Snub icosidodecadodecahedron

Snub icosidodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 104, E = 180 V = 60 (χ = −16)
Faces by sides	(20+60){3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol	| 5/3 3 5
Symmetry group	I, [5,3]+, 532
Index references	U46, C58, W112
Dual polyhedron	Medial hexagonal hexecontahedron
Vertex figure	3.3.3.5.3.5/3
Bowers acronym	Sided

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.

3D model of a snub icosidodecadodecahedron

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 $\phi$ 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 $(1,0,\phi )$ 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 icosidodecadodecahedron. The edge length equals $2{\sqrt {\phi ^{2}\rho ^{2}-2\phi -1}}$, the circumradius equals ${\sqrt {(\phi +2)\rho ^{2}+\rho -3\phi -1}}$, and the midradius equals ${\sqrt {\rho ^{2}+\rho -\phi }}$.

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

Medial hexagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 180 V = 104 (χ = −16)
Symmetry group	I, [5,3]+, 532
Index references	DU46
dual polyhedron	Snub icosidodecadodecahedron

3D model of a medial hexagonal hexecontahedron

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

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

External links

• Weisstein, Eric W. "Medial hexagonal hexecontahedron". MathWorld.