# Small rhombidodecahedron

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Small rhombidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymSird
Coxeter diagramx5/2x5x -12{10/2}
Elements
Faces30 squares, 12 decagons
Edges60+60
Vertices60
Vertex figureButterfly, edge lengths 2 and (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Dihedral angles4–10 #1: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
4–10 #2: ${\displaystyle \arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }}$
Central densityodd
Number of external pieces150
Level of complexity10
Related polytopes
ArmySrid
RegimentSrid
DualSmall rhombidodecacron
ConjugateGreat rhombidodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–18
OrientableNo
Genus20
Properties
SymmetryH3, order 120
Flag orbits4
ConvexNo
NatureTame

The small rhombidodecahedron, or sird, is a uniform polyhedron. It consists of 30 squares and 12 decagons. Two squares and two decagons meet at each vertex..

It is a faceting of the small rhombicosidodecahedron, using its 30 squares along with the 12 decagons of the small dodecicosidodecahedron.

The truncated dodecadodecahedron (x5/2x5x) is a degenerate polyhedron with 12 decagons, 30 squares, and 12 doubly-wound pentagons. If those pentagons are blended out, the result is the small rhombidodecahedron.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the small rhombicosidodecahedron.

## Representations

A small rhombidodecahedron has representations as two reduced Coxeter diagrams:

• x5/2x5x -12{10/2}
• x3/2x5x -20{6/2}