Pentachoric tegum

Pentachoric tegum
Rank	5
Type	CRF
Notation
Bowers style acronym	Penit
Coxeter diagram	oxo3ooo3ooo3ooo&#xt
Elements
Tera	10 pentachora
Cells	5+20 tetrahedra
Faces	10+20 triangles
Edges	10+10
Vertices	2+5
Vertex figures	2 pentachora, edge length 1
	5 skewed tetrahedral tegums, edge length 1
Measures (edge length 1)
Inradius	${\frac {\sqrt {15}}{25}}\approx 0.15492$
Hypervolume	${\frac {\sqrt {3}}{240}}\approx 0.0072169$
Diteral angles	Pen–tet–pen equatorial: $\arccos \left(-{\frac {23}{25}}\right)\approx 156.92608^{\circ }$
	Pen–tet–pen pyramidal: $\arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }$
Central density	1
Related polytopes
Army	Penit
Regiment	Penit
Dual	Semi-uniform pentachoric prism
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A4×A1, order 240
Convex	Yes
Nature	Tame

The pentachoric tegum, also called the pentachoric bipyramid, is a Blind polytope and CRF polyteron with 10 identical regular pentachora as tera. As the name suggests, it is a tegum based on the pentachoron, formed by attaching two regular hexatera at a common facet.

It is part of an infinite family of Blind polytopes known as the simplicial bipyramids. It is one of two non-uniform Blind polytopes in five dimensions, the other being the hexadecachoric pyramid.

Vertex coordinates

The vertices of a pentachoric tegum of edge length 1, centered at the origin, are given by:

• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,0\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,0\right),$
• $\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,0\right),$
• $\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,0\right),$
• $\left(0,\,0,\,0,\,0,\,\pm {\frac {\sqrt {15}}{5}}\right).$

Variations

The pentachoric tegum can have the heights of its pyramids varied while maintaining its full symmetry These variants generally have 10 non-CRF tetrahedral pyramids as tera.

One notable variation can be obtained as the dual of the uniform pentachoric prism, which can be represented by m2m3o3o3o.

External links

• Klitzing, Richard. "penit".