Small rhombated hexateron

Small rhombated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sarx
Coxeter diagram	x3o3x3o3o ()
Elements
Tera	15 tetrahedral prisms, 6 rectified pentachora, 6 small rhombated pentachora
Cells	30 tetrahedra, 60 triangular prisms, 30 octahedra, 15 cuboctahedra
Faces	20+60+120 triangles, 90 squares
Edges	60+180
Vertices	60
Vertex figure	Triangular pyramidal prism, edge lengths 1 (base triangular prism and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{15}}{3} ≈ 1.29099$
Hypervolume	$\frac{193\sqrt3}{120} ≈ 2.78572$
Diteral angles	Rap–tet–tepe: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152^\circ$
	Srip–trip–tepe: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495^\circ$
	Srip–oct–rap: $\arccos\left(-\frac15\right) ≈ 101.53696^\circ$
	Srip–co–srip: $\arccos\left(\frac15\right) ≈ 78.46304^\circ$
Central density	1
Number of external pieces	27
Level of complexity	16
Related polytopes
Army	Sarx
Regiment	Sarx
Dual	Triangular-pyramidal tegmatic hexecontateron
Conjugate	None
Abstract & topological properties
Flag count	11520
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The small rhombated hexateron, or sarx, also called the cantellated 5-simplex, is a convex uniform polyteron. It consists of 15 tetrahedral prisms, 6 rectified pentachora, and 6 small rhombated pentachora. One rectified pentachoron, 2 tetrahedral prisms, and 3 small rhombated pentachora join at each vertex. As the name suggests, it is the cantellation of the hexateron.

Vertex coordinates

The vertices of a small rhombated hexateron of edge length 1 can be given in 6 dimensions as all permutations of:

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

Representations

A small rhombated hexateron has the following Coxeter diagrams:

• x3o3x3o3o (full symmetry)
• oxx3xxo3oox3ooo&#xt (A₄ axial, rectified pentachoron-first)
• x(uo)xo x(ou)xx3o(xo)xo3o(oo)ox&#xt (A₃×A₁ axial, tetrahedral prism-first)
• ox(ou)xx3oo(xo)xo xx(uo)xo3ox(ox)oo&#xt (A₂×A₂ axial, triangle-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

The small rhombated hexateron is the colonel of a regiment that includes 15 uniform members.

External links

• Klitzing, Richard. "sarx".

• Wikipedia Contributors. "Cantellated 5-simplex".