Snub dodecadodecahedron

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In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as Template:Math. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol Template:Math as a snub great dodecahedron.

Cartesian coordinates

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

$p=\left(\begin{array}{c}{\varphi }^{-2}{\xi }^2-{\varphi }^{-2}\xi +{\varphi }^{-1}\\ -{\varphi }^2{\xi }^2+{\varphi }^2\xi +\varphi \\ {\xi }^2+\xi \end{array}\right)$.

Let the matrix $M$ be given by

$M=\left(\begin{array}{ccc}1/2& -\varphi /2& 1/(2\varphi )\\ \varphi /2& 1/(2\varphi )& -1/2\\ 1/(2\varphi )& 1/2& \varphi /2\end{array}\right)$.

$M$ is the rotation around the axis $(1,0,\varphi )$ by an angle of $2\pi /5$, counterclockwise. Let the linear transformations $T_0,\dots ,T_{11}$ be the transformations which send a point $(x,y,z)$ to the even permutations of $(\pm x,\pm y,\pm z)$ with an even number of minus signs. The transformations $T_i$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations $T_i{M}^j$ $(i=0,\dots ,11$, $j=0,\dots ,4)$ 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 $2(\xi +1)\sqrt{{\xi }^2-\xi }$, the circumradius equals $(\xi +1)\sqrt{2{\xi }^2-\xi }$, and the midradius equals ${\xi }^2+\xi$.

For a great snub icosidodecahedron whose edge length is 1, the circumradius is

$R=\frac{1}{2}\sqrt{\frac{2\xi -1}{\xi -1}}\approx 1.2744398820380232$

Its midradius is

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

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

Related polyhedra

Medial pentagonal hexecontahedron

Template:Uniform polyhedra db 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.

See also

• List of uniform polyhedra
• Inverted snub dodecadodecahedron

References

• Template:Citation

External links

• Template:Mathworld
• Template:Mathworld

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