Birectified 6-simplex

Birectified 6-simplex
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Bril
Coxeter diagram	o3o3x3o3o3o ()
Elements
Peta	7 rectified hexatera, 7 dodecatera
Tera	21 pentachora, 21+42 rectified pentachora
Cells	35+105 tetrahedra, 105 octahedra
Faces	140+210 triangles
Edges	210
Vertices	35
Vertex figure	Triangular-tetrahedral duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {42}}{7}}\approx 0.92582$
Hypervolume	${\frac {151{\sqrt {7}}}{2880}}\approx 0.13872$
Dipetal angles	Dot–rap–rix: $\arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }$
	Dot–rap–dot: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
	Rix–pen–rix: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
Height	${\frac {\sqrt {21}}{6}}\approx 0.76376$
Central density	1
Number of external pieces	14
Level of complexity	10
Related polytopes
Army	Bril
Regiment	Bril
Conjugate	None
Abstract & topological properties
Flag count	50400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

The birectified 6-simplex (also called the birectified heptapteon or bril) is a convex uniform 6-polytope. It has 7 rectified hexatera and 7 dodecatera as facets. 3 rectified hexatera and 4 dodecatera join at each triangular-tetrahedral duoprismatic vertex. As the name suggests, it is the birectification of the 6-simplex.

It is also a convex segmentotope, as a rectified hexateron atop dodecateron.

Vertex coordinates

The vertices of a birectified 6-simplex of edge length 1 can be given in seven dimensions as all permutations of:

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

Representations

A birectified 6-simplex has the following Coxeter diagrams:

• o3o3x3o3o3o () (full symmetry)
• oo3xo3ox3oo3oo&#x (A₅ axial, rectified hexateron atop dodecateron)
• oxo xoo3oxo3oox3ooo&#xt (A₄×A₁ axial, pentachoron-first)
• oxoo3ooxo oxoo3ooxo3ooox&#xt (A₃×A₂ axial, vertex-first)

Gallery

• 

  A₅ orthographic projection

• 

  A₄

• 

  A₃

• 

  A₂

Related polytopes

The birectified 6-simplex is the colonel of a regiment of 18 members.

External links

• Klitzing, Richard. "bril".
• Wikipedia contributors. "Birectified 6-simplex".