Snub dodecadodecahedron



In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as $U40$. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol $sr{ 5/2,5},$ 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 $$\phi$$ 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, \phi)$$ by an angle of $$2\pi/5$$, counterclockwise. Let the linear transformations $$T_0, \ldots, 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,\ldots, 11$$, $$j = 0,\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(\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
 * $$R = \frac12\sqrt{\frac{2\xi-1}{\xi-1}} \approx 1.2744398820380232$$

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

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