Small retrosnub icosicosidodecahedron

Small retrosnub 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 | 3/2 3/2 5/2
Symmetry group Ih, [5,3], *532
Index references U72, C91, W118
Dual polyhedron Small hexagrammic hexecontahedron
Vertex figure
(35.5/3)/2
Bowers acronym Sirsid

In geometry, the small retrosnub icosicosidodecahedron (also known as a retrosnub disicosidodecahedron, small inverted retrosnub icosicosidodecahedron, or retroholosnub icosahedron) is a nonconvex uniform polyhedron, indexed as U72. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{⁵/₃,³/₂}.

3D model of a small retrosnub 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.

George Olshevsky nicknamed it the yog-sothoth (after the Cthulhu Mythos deity).[2][3]

Convex hull

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Its convex hull is a nonuniform truncated dodecahedron.

 
Truncated dodecahedron
 
Convex hull
 
Small retrosnub icosicosidodecahedron

Cartesian coordinates

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Let   be the smallest (most 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 snub icosicosidodecahedron.

See also

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References

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  1. ^ Maeder, Roman. "72: small retrosnub icosicosidodecahedron". MathConsult.
  2. ^ Birrell, Robert J. (May 1992). The Yog-sothoth: analysis and construction of the small inverted retrosnub icosicosidodecahedron (M.S.). California State University.
  3. ^ Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.
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