Small cellidodecateron

Small cellidodecateron
(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Scad
Coxeter diagram	x3o3o3o3x ()
Elements
Tera	12 pentachora, 30 tetrahedral prisms, 20 triangular duoprisms
Cells	60 tetrahedra, 120 triangular prisms
Faces	120 triangles, 90 squares
Edges	120
Vertices	30
Vertex figure	Tetrahedral antiprism, edge lengths 1 (base) and √2 (sides)
Measures (edge length 1)
Circumradius	1
Hypervolume	$\frac{21\sqrt3}{40} ≈ 0.90933$
Diteral angles	Tepe–trip–triddip: 135°
	Pen–tet–tepe: $\arccos\left(-\frac{\sqrt{10}}{5}\right) ≈ 129.23152^\circ$
Central density	1
Number of external pieces	62
Level of complexity	8
Related polytopes
Army	Scad
Regiment	Scad
Dual	Tetrahedral-antitegmatic triacontateron
Conjugate	None
Abstract & topological properties
Flag count	11520
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A5×2, order 1440
Convex	Yes
Nature	Tame

The small cellidodecateron, or scad, also called the small cellated hexateron or stericated 5-simplex, is a convex uniform polyteron. It consists of 12 regular pentachora, 30 tetrahedral prisms, and 20 triangular duoprisms. 2 pentachora, 8 tetrahedral prisms, and 6 triangular duoprisms join at each vertex. It can be formed by expanding the facets of the hexateron outward.

Vertex coordinates

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

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

Representations

A small cellated dodecateron has the following Coxeter diagrams:

• x3o3o3o3x (full symmetry)
• xxo3ooo3ooo3oxx&#xt (A₄ axial, pentachoron-first)
• x(xo)o3o(xo)x x(ox)o3o(ox)x&#xt (A₂×A₂ axial, triangular duoprism-first)
• x(uo)x x(ox)o3o(oo)o3o(ox)x&#xt (A₃×A₁ axial, tetrahedral prism-first)

Gallery

• 

  A₄ orthographic projection

• 

  A₃

• 

  A₂

Related polytopes

The small cellidodecateron is the colonel of a regiment with 11 members. Of these 11, 7 have full symmetry, while the other 4 have single A5 symmetry only.

External links

• Bowers, Jonathan. "Category 9: Podiumverts" (#681).

• Klitzing, Richard. "scad".

• Wikipedia Contributors. "Stericated 5-simplex".