Snub dodecadodecahedron

Let $\backslash xi\backslash approx\; 1.2223809502469911$ be the smallest real zero of the polynomial $P=2x^4-5x^3+3x+1$. Denote by $\backslash phi$ the golden ratio. Let the point $p$ be given by

:$p=$

\begin{pmatrix}

\phi^{-2}\xi^2-\phi^{-2}\xi+\phi^{-1}\\

-\phi^{2}\xi^2+\phi^{2}\xi+\phi\\

\xi^2+\xi

\end{pmatrix}

.

Let the matrix $M$ be given by

:$M=$

\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}

.

$M$ is the rotation around the axis $(1,\; 0,\; \backslash phi)$ by an angle of $2\backslash pi/5$, counterclockwise. Let the linear transformations $T\_0,\; \backslash ldots,\; T\_\{11\}$

be the transformations which send a point $(x,\; y,\; z)$ to the even permutations of $(\backslash pm\; x,\; \backslash pm\; y,\; \backslash pm\; z)$ with an even number of minus signs.

The transformations $T\_i$ constitute the group of rotational symmetries of a regular tetrahedron.

The transformations $T\_i\; M^j$ $(i\; =\; 0,\backslash ldots,\; 11$, $j\; =\; 0,\backslash ldots,\; 4)$ constitute the group of rotational symmetries of a regular icosahedron.

Then the 60 points $T\_i\; M^j\; p$ are the vertices of a snub dodecadodecahedron. The edge length equals $2(\backslash xi+1)\backslash sqrt\{\backslash xi^2-\backslash xi\}$, the circumradius equals $(\backslash xi+1)\backslash sqrt\{2\backslash xi^2-\backslash xi\}$, and the midradius equals $\backslash xi^2+\backslash xi$.

For a great snub icosidodecahedron whose edge length is 1,

the circumradius is

:$R\; =\; \backslash frac12\backslash sqrt\{\backslash frac\{2\backslash xi-1\}\{\backslash xi-1\}\}\; \backslash approx\; 1.2744398820380232$

Its midradius is

:$r=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{\backslash xi\}\{\backslash xi-1\}\}\; \backslash approx\; 1.1722614951149297$

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

{{Uniform polyhedra db|Uniform dual polyhedron stat table|Siddid}}

File:Medial pentagonal hexecontahedron.stl

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

• List of uniform polyhedra
• Inverted snub dodecadodecahedron

{{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}}

• {{mathworld | urlname = MedialPentagonalHexecontahedron| title =Medial pentagonal hexecontahedron}}
• {{mathworld | urlname = SnubDodecadodecahedron | title = Snub dodecadodecahedron}}

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Category:Uniform polyhedra

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