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}.

Small snub icosicosidodecahedron
TypeUniform star polyhedron
ElementsF = 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 groupIh, [5,3], *532
Index referencesU32, C41, W110
Dual polyhedronSmall hexagonal hexecontahedron
Vertex figure
35.5/2
Bowers acronymSeside
3D model of a small snub icosicosidodecahedron

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

(±(1-ϕ+α), 0, ±(3+ϕα))
(±(ϕ-1+α), ±2, ±(2ϕ-1+ϕα))
(±(ϕ+1+α), ±2(ϕ-1), ±(1+ϕα))

where ϕ = (1+5)/2 is the golden ratio and α = 3ϕ−2.

See also

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


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