Dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton

Dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton
(OFF file)
Rank	8
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Fy
Coxeter diagram	o3o3o3o *c3o3o3o3x ()
Elements
Zetta	17280 octaexa, 2160 hecatonicosoctaexa
Exa	69120+138240 heptapeta
Peta	483840 hexatera
Tera	483840 pentachora
Cells	241920 tetrahedra
Faces	60480 triangles
Edges	6720
Vertices	240
Vertex figure	Hecatonicosihexapentacosiheptacontahexaexon
Measures (edge length 1)
Circumradius	1
Hypervolume	$\frac{57}{112} ≈ 0.50893$
Dizettal angles	oca–hop-zee: $\arccos\left(-\frac{5\sqrt2}{8}\right) ≈ 152.11443°$
	zee–hop-zee: $\arccos\left(-\frac34\right) ≈ 138.59038°$
Central density	1
Number of pieces	19440
Level of complexity	3
Related polytopes
Army	Fy
Regiment	Fy
Dual	Semistellatotriacontaditeri-icosiheptapeto-pentacontoctaexic diacositetracontazetton
Conjugate	None
Abstract properties
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	E8, order 696729600
Convex	Yes
Nature	Tame

The dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton, abbreviated as 2160-17280-zetton, also known as fy, the E8 polytope, or the 4₂₁ polytope, is a unifrom polyzetton. It consists of 2160 hecatonicosoctaexa and 17280 octaexa as facets, with 126 hecatonicosoctaexa and 576 octaexa as facets forming a hecatonicosihexapentacosiheptacontahexaexon as the vertex figure.

The dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton contains the vertices and edges of the birectified enneazetton, small exioctadecazetton, demiocteract, rectified diacosipentacontahexazetton, hecatonicosihexapentacosiheptacontahexaexic prism, triangular-icosiheptaheptacontidipetic duoprism, pentachoric-rectified pentachoric duoprism, tetrahedral-demipenteractic duoprism, hexadecachoric duoprism, octahedral-triacontiditeric duoprism, square-hexacontatetrapetic duoprism, rectified octaexic prism, and triangular-hexateric duoprismatic prism.

Vertex coordinates

Coordinates for the vertices of a dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton with edge length 1 are given by all permutations and sign changes of

• $\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0\right),$

as well as all even sign changes of

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

Representations

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

• o3o3o3o *c3o3o3o3x (full symmetry)
• ooxoo3ooooo3ooooo3ooooo *c3ooooo3ooooo3oxoxo&#xt (E7 axial, vertex-first)
• oxooo3ooooo3oooxo *b3ooooo3ooooo3ooxoo3xooox&#xt (D7 axial, hecatonicosoctaexon-first)
• xo3oo3oo *b3oo3oo3oo3ox3oo&#zx (D8 subsymmetry)
• oxo3ooo3xoo3ooo3ooo3oox3ooo3oxo&#zx (A8 subsymmetry)
• xooxooo3oooooxo3ooxoooo3ooooooo3ooooxoo3oxooooo3oooxoox&#xt (A7 axial, octaexon-first)
• xoxooo3ooooxo3ooooox3xooxoo oxooox3ooxooo3oooxoo3oxooxo&#zx (A4×A4 subsymmetry)
• ooxoo3xoooo3oooxo *b3oooox ooxoo3oxooo3oooxo *f3oooox&#zx (D4×D4 subsymmetry)
• xoxo3xoox ooxo3oooo3oooo3oooo3ooox *e3oxoo&#zx (A6×A2 subsymmetry)

External links

• Klitzing, Richard. "Fy".

• Wikipedia Contributors. "4₂₁ polytope".