Biaugmented triangular prism

Biaugmented triangular prism
Rank3
TypeCRF
Notation
Bowers style acronymBautip
Elements
Faces
Edges1+2+2+4+4+4
Vertices2+2+4
Vertex figures2 squares, edge length 1
4 trapezoids, edge lengths 1, 1, 1, 2
2 pentagons, edge length 1
Measures (edge length 1)
Volume${\displaystyle {\frac {4{\sqrt {2}}+3{\sqrt {3}}}{12}}\approx 0.90442}$
Dihedral angles3–3 double join: ${\displaystyle \arccos \left(-{\frac {1+2{\sqrt {6}}}{6}}\right)\approx 169.47122^{\circ }}$
3–3 single join: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–4 join: ${\displaystyle \arccos \left(-{\frac {3{\sqrt {2}}-{\sqrt {3}}}{6}}\right)\approx 114.73561^{\circ }}$
3–3 pyramidal: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
3–4 prismatic: 90°
Central density1
Number of external pieces11
Level of complexity17
Related polytopes
ArmyBautip
RegimentBautip
DualLaterobitruncated triangular tegum
ConjugateBiaugmented triangular prism
Abstract & topological properties
Flag count68
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
Flag orbits17
ConvexYes
NatureTame

The biaugmented triangular prism (OBSA: bautip) is one of the 92 Johnson solids (J50). It consists of 2+2+2+4 triangles and 1 square. It can be constructed by attaching square pyramids to two of the square faces of the triangular prism.

Vertex coordinates

A biaugmented triangular prism of edge length 1 has the following vertices:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {6}}}{4}},\,{\frac {3{\sqrt {2}}+{\sqrt {3}}}{12}},\,0\right)}$.

In vertex figures

Variants of the biaugmented triangular prism (in its symmetry axis, with non-base connected edge lengths of 2) by changing the edge opposite to the square appear as the vertex figure of the nonuniform antiditetragoltriate, and has no corealmic realization.