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.^[1] As the name indicates, it belongs to the family of snub polyhedra.

The circumradius of the snub icosidodecadodecahedron with unit edge length is ${\frac {1}{2}}{\sqrt {\frac {2\rho -1}{\rho -1}}},$ where $ρ$ is the plastic constant, or the unique real root of $ρ 3 = ρ + 1$ .^[2]

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