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

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}.

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

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Its convex hull is a nonuniform truncated icosahedron.

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

Cartesian coordinates

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Let   be largest (least negative) zero of the polynomial  , where   is the golden ratio. Let the point   be given by

 .

Let the matrix   be given by

 .

  is the rotation around the axis   by an angle of  , counterclockwise. Let the linear transformations   be the transformations which send a point   to the even permutations of   with an even number of minus signs. The transformations   constitute the group of rotational symmetries of a regular tetrahedron. The transformations    ,   constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points   are the vertices of a small snub icosicosidodecahedron. The edge length equals  , the circumradius equals  , and the midradius equals  .

For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is

 

Its midradius is

 

The other zero of   plays a similar role in the description of the small retrosnub icosicosidodecahedron.

See also

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