# Augmented triangular prism

Augmented triangular prism
Rank3
TypeCRF
Notation
Bowers style acronymAutip
Coxeter diagramoxx oxo&#xt
Elements
Faces
Edges1+2+2+4+4
Vertices1+2+4
Vertex figures1 square, edge length 1
4 trapezoids, edge lengths 1, 1, 1, 2
2 isosceles triangles, edge lengths 1, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {2{\sqrt {2}}+3{\sqrt {3}}}{12}}\approx 0.66871}$
Dihedral angles3–3 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°
4–4: 60°
Central density1
Number of external pieces8
Level of complexity13
Related polytopes
ArmyAutip
RegimentAutip
DualLateromonotruncated triangular tegum
ConjugateExcavated triangular prism
Abstract & topological properties
Flag count52
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
Flag orbits13
ConvexYes
NatureTame

The augmented triangular prism (OBSA: autip) is one of the 92 Johnson solids (J49). It consists of 2+2+2 triangles and 2 squares. It can be constructed by attaching a square pyramid to one of the square faces of the triangular prism.

The augmented triangular prism (with theoretical edge length 1) is the vertex figure of the digonal-triangular duoantiprism, which cannot be made uniform, because the Johnson solid variant is not circumscribable.

## Vertex coordinates

An augmented triangular prism of edge length 1 has the following vertices:

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

## In vertex figures

The augmented triangular prism appears as the vertex figure of the nonuniform digonal-triangular duoantiprism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson augmented triangular prism has no circumscribed sphere.

Variants made by changing the edge opposite to the vertex also appear as the vertex figure of the nonuniform duoantiprisms made out of a dyad and a regular polygon.