# Small snub icosicosidodecahedron: Difference between revisions

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Small snub icosicosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymSeside
Coxeter diagrams5/2s3s3*a ()
Elements
Faces60 triangles, 40 triangles as 20 hexagrams, 12 pentagrams
Edges60+60+60
Vertices60
Vertex figureMirror-symmetric hexagon, edge lengths 1, 1, 1, 1, 1, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+3{\sqrt {5}}+{\sqrt {102+46{\sqrt {5}}}}}}{4}}\approx 1.45819}$
Volume${\displaystyle {\frac {45+39{\sqrt {5}}+5{\sqrt {302+150{\sqrt {5}}}}}{12}}\approx 21.53680}$
Dihedral angles5/2–3: ${\displaystyle \arccos \left(-{\sqrt {\frac {15-2{\sqrt {5}}+2{\sqrt {30{\sqrt {5}}-65}}}{15}}}\right)\approx 161.02258^{\circ }}$
3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {3+2{\sqrt {5}}}}{3}}\right)\approx 155.66804^{\circ }}$
Central density2
Number of external pieces212
Level of complexity15
Related polytopes
ArmySemi-uniform ti, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons), ${\displaystyle {\frac {1-{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{4}}}$ (between ditrigons)
RegimentSeside
DualSmall hexagonal hexecontahedron
ConjugateSmall retrosnub icosicosidodecahedron
Convex coreTruncated pentakis dodecahedron
Abstract & topological properties
Flag count720
Euler characteristic-8
OrientableYes
Genus5
Properties
SymmetryH3, order 120
Flag orbits6
ConvexNo
NatureTame

The small snub icosicosidodecahedron, or seside, is a uniform polyhedron. It consists of 60 snub triangles, 40 more triangles that create 20 hexagrams due to pairs falling in the same plane, and 12 pentagrams. Five triangles and one pentagram meet at each vertex.

It can be obtained as a holosnub truncated icosahedron, after adjusting all edge lengths to be equal.

## Vertex coordinates

A small snub icosicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {1-{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {{\sqrt {5}}+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {{\sqrt {5}}-1+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {3+2{\sqrt {5}}}}}{4}},\,\pm {\frac {3+{\sqrt {5}}+{\sqrt {6{\sqrt {5}}-2}}}{8}}\right).}$

## Representations

A small snub icosicosidodecahedron has the following Coxeter diagrams:

• s5/2s3s3*a
• o5ß3ß (as holosnub)

## Related polyhedra

o5/2o3o3*a truncations
Name OBSA CD diagram Picture
Small ditrigonary icosidodecahedron sidtid x5/2o3o3*a ()
(degenerate, double cover of id) x5/2x3o3*a ()
(degenerate, double cover of ike) o5/2o3x3*a ()
Small icosicosidodecahedron siid x5/2o3x3*a ()
(degenerate, double cover of ti) x5/2x3x3*a ()
Small snub icosicosidodecahedron seside s5/2s3s3*a ()