# Truncated great dodecahedron

Truncated great dodecahedron
Rank3
TypeUniform
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${\displaystyle {\sqrt {\frac {17+5{\sqrt {5}}}{8}}}\approx 1.87684}$
Volume${\displaystyle 11{\frac {5+3{\sqrt {5}}}{4}}\approx 32.19756}$
Dihedral angles10–5/2: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
10–10: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density3
Number of external pieces72
Level of complexity7
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons) and 1 (between ditrigons)
RegimentTigid
DualSmall stellapentakis dodecahedron
ConjugateQuasitruncated small stellated dodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count360
Euler characteristic-6
OrientableYes
Genus4
Properties
SymmetryH3, order 120
Flag orbits3
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:

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,

plus all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$.