# Tetrahedral prism

Tetrahedral prism Rank4
TypeUniform
SpaceSpherical
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$\frac{\sqrt{10}}{4} ≈ 0.79057$ Hypervolume$\frac{\sqrt2}{12} ≈ 0.11785$ Dichoral anglesTet–3-trip: 90°
Trip–4–trip: $\arccos\left(\frac13\right) ≈ 70.52877^\circ$ HeightsTet atop tet: 1
Dyad atop trip: $\frac{\sqrt6}{3} ≈ 0.81650$ Square atop ortho square: $\frac{\sqrt2}{2} ≈ 0.70711$ 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:

• $\left(\frac{\sqrt2}4,\,\frac{\sqrt2}4,\,\frac{\sqrt2}4,\,±\frac12\right).$ ## 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)