Celliprismated hexateron

Celliprismated hexateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Cappix
Coxeter diagram	x3x3o3o3x ()
Elements
Tera	15 tetrahedral prisms, 20 triangular-hexagonal duoprisms, 6 truncated pentachora, 15 truncated tetrahedral prisms, 6 small prismatodecachora
Cells	30+30 tetrahedra, 60+60+60 triangular prisms, 60 hexagonal prisms, 30 truncated tetrahedra
Faces	120+120 triangles, 90+180 squares, 60 hexagons
Edges	60+180+180
Vertices	120
Vertex figure	Skew triangular antiprismatic pyramid, edge lengths 1 (base triangles), √2 (sides of antiprism and lacings to one base), and √3 (lacings to other base)
Measures (edge length 1)
Circumradius	$\frac{2\sqrt6}{3} ≈ 1.63299$
Hypervolume	$\frac{1297\sqrt3}{120} ≈ 18.72058$
Diteral angles	Tuttip–hip–thiddip: 135°
	Tepe–trip–thiddip: 135°
	Tip–tut–tuttip: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Spid–tet–tepe: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152°$
	Spid–trip–thiddip: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
	Spid–trip–tuttip: $\arccos\left(-\frac{\sqrt{10}}{10}\right) ≈ 108.43495°$
	Spid–tet–tip: $\arccos\left(-\frac15\right) ≈ 101.53696°$
Central density	1
Number of external pieces	62
Level of complexity	40
Related polytopes
Army	Cappix
Regiment	Cappix
Dual	Triangular-antitegmatic pyramidal hecatonicosateron
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5, order 720
Convex	Yes
Nature	Tame

The celliprismated hexateron, or cappix, also called the steritruncated 5-simplex, is a convex uniform polyteron. It consists of 15 tetrahedral prisms, 20 triangular-hexagonal duoprisms, 15 truncated tetrahedral prisms, 6 truncated pentachora, and 6 small prismatodecachora. One small prismatodecachoron, 1 tetrahedral prism, 1 truncated pentachoron, 3 truncated tetrahedral prisms, and 3 triangular-hexagonal duoprisms join at each vertex. As the name suggests, it is the steritruncation of the hexateron.

Vertex coordinates

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

• $\left(\frac{3\sqrt2}{2},\,\sqrt2,\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0\right).$

Representations

A celliprismated hexateron has the following Coxeter diagrams:

• x3x3o3o3x (full symmetry)
• xuxx3ooxx3oooo3xxxo&#xt (A₄ axial, small prismatodecachoron-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

The celliprismated hexateron is the colonel of a regiment of 7 members.

External links

• Klitzing, Richard. "cappix".

• Wikipedia Contributors. "Steritruncated 5-simplex".