Small snub icosicosidodecahedron

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Small snub icosicosidodecahedron
Small snub icosicosidodecahedron.png
Type Uniform star polyhedron
Elements F = 112, E = 180
V = 60 (χ = 8)
Faces by sides(40+60){3}+12{5/2}
Coxeter diagram CDel label5-2.pngCDel branch hh.pngCDel split2.pngCDel node h.png
Wythoff symbol | 5/2 3 3
Symmetry group Ih, [5,3], *532
Index references U 32, C 41, W 110
Dual polyhedron Small hexagonal hexecontahedron
Vertex figure Small snub icosicosidodecahedron vertfig.png
35.5/2
Bowers acronym Seside
3D model of a small snub icosicosidodecahedron Small snub icosicosidododecahedron.stl
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}.

Contents

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

Cartesian coordinates

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