecahedron. The edge length equals <math>2(\xi+1)\sqrt{\xi^2-\xi}</math>, the circumradius equals <math>(\xi+1)\sqrt{2\xi^2-\xi}</math>, and the midradius equals <math>\xi^2+\xi</math>.

For a great snub icosidodecahedron whose edge length is 1,
the circumradius is
:<math>R = \frac12\sqrt{\frac{2\xi-1}{\xi-1}} \approx 1.2744398820380232</math> 
Its midradius is
:<math>r=\frac{1}{2}\sqrt{\frac{\xi}{\xi-1}} \approx 1.1722614951149297</math>

The other real root of P plays a similar role in the description of the [[Inverted snub dodecadodecahedron]]

== Related polyhedra ==

=== Medial pentagonal hexecontahedron ===
{{Uniform polyhedra db|Uniform dual polyhedron stat table|Siddid}}
[[File:Medial pentagonal hexecontahedron.stl|thumb|3D model of a medial pentagonal hexecontahedron]]
The [[medial pentagonal hexecontahedron]] is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]]. It is the [[Dual polyhedron|dual]] of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

== See also ==
* [[List of uniform polyhedra]]
* [[Inverted snub dodecadodecahedron]]

==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 | doi=10.1017/CBO9780511569371}}

== External links ==
* {{mathworld | urlname = MedialPentagonalHexecontahedron| title =Medial pentagonal hexecontahedron}}
* {{mathworld | urlname = SnubDodecadodecahedron | title = Snub dodecadodecahedron}}

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[[Category:Uniform polyhedra]]


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