# Parabiaugmented truncated dodecahedron

Parabiaugmented truncated dodecahedron
Rank3
TypeCRF
Notation
Bowers style acronymPabautid
Elements
Faces
Edges6×10+3×20
Vertices3×10+20+20
Vertex figures10 isosceles trapezoids, edge length 1, 2, (1+5)/2, 2
20 irregular tetragons, edge length 1, 2, 1, (5+5)/2
40 isosceles triangles, edge lengths 1, 2+2, 2+2
Measures (edge length 1)
Volume${\displaystyle {\frac {515+251{\sqrt {5}}}{12}}\approx 89.68776}$
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 pieces52
Level of complexity24
Related polytopes
ArmyPabautid
RegimentPabautid
DualParabirhombirhombistellated triakis icosahedron
ConjugateParabiaugmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(10)×A1)/2, order 20
Flag orbits24
ConvexYes
NatureTame

The parabiaugmented truncated dodecahedron (OBSA: pabautid) is one of the 92 Johnson solids (J69). It consists of 10+10+10 triangles, 10 squares, 2 pentagons, and 10 decagons. It can be constructed by attaching two pentagonal cupolas to two opposite decagonal faces of the truncated dodecahedron.

## Vertex coordinates

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

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

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