Runcitruncated 6-simplex

Runcitruncated 6-simplex
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Patal
Coxeter diagram	x3x3o3x3o3o ()
Elements
Peta	21 rectified pentachoric prisms 35 hexagonal-tetrahedral duoprisms 7 small rhombated hexatera 7 prismatotruncated hexatera
Tera	105+105 tetrahedral prisms 42 rectified pentachora 105 octahedral prisms 140 triangular-hexagonal duoprisms 42 small rhombated pentachora 21 prismatorhombated pentachora
Cells	210 tetrahedra 210+420+420 triangular prisms 210 octahedra 210 hexagonal prisms 35 truncated tetrahedra 105 cuboctahedra
Faces	840+420+140 triangles 630+630 squares 140 hexagons
Edges	1260+420+210
Vertices	420
Vertex figure	Triangular pyramidal prismatic pyramid, edge lengths 1 (base triangular prism and top edge of base), √2 (sides of base and lacings to prism), and √3 (lacings to top edge)
Measures (edge length 1)
Circumradius	2
Hypervolume	${\frac {41867{\sqrt {7}}}{2880}}\approx 38.46169$
Dipetal angles	Rappip–tepe–hatet: $\arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }$
	Sarx–rap–rappip: $\arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }$
	Sarx–tepe–hatet: $\arccos \left(-{\frac {\sqrt {2}}{3}}\right)\approx 118.12551^{\circ }$
	Pattix–thiddip–hatet: $\arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }$
	Pattix–ope–rappip: $\arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }$
	Pattix–srip–sarx: $\arccos \left(-{\frac {1}{6}}\right)\approx 101.53696^{\circ }$
	Pattix–prip–pattix: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
Central density	1
Number of external pieces	70
Level of complexity	96
Related polytopes
Army	Patal
Regiment	Patal
Conjugate	None
Abstract & topological properties
Flag count	483840
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

The runcitruncated 6-simplex (also called the runcitruncated heptapeton, prismatotruncated heptapeton or patal) is a convex uniform 6-polytope. It consists of 7 prismatotruncated hexatera, 7 small rhombated hexatera, 21 rectified pentachoric prisms, and 35 hexagonal-tetrahedral duoprisms as facets. 3 prismatotruncated hexatera, 1 small rhombated hexateron, 1 rectified pentachoric prism, and 2 hexagonal-tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcitruncation of the 6-simplex.

Gallery

• 

  A₅ orthographic projection

• 

  A₄

• 

  A₃

• 

  A₂

Vertex coordinates

The vertices of a prismatotruncated heptapeton of edge length 1 can be given in 7 dimensions as all permutations of:

• $\left({\frac {3{\sqrt {2}}}{2}},\,{\sqrt {2}},\,{\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {2}}{2}},\,0,\,0,\,0\right).$

Related polytopes

The regiment of the prismatotruncated heptapeton contains 15 uniform members.

External links

• Klitzing, Richard. "patal".
• Wikipedia contributors. "Runcitruncated 6-simplex".