Cantellated 6-orthoplex

Cantellated 6-orthoplex
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Srog
Coxeter diagram	o4o3o3x3o3x ()
Elements
Peta	60 hexadecachoric prisms 64 cantellated 5-simplices 12 rectified 5-orthoplexes
Tera	960 tetrahedral prisms 120 hexadecachora 384 rectified pentachora 192 small rhombated pentachora
Cells	1920 tetrahedra 1920 triangular prisms 960 octahedra 240 cuboctahedra
Faces	160+960+3840 triangles, 1440 squares
Edges	480+2880
Vertices	480
Vertex figure	Octahedral-pyramidal prism, edge lengths 1 (base octahedral prism and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {3}}\approx 1.73205$
Hypervolume	${\frac {3637}{90}}\approx 40.41111$
Dipetal angles	Rat–hex–hexip: 135°
	Sarx–tepe–hexip: $\arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }$
	Sarx–srip–sarx: $\arccos \left(-{\frac {2}{3}}\right)\approx 131.81032^{\circ }$
	Rat–rap–sarx: $\arccos \left(-{\frac {\sqrt {6}}{6}}\right)\approx 114.09484^{\circ }$
Central density	1
Number of external pieces	136
Level of complexity	25
Related polytopes
Army	Srog
Regiment	Srog
Conjugate	None
Abstract & topological properties
Flag count	1152000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B6, order 46080
Convex	Yes
Nature	Tame

The cantellated 6-orthoplex, also called the cantellated hexacontatetrapeton, small rhombated hexacontatetrapeton, or srog, is a convex uniform 6-polytope. It consists of 64 cantellated 5-simplices, 12 rectified 5-orthoplexes, and 60 hexadecachoric prisms. 8 small rhombated 5-simplices, 1 rectified 5-orthoplex, and 2 hexadecachoric prisms join at each vertex. As the name suggests, it is the cantellation of the 6-orthoplex.

The cantellated 6-orthoplex contains the vertices and edges of the square-truncated hexadecachoric duoprism and the octahedral-truncated octahedral duoprism.

Vertex coordinates

The vertices of a cantellated 6-orthoplex of edge length 1 are given by all permutations of:

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

Representations

A cantellated 6-orthoplex has the following Coxeter diagrams:

• o4o3o3x3o3x () (full symmetry)
• o3o3x3o3x *b3o () (D₆ symmetry)

Gallery

• 

  B₅ orthographic projection

• 

  B₄

• 

  B₃

• 

  B₂

• 

  A₅

• 

  A₃

Related polytopes

The regiment of the cantellated 6-orthoplex contains 290 uniform members, plus 63 fissaries and two compounds.

External links

• Klitzing, Richard. "srog".
• Wikipedia contributors. "Cantellated 6-orthoplex".