Tetrahedral tegum

Tetrahedral tegum
(OFF file)
Rank	4
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Tete
Coxeter diagram	oxo3ooo3ooo&#xt
Elements
Cells	8 tetrahedra
Faces	4+12 triangles
Edges	6+8
Vertices	2+4
Vertex figures	2 tetrahedra, edge length 1
	4 skewed triangular tegums, edge length 1
Measures (edge length 1)
Inradius	$\frac{\sqrt{10}}{16} ≈ 0.19764$
Hypervolume	$\frac{\sqrt5}{48} ≈ 0.046584$
Dichoral angles	Tet–3–tet equatorial: $\arccos\left(-\frac78\right) ≈ 151.04498°$
	Tet–3–tet pyramidal: $\arccos\left(\frac14\right) ≈ 52.52249°$
Height	$\frac{\sqrt{10}}{2} ≈ 1.58114$
Central density	1
Related polytopes
Army	Tete
Regiment	Tete
Dual	Sem-uniform Tetrahedral prism
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3×A1, order 48
Convex	Yes
Nature	Tame

The tetrahedral tegum or tete, 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:

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

Representations

A tetrahedral tegum has the following Coxeter diagrams:

• oxo3ooo3ooo&#xt
• yo ox3oo3oo&#xt (y = $\frac{\sqrt{10}}{2}$, 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 $\frac12 = 0.5$ times the length of the edges of the base tetrahedron, and all the dichoral angles are $\arccos\left(-\frac15\right) ≈ 101.53696°$.

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "tete".