Birectified 7-cube

Birectified 7-cube
Rank	7
Type	Uniform
Notation
Bowers style acronym	Bersa
Coxeter diagram	o4o3x3o3o3o3o ()
Elements
Exa	128 rectified heptapeta, 14 birectified hexeracts
Peta	448 hexatera, 896 rectified hexatera, 84 penteractitriacontaditera
Tera	2688 pentachora, 2688 rectified pentachora, 280 icositetrachora
Cells	6720 tetrahedra, 560+4480 octahedra
Faces	4480+8960 triangles
Edges	6720
Vertices	672
Vertex figure	Square-pentachoric duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{2}}\approx 1.58114$
Hypervolume	${\frac {4919{\sqrt {2}}}{630}}\approx 11.04209$
Diexal angles	Ril–hix–ril: $\arccos \left(-{\frac {5}{7}}\right)\approx 135.58470^{\circ }$
	Brox–rix–ril: $\arccos \left(-{\frac {\sqrt {7}}{7}}\right)\approx 112.20765^{\circ }$
	Brox–nit–brox: 90°
Central density	1
Number of external pieces	142
Level of complexity	15
Related polytopes
Army	Bersa
Regiment	Bersa
Conjugate	None
Abstract & topological properties
Flag count	9676800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B7, order 645120
Convex	Yes
Nature	Tame

The birectified hepteract, or bersa, also called the birectified 7-cube, is a convex uniform polyexon. It consists of 14 birectified hexeracts and 128 rectified heptapeta. 4 rectified heptapeta and 5 birectified hexeracts join at each square-pentachoric duoprismatic vertex. As the name suggests, it is the birectification of the hepteract. It is also the rectified demihepteract.

The birectified hepteract contains the vertices of an inscribed square-rectified penteractic duoprism and cuboctahedral-rectified tesseractic duoprism.

Vertex coordinates

The vertices of a birectified hepteract of edge length 1 are given by all permutations of:

• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right).$

Representations

A birectified hepteract has the following Coxeter diagrams:

• o4o3x3o3o3o3o (full symmetry)
• o3x3o *b3o3o3o3o (D7 symmetry, rectified demihepteract)
• ooo4oxo3xox3ooo3ooo3ooo&#xt (B6 axial, birectified hexeract-first)

External links

• Klitzing, Richard. "bersa".
• Wikipedia contributors. "Birectified 7-cube".