Tetrahedral prism

Tetrahedral prism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Tepe
Coxeter diagram	x x3o3o ()
Tapertopic notation	121
Elements
Cells	2 tetrahedra, 4 triangular prisms
Faces	8 triangles, 6 squares
Edges	4+12
Vertices	8
Vertex figure	Triangular pyramid, edge lengths 1 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{4}}\approx 0.79057$
Hypervolume	${\frac {\sqrt {2}}{12}}\approx 0.11785$
Dichoral angles	Tet–3-trip: 90°
	Trip–4–trip: $\arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }$
Heights	Tet atop tet: 1
	Dyad atop trip: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Square atop ortho square: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	6
Level of complexity	4
Related polytopes
Army	Tepe
Regiment	Tepe
Dual	Tetrahedral tegum
Conjugate	None
Abstract & topological properties
Flag count	192
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3×A1, order 48
Convex	Yes
Nature	Tame

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).

Gallery

• 

  Segmentochoron display, tet atop tet

• 

  Segmentochoron display, dyad atop trip

• 

  Segmentochoron display, square atop orthogonal square

• 

  Net

• 

  Card with cell counts, verf, and cross-sections

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:

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

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)

External links

• Bowers, Jonathan. "Category 19: Prisms" (#889).

• Klitzing, Richard. "Tepe".
• Wikipedia contributors. "Tetrahedral prism".