2 41 polytope

2 41 polytope
(OFF file)
Rank	8
Type	Uniform
Notation
Bowers style acronym	Bay
Coxeter diagram	o3o3o3o3o3o3x *e3o ()
Elements
Zetta	17280 7-simplices 240 231 polytopes
Exa	138240 6-simplices 6720 221 polytopes
Peta	483840 5-simplices 60480 5-orthoplexes
Tera	241920+967680 pentachora
Cells	1209600 tetrahedra
Faces	483840 triangles
Edges	69120
Vertices	2160
Vertex figure	Demihepteract, edge length 1
Measures (edge length 1)
Circumradius	${\sqrt {2}}\approx 1.41421$
Hypervolume	${\frac {1791}{112}}\approx 15.99107$
Dizettal angles	Laq–hop–oca: $\arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }$
	Laq–jak–laq: 120°
Central density	1
Number of external pieces	17520
Level of complexity	5
Related polytopes
Army	Bay
Regiment	Bay
Conjugate	None
Abstract & topological properties
Flag count	3483648000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	E8, order 696729600
Flag orbits	5
Convex	Yes
Nature	Tame

The 2₄₁ polytope, also known as the diacositetraconta-myriaheptachiliadiacosioctaconta-zetton (abbreviated as 240-17280-zetton) or bay, is a uniform 8-polytope. It consists of 17280 7-simplices and 240 2₃₁ polytopes, with 14 2₃₁ polytopes and 64 7-simplices joining at each vertex forming a 7-demicube as the vertex figure.

Vertex coordinates

Coordinates for a diacositetraconta-myriaheptachiliadiacosioctaconta-zetton with edge length 1 are given by all permutations of

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

all permutations of

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

and all permutations and odd sign changes of

• $\left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$.

Related 8-polytopes

The diacositetraconta-myriaheptachiliadiacosioctaconta-zetton contains the vertices and edges of the truncated 8-simplex, stericated 8-simplex, bipentellated 8-simplex, trirectified 8-orthoplex, small petidemiocteract, 1₃₂ polytope prism, triangular-rectified 2₂₁ polytope duoprism, hexagonal-1₂₂ polytope duoprism, pentachoric-truncated pentachoric duoprism, rectified pentachoric-small rhombated pentachoric duoprism, small prismatodecachoric duoprism, square-birectified 6-orthoplex duoprism, octahedral-birectified 5-cubic duoprism, cuboctahedral-rectified 5-orthoplex duoprism, hexadecachoric-rectified tesseractic duoprism, icositetrachoric duoprism, tetrahedral-steric 5-cubic duoprism, truncated tetrahedral-5-demicubic duoprism, stericated 7-simplicial prism, triangular-runcinated 5-simplicial duoprismatic prism, hexagonal-birectified 5-simplicial duoprismatic prism, triangular-triangular-triangular-hexagonal tetraprism, tesseractic-icositetrachoric duoprism, square-octahedral-cuboctahedral trioprism, and the 8-cube.

Representations

A diacositetraconta-myriaheptachiliadiacosioctaconta-zetton has the following Coxeter diagrams:

• o3o3o3o3o3o3x *e3o () (full symmetry)
• xooox3ooooo3ooooo3oxoxo *c3ooooo3ooxoo3ooooo&#xt (E₇ axial, pentacontahexapentacosiheptacontahexaexon-first)
• oxo3ooo3ooo *b3ooo3xoo3ooo3ooo3oxu&#zx (D₈ subsymmetry)
• xxooo3xoxoo3ooooo3oooxo3oxooo3ooooo3ooxox3oooxx&#zx (A₈ subsymmetry)

External links

• Klitzing, Richard. "bay".
• Wikipedia contributors. "2₄₁ polytope".