# Parabiaugmented hexagonal prism

Parabiaugmented hexagonal prism
Rank3
TypeCRF
Notation
Bowers style acronymPabauhip
Coxeter diagramoxxxo oxuxo&#xt
Elements
Faces
Edges2+4+4+8+8
Vertices2+4+8
Vertex figures2 squares, edge length 1
8 irregular tetragons, edge lengths 1, 1, 2, 3
4 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {2{\sqrt {2}}+9{\sqrt {3}}}{6}}\approx 3.06948}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {7+2{\sqrt {6}}}{12}}}\right)\approx 174.73561^{\circ }}$
3–6: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 120°
3–3: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
4–6: 90°
Central density1
Number of external pieces14
Level of complexity13
Related polytopes
ArmyPabauhip
RegimentPabauhip
DualParalaterobitruncated hexagonal tegum
ConjugateParabiaugmented hexagonal prism
Abstract & topological properties
Flag count104
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK3, order 8
Flag orbits13
ConvexYes
NatureTame

The parabiaugmented hexagonal prism (OBSA: pabauhip) is one of the 92 Johnson solids (J55). It consists of 4+4 triangles, 4 squares, and 2 hexagons. It can be constructed by attaching square pyramids to two opposite square faces of the hexagonal prism.

## Vertex coordinates

A parabiaugmented hexagonal prism of edge length 1 has the following vertices:

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