# Triaugmented truncated dodecahedron

Triaugmented truncated dodecahedron Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymTautid
Elements
Faces1+1+3+3+3+6+6+6+6 triangles, 3+6+6 squares, 3 pentagons, 3+3+3 decagons
Edges9×3+18×6
Vertices3+3+3+3+3+6+6+6+6+6+6+6+6+6+6
Vertex figures15 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
30 irregular tetragons, edge length 1, 2, 1, (5+5)/2
30 isosceles triangles, edge lengths 1, 2+2, 2+2
Measures (edge length 1)
Volume$7\frac{75+37\sqrt5}{12} ≈ 92.01180$ 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
ArmyTautid
RegimentTautid
DualTrirhombirhombistellated triakis icosahedron
ConjugateTriaugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×I, order 6
ConvexYes
NatureTame

The triaugmented truncated dodecahedron is one of the 92 Johnson solids (J71). It consists of 1+1+3+3+3+6+6+6+6 triangles, 3+6+6 squares, 3 pentagons, and 3+3+3 decagons. It can be constructed by attaching pentagonal cupolas to three mutually non-adjacent decagonal faces of the truncated dodecahedron.

## Vertex coordinates

A triaugmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of:

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