# Icosidodecatruncated icosidodecahedron

The icosidodecatruncated icosidodecahedron or idtid, also called the icositruncated dodecadodecahedron, is a uniform polyhedron. It consists of 12 decagrams, 12 decagons, and 20 hexagons, with one of each type of face meeting per vertex.

Icosidodecatruncated icosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymIdtid
Coxeter diagramx5/3x3x5*a ()
Elements
Faces20 hexagons, 12 decagons, 12 decagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle, edge lengths 3, (5+5)/2, (5–5)/2
Measures (edge length 1)
Volume80
Dihedral angles10/3–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263^\circ}$
10–10/3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505^\circ}$
10–6: ${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ}$
Central density4
Number of external pieces152
Level of complexity15
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle \frac{3-\sqrt5}{2}}$ (decagons), 1 (ditrigon-rectangle)
RegimentIdtid
DualTridyakis icosahedron
ConjugateIcosidodecatruncated icosidodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count720
Euler characteristic-16
OrientableYes
Genus9
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

It can be alternated into the snub icosidodecadodecahedron after equalizing edge lengths.

## Vertex coordinates

An icosidodecatruncated icosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(±\frac12,\,±\frac{3-\sqrt5}{4},\,±\frac{1+3\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{5+\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±1,\,±\frac{\sqrt5-1}{2},\,±\frac{1+\sqrt5}{2}\right),}$
• ${\displaystyle \left(±\frac32,\,±\frac{3-\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{3\sqrt5-1}{4},\,±\frac{3+\sqrt5}{4}\right).}$

## Related polyhedra

o5/3o3o5*a truncations
Name OBSA CD diagram Picture