# Augmented truncated dodecahedron

Augmented truncated dodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymAutid
Elements
Faces
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${\displaystyle {\frac {505+243{\sqrt {5}}}{12}}\approx 87.37361}$
Dihedral angles3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {23+3{\sqrt {5}}}{30}}}\right)\approx 174.34011^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
3–10 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {65-2{\sqrt {5}}}{75}}}\right)\approx 153.94242^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
3–10 tid: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
10–10: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces42
Level of complexity42
Related polytopes
ArmyAutid
RegimentAutid
DualRhombirhombistellated triakis icosahedron
ConjugateAugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Flag count420
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
Flag orbits42
ConvexYes
NatureTame

The augmented truncated dodecahedron (OBSA: autid) 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:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)}$,

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