Runcinated 6-simplex

Runcinated 6-simplex
(OFF file)
Rank	6
Type	Uniform
Notation
Bowers style acronym	Spil
Coxeter diagram	x3o3o3x3o3o ()
Elements
Peta	35 triangular-tetrahedral duoprisms 21 rectified pentachoric prisms 7 dodecatera 7 small prismated hexatera
Tera	105 tetrahedral prisms 140 triangular duoprisms 42+42 rectified pentachora 105 octahedral prisms 21 small prismatodecachora
Cells	35+105+105 tetrahedra 210+210+420 triangular prisms 210 octahedra
Faces	140+420+420 triangles, 630 squares
Edges	210+630
Vertices	140
Vertex figure	Triangle atop alternate triangular duoprism, edge lengths 1 (base duoprism and top triangle) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {2{\sqrt {21}}}{7}}\approx 1.30931$
Hypervolume	${\frac {3847{\sqrt {7}}}{2880}}\approx 3.53410$
Dipetal angles	Rappip–tepe–tratet: $\arccos \left(-{\frac {2{\sqrt {30}}}{15}}\right)\approx 136.91128^{\circ }$
	Dot–rap–rappip: $\arccos \left(-{\frac {\sqrt {15}}{6}}\right)\approx 130.20297^{\circ }$
	Spix–triddip–tratet: $\arccos \left(-{\frac {\sqrt {2}}{4}}\right)\approx 110.70481^{\circ }$
	Spix–ope–rappip: $\arccos \left(-{\frac {\sqrt {15}}{15}}\right)\approx 104.96322^{\circ }$
	Spix–rap–dot: $\arccos \left(-{\frac {1}{6}}\right)\approx 99.59407^{\circ }$
	Spix–spid–spix: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
Central density	1
Number of external pieces	70
Level of complexity	56
Related polytopes
Army	Spil
Regiment	Spil
Conjugate	None
Abstract & topological properties
Flag count	282240
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A6, order 5040
Convex	Yes
Nature	Tame

The runcinated 6-simplex (also called the small prismated heptapeton or spil) is a convex uniform 6-polytope. It consists of 7 dodecatera, 7 small prismated hexatera, 21 rectified pentachoric prisms, and 35 triangular-tetrahedral duoprisms as facets. 1 dodecateron, 3 small prismated hexatera, 3 rectified pentachoric prisms, and 3 triangular-tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcination of the 6-simplex.

Vertex coordinates

The vertices of a runcinated 6-simplex of edge length 1 can be given in seven dimensions as all permutations of:

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

Representations

A runcinated 6-simplex has the following Coxeter diagrams:

• x3o3o3x3o3o () (full symmetry)
• ooo3oox3xx3o3ooo3oxx&#xt (A₅ axial, dodecateron-first)

Gallery

• 

  A₅ orthographic projection

• 

  A₄

• 

  A₃

• 

  A₂

Related polytopes

The runcinated 6-simplex is the colonel of a regiment with 138 uniform members plus 15 fissaries.

External links

• Klitzing, Richard. "spil".
• Klitzing, Richard. "spil-verf".
• Wikipedia contributors. "Runcinated 6-simplex".