# Small inverted retrosnub icosicosidodecahedral prism

Small inverted retrosnub icosicosidodecahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSirsiddip
Coxeter diagramx2s5/2s3/2s3/2*b ()
Elements
Cells60 triangular prisms, 40 triangular prisms as 20 hexagrammic prisms, 12 pentagrammic prisms, 2 small inverted retrosnub icosicosidodecahedra
Faces120 triangles, 80 triangles as 40 hexagrams, 60+60+60 squares, 24 pentagrams
Edges60+120+120+120
Vertices120
Vertex figureMirror-symmetric hexagonal pyramid, edge lengths 1, 1, 1, 1, 1, (5–1)/2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{17+3\sqrt5-\sqrt{102+46\sqrt5}}}{4} ≈ 0.76629}$
Hypervolume${\displaystyle \frac{45+39\sqrt5-5\sqrt{302+150\sqrt5}}{12} ≈ 0.49764}$
Dichoral anglesSirsid–5/2–stip: 90°
Sirsid–3–trip: 90°
Stip–4–trip: ${\displaystyle \arccos\left(\sqrt{\frac{15-2\sqrt5-2\sqrt{30\sqrt5-65}}{15}}\right) \approx 44.45753^\circ}$
Trip–4–trip: ${\displaystyle \arccos\left(\frac{\sqrt{3+2\sqrt5}}{3}\right) ≈ 24.33196^\circ}$
Height1
Central density38
Number of pieces3062
Related polytopes
ArmySemi-uniform Tiddip
RegimentSirsiddip
DualSmall hexagrammic hexecontahedral tegum
Abstract properties
Euler characteristic–30
Topological properties
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexNo
NatureTame

The small inverted retrosnub icosicosidodecahedral prism or sirsiddip is a prismatic uniform polychoron that consists of 2 small inverted retrosnub icosicosidodecahedra, 12 pentagrammic prisms, and 40+60 triangular prisms (40 of which form compounds in the same hyperplane, with bases combining into hexagrams). Each vertex joins 1 small inverted retrosnub icosicosidodecahedron, 1 pentagrammic prism, and 5 triangular prisms. As the name suggests, it is a prism based on the small inverted retrosnub icosicosidodecahedron.

## Vertex coordinates

A small inverted retrosnub icosicosidodecahedral prism of edge length 1 has vertex coordinates given by all even permutations of the first three coordinates of:

• ${\displaystyle \left(0,\,±\frac{3-\sqrt{3+2\sqrt5}}{4},\,±\frac{\sqrt5-1+\sqrt{6\sqrt5-2}}{8},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt{3+2\sqrt5}-\sqrt5}{4},\,±\frac{1-\sqrt5+\sqrt{6\sqrt5-2}}{8},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt{3+2\sqrt5}-1}{4},\,±\frac{3+\sqrt5-\sqrt{6\sqrt5-2}}{8},\,±\frac12\right).}$