Small snub icosicosidodecahedron

Small snub icosicosidodecahedron

Type	Uniform star polyhedron
Elements	F = 112, E = 180 V = 60 (χ = −8)
Faces by sides	(40+60){3}+12{5/2}
Coxeter diagram
Wythoff symbol	\| 5/2 3 3
Symmetry group	I_h, [5,3], *532
Index references	U₃₂, C₄₁, W₁₁₀
Dual polyhedron	Small hexagonal hexecontahedron
Vertex figure	3⁵.5/2
Bowers acronym	Seside

In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U₃₂. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, $ß{3,5}.$

The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.

Convex hull

Its convex hull is a nonuniform truncated icosahedron.

Truncated icosahedron (regular faces)	Convex hull (isogonal hexagons)	Small snub icosicosidodecahedron

Cartesian coordinates

Let $\xi =-{\frac {3}{2}}+{\frac {1}{2}}{\sqrt {1+4\phi }}\approx -0.1332396008261379$ be largest (least negative) zero of the polynomial $P=x^{2}+3x+\phi ^{-2}$ , where $\phi$ is the golden ratio. Let the point $p$ be given by

p={\begin{pmatrix}\phi ^{-1}\xi +\phi ^{-3}\\\xi \\\phi ^{-2}\xi +\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 small snub icosicosidodecahedron. The edge length equals $-2\xi$ , the circumradius equals ${\sqrt {-4\xi -\phi ^{-2}}}$ , and the midradius equals ${\sqrt {-\xi }}$ .