Birectified 5-cube

Birectified 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Nit
Coxeter diagram	o4o3x3o3o ()
Elements
Tera	32 rectified pentachora, 10 icositetrachora
Cells	80 tetrahedra, 40+160 octahedra
Faces	320+320 triangles
Edges	480
Vertices	80
Vertex figure	Triangular-square duoprism, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {6}}{2}}\approx 1.22475$
Hypervolume	${\frac {31{\sqrt {2}}}{10}}\approx 4.38406$
Diteral angles	Rap–tet–rap: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Rap–oct–ico: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Ico–oct–ico: 90°
Central density	1
Number of external pieces	42
Level of complexity	6
Related polytopes
Army	Nit
Regiment	Nit
Dual	Triangular-square duotegmatic octacontateron
Conjugate	None
Abstract & topological properties
Flag count	23040
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The birectified 5-cube, also called the penteractitriacontaditeron or nit, is a convex uniform 5-polytope. It consists of 10 icositetrachora, and 32 rectified pentachora. 3 icositetrachora and 4 rectified pentachora join at each triangular-square duoprismatic vertex. It is the middle stage in the series of truncations between a regular 5-cube and its dual 5-orthoplex. It is also the rectified 5-demicube.

The birectified 5-cube contains the vertices of a square-cuboctahedral duoprism.

Vertex coordinates

The vertices of a birectified 5-cube of edge length 1 are given by all permutations of:

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

Representations

A birectified 5-cube has the following Coxeter diagrams:

• o4o3x3o3o () (full symmetry)
• o3x3o3o *b3o () (D₅ symmetry, rectified demipenteract)
• ooo4oxo3xox3ooo&#xt (B₄ axial, icositetrachoron-first)
• oxo3xox3oxo *b3ooo&#xt (D₄ axial, icositetrachoron-first)
• oxoo3xoxo3oxox3ooxo&#xt (A₄ axial, rectified pentachoron-first)
• ox(uoo)xo3xo(xox)ox3ox(oou)xo ox(ouo)xo&#xt (A₃×A₁ axial)
• qoo4oxo3oox oxo4ooq&#zx (B₂×B₂ symmetry)
• ox(ou)x(xo)oo3oo(xo)x(ou)xo ox(oo)x(oo)xo4oo(qo)o(qo)oo&#xt (B₃×A₂ axial, vertex-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The birectified 5-cube is the colonel of the largest known regiment of nonprismatic uniform polytera. Its regiment includes 149 main members, 138 fissaries, and one compound. Of these, 24 main members, 10 fissaries, and the compound have full symmetry, the rest have D₅ symmetry only.

External links

• Klitzing, Richard. "nit".
• Wikipedia contributors. "Birectified 5-cube".