Cantellated 5-orthoplex

Cantellated 5-orthoplex
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Sart
Coxeter diagram	o4o3x3o3x ()
Elements
Tera	40 octahedral prisms, 10 icositetrachora, 32 small rhombated pentachora
Cells	320 triangular prisms, 80+160 octahedra, 80 cuboctahedra
Faces	80+320+640 triangles, 480 squares
Edges	240+960
Vertices	240
Vertex figure	Square pyramidal prism, edge lengths 1 (base cube and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {3}}\approx 1.73205$
Hypervolume	${\frac {412{\sqrt {2}}}{15}}\approx 38.84373$
Diteral angles	Ico–oct–ope: 135°
	Srip–trip–ope: $\arccos \left(-{\frac {\sqrt {10}}{5}}\right)\approx 129.23152^{\circ }$
	Srip–co–srip: $\arccos \left(-{\frac {3}{5}}\right)\approx 126.86990^{\circ }$
	Ico–oct–srip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Central density	1
Number of external pieces	82
Level of complexity	16
Related polytopes
Army	Sart
Regiment	Sart
Dual	Square-pyramidal tegmatic diacositetracontateron
Conjugate	None
Abstract & topological properties
Flag count	61440
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B5, order 3840
Convex	Yes
Nature	Tame

The cantellated 5-orthoplex, also called the small rhombated triacontaditeron or sart, is a convex uniform 5-polytope. It consists of 40 octahedral prisms, 32 small rhombated pentachora, and 10 icositetrachora. One icositetrachoron, 2 octahedral prisms, and 4 small rhombated pentachora join at each vertex. As the name suggests, it is the cantellation of the 5-orthoplex.

The cantellated 5-orthoplex contains the vertices of a square-truncated octahedral duoprism.

Vertex coordinates

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

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

Representations

A cantellated 5-orthoplex has the following Coxeter diagrams:

• o4o3x3o3x () (full symmetry)
• o3x3o3x *b3o () (D₅ symmetry)
• ooooo4ooxoo3xxoxx3oxxxo&#xt (B₄ axial, icositetrachoron-first)
• ooxoo3xxoxx3ooxoo *b3oxxxo&#xt (D₄ axial, icositetrachoron-first)

Gallery

• 

  B₄ or D₅ orthographic projection

• 

  B₃, D₄, or A₂

• 

  B₂

• 

  A₃

Related polytopes

The cantellated 5-orthoplex is the colonel of a regiment that contains 92 members and one fissary. Of these, 28, plus the fissary, have full symmetry, while the rest have only D5 symmetry. This makes it the third-largest regiment of non-prismatic uniform polytera.

External links

• Klitzing, Richard. "sart".
• Wikipedia contributors. "Cantellated 5-orthoplex".