# Tetrahedral tegum

Tetrahedral tegum
Rank4
TypeCRF
Notation
Bowers style acronymTete
Coxeter diagramoxo3ooo3ooo&#xt
Elements
Cells8 tetrahedra
Faces4+12 triangles
Edges6+8
Vertices2+4
Vertex figure
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {10}}{16}}\approx 0.19764}$
Hypervolume${\displaystyle {\frac {\sqrt {5}}{48}}\approx 0.046584}$
Dichoral anglesTet–3–tet equatorial: ${\displaystyle \arccos \left(-{\frac {7}{8}}\right)\approx 151.04498^{\circ }}$
Tet–3–tet pyramidal: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 52.52249^{\circ }}$
Height${\displaystyle {\frac {\sqrt {10}}{2}}\approx 1.58114}$
Central density1
Related polytopes
ArmyTete
RegimentTete
DualSemi-uniform tetrahedral prism
ConjugateNone
Abstract & topological properties
Flag count192
Euler characteristic0
OrientableYes
Properties
SymmetryA3×A1, order 48
Flag orbits4
ConvexYes
NatureTame

The tetrahedral tegum, also called the tetrahedral bipyramid, is a CRF polychoron with 8 identical regular tetrahedra as cells. As such it is also a Blind polytope. As the name suggests, it is a tegum based on the tetrahedron, formed by attaching two regular pentachora at a common cell.

## Vertex coordinates

The vertices of a tetrahedral tegum of edge length 1 are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,0\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,0\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,0\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,\pm {\frac {\sqrt {10}}{4}}\right)}$.

## Representations

A tetrahedral tegum has the following Coxeter diagrams:

• oxo3ooo3ooo&#xt
• yo ox3oo3oo&#xt (y = ${\displaystyle {\frac {\sqrt {10}}{2}}}$, as full tegum)
• oyo oox3ooo&#xt (as triangular pyramidal tegum)

## Variations

The tetrahedral tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 8 non-CRF triangular pyramids as cells.

One notable variation can be obtained as the dual of the uniform tetrahedral prism, which can be represented by m2m3o3o (). In this variation the height between the top and bottom vertices of the tegum is exactly ${\displaystyle {\frac {1}{2}}=0.5}$ times the length of the edges of the base tetrahedron, and all the dichoral angles are ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$.