# Small dodecicosidodecahedron

Small dodecicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Coxeter diagramx3/2o5x5*a ()
Elements
Faces20 triangles, 12 pentagons, 12 decagons
Edges60+60
Vertices60
Vertex figureCrossed isosceles trapezoid, edge lengths 1, (5+5)/2, (1+5)/2, (5+5)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295}$
Volume${\displaystyle 2\frac{45+17\sqrt5}{3} ≈ 55.34210}$
Dihedral angles5–10: ${\displaystyle \arccos\left(\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ}$
3–10: ${\displaystyle \arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737^\circ}$
Central density2
Number of external pieces152
Level of complexity9
Related polytopes
ArmySrid
RegimentSrid
DualSmall dodecacronic hexecontahedron
ConjugateGreat dodecicosidodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small dodecicosidodecahedron, or saddid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagons, and 12 decagons. One triangle, one pentagon, and two decagons join at each vertex.

It is a faceting of the small rhombicosidodecahedron, using its 12 pentagons and 20 triangles along with 12 additional decagons.

## Vertex coordinates

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

## Representations

A small dodecicosidodecahedron has the following Coxeter diagrams:

• x3/2o5x5*a
• x5o3ß (, as holosnub)