# Triaugmented triangular prism

Triaugmented triangular prism
Rank3
TypeCRF
Notation
Bowers style acronymTautip
Conway notationk4P3
Elements
Faces2+6+6 triangles
Edges3+6+12
Vertices3+6
Vertex figures3 squares, edge length 1
6 pentagons, edge length 1
Measures (edge length 1)
Volume${\displaystyle {\frac {2{\sqrt {2}}+{\sqrt {3}}}{4}}\approx 1.14012}$
Dihedral angles3–3 double join: ${\displaystyle \arccos \left(-{\frac {1+2{\sqrt {6}}}{6}}\right)\approx 169.47122^{\circ }}$
3–3 single join: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–3 pyramidal: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density1
Number of external pieces14
Level of complexity7
Related polytopes
ArmyTautip
RegimentTautip
DualOrder-4 truncated triangular tegum
ConjugateTriaugmented triangular prism
Abstract & topological properties
Flag count84
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
SkeletonFritsch graph
Properties
SymmetryA2×A1, order 12
Flag orbits7
ConvexYes
NatureTame

The triaugmented triangular prism (OBSA: tautip) is one of the 92 Johnson solids (J51). It consists of 2+6+6 triangles as faces. It can be constructed by attaching square pyramids to all three of the square faces of the triangular prism.

Its dual is one possible realization of the 3D associahedron.

## Vertex coordinates

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

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