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	Ih, [5,3], *532
Index references	U32, C41, W110
Dual polyhedron	Small hexagonal hexecontahedron
Vertex figure	35.5/2
Bowers acronym	Seside

3D model of a small snub icosicosidodecahedron

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

Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the even permutations of

$${\begin{array}{clllc}{\Bigl (}&\pm {\bigl [}1-\varphi +\alpha {\bigr ]},&\ \ \ \,0,&\pm {\bigl [}3+\varphi \alpha {\bigr ]}&{\Bigr )}\\{\Bigl (}&\pm {\bigl [}\varphi -1+\alpha {\bigr ]},&\pm \,2,&\pm {\bigl [}2\varphi -1+\varphi \alpha {\bigr ]}&{\Bigr )}\\{\Bigl (}&\pm {\bigl [}\varphi +1+\alpha {\bigr ]},&\pm \,2{\bigl [}\varphi -1{\bigr ]},&\pm {\bigl [}1+\varphi \alpha {\bigr ]}&{\Bigr )}\end{array}}$$

where $\varphi ={\tfrac {1+{\sqrt {5}}}{2}}$ is the golden ratio and $\alpha ={\sqrt {3\varphi -2}}.$

See also

• List of uniform polyhedra
• Small retrosnub icosicosidodecahedron

External links

• Weisstein, Eric W. "Small snub icosicosidodecahedron". MathWorld.
• Klitzing, Richard. "3D star small snub icosicosidodecahedron".