Snub icosidodecadodecahedron

Template:Short description Template:Uniform polyhedra db File:Snub icosidodecadodecahedron.stl In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₄₆. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.

Cartesian coordinates

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

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

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

For a snub icosidodecadodecahedron whose edge length is 1, the circumradius is

$R=\frac{1}{2}\sqrt{{\rho }^2+\rho +2}\approx 1.126897912799939$

Its midradius is

$r=\frac{1}{2}\sqrt{{\rho }^2+\rho +1}\approx 1.0099004435452335$

Related polyhedra

Medial hexagonal hexecontahedron

Template:Uniform polyhedra db File:Medial hexagonal hexecontahedron.stl The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

See also

• List of uniform polyhedra

References

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External links

• Template:Mathworld

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