Small dodecicosidodecahedron

Small dodecicosidodecahedron
Rank3
TypeUniform
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${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle 2{\frac {45+17{\sqrt {5}}}{3}}\approx 55.34210}$
Dihedral angles5–10: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
3–10: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
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
Flag orbits4
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)