# Augmented truncated dodecahedron

Augmented truncated dodecahedron Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAutid
Elements
Faces5+5+5+5+5 triangles, 5 squares, 1 pentagon, 1+5+5 decagons
Edges5+5+5+5+5+5+5+5+5+10+10+10+10+10+10
Vertices5+5+5+5+5+10+10+10+10
Vertex figures5 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
10 irregular tetragons, edge length 1, 2, 1, (5+5)/2
50 isosceles triangles, edge lengths 1, 2+2, 2+2
Measures (edge length 1)
Volume$\frac{505+243\sqrt5}{12} ≈ 87.37361$ Dihedral angles3–4 join: $\arccos\left(-\sqrt{\frac{23+3\sqrt5}{30}}\right) ≈ 174.34011°$ 3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$ 3–10 join: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$ 4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ 3–10 tid: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$ 10–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ Central density1
Related polytopes
ArmyAutid
RegimentAutid
DualRhombirhombistellated triakis icosahedron
ConjugateAugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The augmented truncated dodecahedron is one of the 92 Johnson solids (J68). It consists of 5+5+5+5+5 triangles, 5 squares, 1 pentagon, and 1+5+5 decagons. It can be constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron.

## Vertex coordinates

An augmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{5+3\sqrt5}{4}\right),$ • $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$ • $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$ • $\left(±\frac12,\,\frac{15+13\sqrt5}{20},\,3\frac{5+\sqrt5}{10}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\frac{25+13\sqrt5}{20},\,\frac{25+\sqrt5}{20}\right),$ • $\left(0,\,\frac{10+9\sqrt5}{10},\,\frac{15+\sqrt5}{20}\right).$ 