Cantellated 5-cube

Cantellated 5-cube
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Sirn
Coxeter diagram	x4o3x3o3o ()
Elements
Tera	80 tetrahedral prisms, 32 rectified pentachora, 10 small rhombated tesseracts
Cells	160 tetrahedra, 320 triangular prisms, 160 octahedra, 40 small rhombicuboctahedra
Faces	320+640 triangles, 80+480 squares
Edges	320+960
Vertices	320
Vertex figure	Triangular pyramidal prism, edge lengths 1 (base triangular prism) and √2 (sides and top edge)
Measures (edge length 1)
Circumradius	${\frac {3+{\sqrt {2}}}{2}}\approx 2.20711$
Hypervolume	${\frac {1205+843{\sqrt {2}}}{30}}\approx 79.90607$
Diteral angles	Rap–tet–tepe: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	Srit–trip–tepe: 120°
	Srit–oct–rap: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Srit–sirco–srit: 90°
Central density	1
Number of external pieces	122
Level of complexity	16
Related polytopes
Army	Sirn
Regiment	Sirn
Dual	Triangular-pyramidal tegmatic triacosiicosateron
Conjugate	Quasirhombated penteract
Abstract & topological properties
Flag count	61440
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The cantellated 5-cube, also called the small rhombated penteract or sirn, is a convex uniform 5-polytope. It consists of 80 tetrahedral prisms, 32 rectified pentachora, and 10 small rhombated tesseracts. One rectified pentachoron, 2 tetrahedral prisms, and 3 small rhombated tesseracts join at each vertex. As the name suggests, it is the cantellation of the 5-cube.

The cantellated 5-cube contains the vertices of a truncated tesseractic prism and octagonal-truncated cubic duoprism.

Vertex coordinates

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

• $\left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)$.

Representations

A cantellated 5-cube has the following Coxeter diagrams:

• x4o3x3o3o () (full symmetry)
• xxxx4oxxo3xoox3oooo&#xt (B₄ axial, small rhombated tesseract-first)
• oxo4wxx xoo3oxo4xxw&#zx (B₃×B₂ symmetry)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The cantellated 5-cube is the colonel of a regiment that includes 15 uniform members.

External links

• Klitzing, Richard. "sirn".
• Wikipedia contributors. "Cantellated 5-cube".