Cantellated 7-simplex

Cantellated 7-simplex
Rank	7
Type	Uniform
Notation
Bowers style acronym	Saro
Coxeter diagram	x3o3x3o3o3o3o ()
Elements
Exa	28 5-simplicial prisms 8 rectified 6-simplices 8 cantellated 6-simplices
Peta	56 5-simplices 168 pentachoric prisms 56 rectified 5-simplices 28 cantellated 5-simplices
Tera	336 pentachora 420 tetrahedral prisms 168 rectified pentachora 56 small rhombated pentachora
Cells	840 tetrahedra 560 triangular prisms 280 octahedra 70 cuboctahedra
Faces	56+280+1120 triangles 420 squares
Edges	168+840
Vertices	168
Vertex figure	Pentachoric pyramidal prism, edge lengths 1 (base and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {2}}\approx 1.41421$
Hypervolume	${\frac {211}{280}}\approx 0.75357$
Diexal angles	Ril–hix–hixip: $\arccos \left(-{\frac {\sqrt {21}}{7}}\right)\approx 130.89339^{\circ }$
	Sril–penp–hixip: $\arccos \left(-{\frac {\sqrt {21}}{21}}\right)\approx 102.60438^{\circ }$
	Sril–rix–ril: $\arccos \left(-{\frac {1}{7}}\right)\approx 98.21321^{\circ }$
	Sril–sarx–sril: $\arccos \left({\frac {1}{7}}\right)\approx 81.78679^{\circ }$
Central density	1
Number of external pieces	44
Level of complexity	36
Related polytopes
Army	Saro
Regiment	Saro
Conjugate	None
Abstract & topological properties
Flag count	1451520
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A7, order 40320
Convex	Yes
Nature	Tame

The cantellated 7-simplex, also called the cantellated octaexon or small rhombated octaexon, is a convex uniform 7-polytope. It consists of 8 rectified 6-simplices, 8 cantellated 6-simplices, and 28 5-simplicial prisms. 1 rectified 6-simplex, 5 cantellated 6-simplices, and 2 5-simplicial prisms join at each vertex. As the name suggests, it is the cantellation of the 7-simplex.

The cantellated 7-simplex can be vertex-inscribed into the rectified hecatonicosihexapentacosiheptacontahexaexon.

Vertex coordinates

The vertices of a cantellated 7-simplex of edge length 1 can be given in eight dimensions as all permutations of:

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

External links

• Klitzing, Richard. "saro".
• Wikipedia contributors. "Cantellated 7-simplex".