Rectified hexateron

Rectified hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rix
Coxeter diagram	o3x3o3o3o ()
Elements
Tera	6 pentachora, 6 rectified pentachora
Cells	30 tetrahedra, 15 octahedra
Faces	20+60 triangles
Edges	60
Vertices	15
Vertex figure	Tetrahedral prism, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{3} ≈ 0.81650$
Hypervolume	$\frac{13\sqrt3}{240} ≈ 0.093819$
Diteral angles	Rap–tet–pen: $\arccos\left(-\frac15\right) ≈ 101.53696^\circ$
	Rap–oct–rap: $\arccos\left(\frac15\right) ≈ 78.46304^\circ$
Height	$\frac{\sqrt{15}}{5} ≈ 0.77460$
Central density	1
Number of external pieces	12
Level of complexity	4
Related polytopes
Army	Rix
Regiment	Rix
Dual	Joined hexateron
Conjugate	None
Abstract & topological properties
Flag count	2880
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The rectified hexateron, or rix, also called the rectified 5-simplex, is a convex uniform polyteron. It consists of 6 regular pentachora and 6 rectified pentachora. Two pentachora and 4 rectified pentachora join at each tetrahedral prismatic vertex. As the name suggests, it is the rectification of the hexateron.

It is also a convex segmentoteron, as a pentachoron atop rectified pentachoron.

Vertex coordinates

The vertices of a rectified hexateron of edge length 1 are given by:

• $\left(-\frac{2\sqrt{15}}{15},\,-\frac{\sqrt{10}}{5},\,0,\,0,\,0\right),$
• $\left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),$
• $\left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(-\frac{2\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),$
• $\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right),$
• $\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right),$
• $\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac12\right).$

Much simpler coordinates can be given in six dimensions, as all permutations of:

• $\left(\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0,\,0,\,0,\,0\right).$

Representations

A rectified hexateron has the following Coxeter diagrams:

• o3x3o3o3o (full symmetry)
• xo3ox3oo3oo&#x (A₄ axial, pentachoron atop rectified pentachoron)
• oxo oxo3oox3ooo&#xt (A₃×A₁ symmetry, vertex-first)
• x(oo)x3o(ox)o3o(oo)o&#xt (A₃ symmetry, tetrahedron-first)
• oooo3ooxo3oxox&#xr (A₃ symmetry)
• xoo3oxo oxo3oox&#xt (A₂×A₂ axial, triangle-first)
• oxoox ooxoo3oxoxo&#xr (A₂×A₁ symmetry)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

A rectified hexateron can be diminished by removing tetrahedral prismatic pyramids. 3 such pyramids from non-adjacent vertices can be removed to form a scaliform tridiminished rectified hexateron.

The rectified hexateron is the colonel of a 7-member regiment. Its other members include the cellibiprismatointercepted hexateron, facetorectified hexateron, cellihexateron, biprismatodishexateron, cellintercepted dishexateron, and spinobiprismatohexateron.

External links

• Bowers, Jonathan. "Category 3: Rectates" (#25).

• Klitzing, Richard. "rix".

• Wikipedia Contributors. "Rectified 5-simplex".