snub icosidodecadodecahedron

Let $\backslash rho\backslash approx\; 1.3247179572447454$ be the real zero of the polynomial $x^3-x-1$. The number $\backslash rho$ is known as the plastic ratio. Denote by $\backslash phi$ the golden ratio. Let the point $p$ be given by

:$p=$

\begin{pmatrix}

\rho \\

\phi^2\rho^2-\phi^2\rho-1\\

-\phi\rho^2+\phi^2

\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 icosidodecadodecahedron. The edge length equals $2\backslash sqrt\{\backslash phi^2\backslash rho^2-2\backslash phi-1\}$, the circumradius equals $\backslash sqrt\{(\backslash phi+2)\backslash rho^2+\backslash rho-3\backslash phi-1\}$, and the midradius equals $\backslash sqrt\{\backslash rho^2+\backslash rho-\backslash phi\}$.

For a snub icosidodecadodecahedron whose edge length is 1,

the circumradius is

:$R\; =\; \backslash frac12\backslash sqrt\{\backslash rho^2+\backslash rho+2\}\; \backslash approx\; 1.126897912799939$

Its midradius is

:$r\; =\; \backslash frac12\backslash sqrt\{\backslash rho^2+\backslash rho+1\}\; \backslash approx\; 1.0099004435452335$

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

File:Medial hexagonal hexecontahedron.stl

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

• List of uniform polyhedra

{{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}}

• {{mathworld | urlname = MedialHexagonalHexecontahedron| title =Medial hexagonal hexecontahedron}}

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Category:Uniform polyhedra

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