# Small inverted retrosnub icosicosidodecahedron

Small inverted retrosnub icosicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSirsid
Coxeter diagrams5/2s3/2s3/2*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${\displaystyle \frac{\sqrt{13+3\sqrt5-\sqrt{102+46\sqrt5}}}{4} \approx 0.58069}$
Volume${\displaystyle \frac{45+39\sqrt5-5\sqrt{302+150\sqrt5}}{12} \approx 0.49764}$
Dihedral angles5/2–3: ${\displaystyle \arccos\left(\sqrt{\frac{15-2\sqrt5-2\sqrt{30\sqrt5-65}}{15}}\right) \approx 44.45753^\circ}$
3–3: ${\displaystyle \arccos\left(\frac{\sqrt{3+2\sqrt5}}{3}\right) \approx 24.33196^\circ}$
Central density38
Number of external pieces3060
Level of complexity213
Related polytopes
ArmySemi-uniform Tid, edge lengths ${\displaystyle \frac{\sqrt{3+2\sqrt5}-\sqrt5}{2}}$ (triangles), ${\displaystyle \frac{3-\sqrt{3+2\sqrt5}}{2}}$ (between dipentagons)
RegimentSirsid
DualSmall hexagrammic hexecontahedron
ConjugateSmall snub icosicosidodecahedron
Convex coreOrder-6 truncated pentakis dodecahedron
Abstract & topological properties
Flag count720
Euler characteristic–8
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

In terms of level of complexity, this is the most complex uniform polyhedron.

## Vertex coordinates

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

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

## Representations

A small inverted retrosnub icosicosidodecahedron has the following Coxeter diagrams:

• s3/2s3/2s5/2*a
• o5ß3/2ß (as holosnub)