# Tetrahedral prism

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Tetrahedral prism
Rank4
TypeUniform
Notation
Bowers style acronymTepe
Coxeter diagramx x3o3o ()
Tapertopic notation121
Elements
Cells2 tetrahedra, 4 triangular prisms
Faces8 triangles, 6 squares
Edges4+12
Vertices8
Vertex figureTriangular pyramid, edge lengths 1 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{12}}\approx 0.11785}$
Dichoral anglesTet–3-trip: 90°
Trip–4–trip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
HeightsTet atop tet: 1
Dyad atop trip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Square atop ortho square: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces6
Level of complexity4
Related polytopes
ArmyTepe
RegimentTepe
DualTetrahedral tegum
ConjugateNone
Abstract & topological properties
Flag count192
Euler characteristic0
OrientableYes
Properties
SymmetryA3×A1, order 48
ConvexYes
NatureTame

The tetrahedral prism or tepe is a prismatic uniform polychoron that consists of 2 tetrahedra and 4 triangular prisms. Each vertex joins 1 tetrahedron and 3 triangular prisms. As the name suggests, it is a prism based on a tetrahedron, and as such is also a segmentochoron (designated K-4.9 in Richard Klitzing's list).

## Vertex coordinates

The vertices of a tetrahedral prism of edge length 1 are given by all even sign changes of the first three coordinates of:

• ${\displaystyle \left({\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A tetrahedral prism has the following Coxeter diagrams:

• x x3o3o () (full symmetry)
• x2s4o3o () (prism of alternated cube)
• x2s2s4o () (prism of alternated square prism)
• x2s2s2s () (prism of alternated cuboid)
• xx3oo3oo&#x (bases considered separate)
• xx ox3oo&#x (A2×A1 axial, dyad atop triangular prism)
• xx xo ox&#x (A1×A1×A1 axial, square atop orthogonal square)
• oox xxx&#x (base has one symmetry axis only)
• xxxx&#x (irregular bases)
• xxoo ooxx&#xr (A1×A1 axial)
• oxxo3oooo&#xr (A2 axial)