# Icosidodecatruncated icosidodecahedron

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: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$ 10–10/3: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ 10–6: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232°$ Central density4
Number of pieces152
Level of complexity15
Related polytopes
ArmySemi-uniform Grid
RegimentIdtid
DualTridyakis icosahedron
ConjugateIcosidodecatruncated icosidodecahedron
Convex coreDodecahedron
Abstract properties
Euler characteristic-16
Topological properties
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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.

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:

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

o5/3o3o5*a truncations
Name OBSA CD diagram Picture
Ditrigonary dodecadodecahedron ditdid   Small complex icosidodecahedron (degenerate, ike+gad) cid   Great complex icosidodecahedron (degenerate, sissid+gike) gacid   Icosidodecadodecahedron ided   Small ditrigonal dodecicosidodecahedron sidditdid   Great ditrigonal dodecicosidodecahedron gidditdid   Icosidodecatruncated icosidodecahedron idtid   Snub icosidodecadodecahedron sided   