# Truncated great dodecahedron

Truncated great dodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTigid
Coxeter diagramo5/2x5x (       )
Elements
Faces12 pentagrams, 12 decagons
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths (5–1)/2, (5+5)/2, (5+5)/2 Measures (edge length 1)
Circumradius$\sqrt{\frac{17+5\sqrt5}{8}} ≈ 1.87684$ Volume$11\frac{5+3\sqrt5}{4} ≈ 32.19756$ Dihedral angles10–5/2: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ 10–10: $\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°$ Central density3
Number of pieces72
Level of complexity7
Related polytopes
ArmySemi-uniform Ti
RegimentTigid
DualSmall stellapentakis dodecahedron
ConjugateQuasitruncated small stellated dodecahedron
Convex coreDodecahedron
Abstract properties
Euler characteristic-6
Topological properties
OrientableYes
Genus4
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The truncated great dodecahedron, or tigid, also called the great truncated dodecahedron, is a uniform polyhedron. It consists of 12 pentagrams and 12 decagons. Each vertex joins one pentagram and two decagons. As the name suggests, it can be obtained by the truncation of the great dodecahedron.

## Vertex coordinates

A truncated great dodecahedron of edge length 1 has vertex coordinates given by all permutations of:

• $\left(±\frac{\sqrt5-1}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$ plus all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{5+\sqrt5}{4}\right),$ • $\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right).$ ## Related polyhedra

o5o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great dodecahedron gad {5,5/2} x5o5/2o (     )
Truncated great dodecahedron tigid t{5,5/2} x5x5/2o (     )
Dodecadodecahedron did r{5,5/2} o5x5/2o (     )
Truncated small stellated dodecahedron (degenerate, triple cover of doe) t{5/2,5} o5x5/2x (     )
Small stellated dodecahedron sissid {5/2,5} o5o5/2x (     )
Rhombidodecadodecahedron raded rr{5,5/2} x5o5/2x (     )
Truncated dodecadodecahedron (degenerate, sird+12(10/2)) tr{5,5/2} x5x5/2x (     )
Snub dodecadodecahedron siddid sr{5,5/2} s5s5/2s (     )