1 42 polytope

1 42 polytope
Rank	8
Type	Uniform
Notation
Bowers style acronym	Bif
Coxeter diagram	o3o3o3o3o3o3o *e3x ()
Elements
Zetta	2160 demihepteracts, 240 pentacontahexahecatonicosihexaexa
Exa	69120 heptapeta, 30240 demihexeracts, 6720 pentacontatetrapeta
Peta	483840 hexatera, 60480+181440 demipenteracts
Tera	241920+1451520 pentachora, 604800 hexadecachora
Cells	1209600+2419200 tetrahedra
Faces	2419200 triangles
Edges	483840
Vertices	17280
Vertex figure	Birectified octaexon, edge length 1
Measures (edge length 1)
Circumradius	2
Hypervolume	${\frac {44985}{112}}\approx 401.65179$
Dizettal angles	Hesa–hop–hesa: $\arccos \left(-{\frac {3}{4}}\right)\approx 138.59038^{\circ }$
	Lin–hax–hesa: 135°
	Lin–mo–lin: 120°
Central density	1
Number of external pieces	2400
Level of complexity	15
Related polytopes
Army	Bif
Regiment	Bif
Conjugate	None
Abstract & topological properties
Flag count	10450944000
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	E8, order 696729600
Convex	Yes
Nature	Tame

The diacositetraconta-dischiliahecatonhexaconta-zetton, abbreviated as 240-2160-zetton, also known as bif or the 1₄₂ polytope, is a convex uniform polyzetton. It consists of 2160 demihepteracts and 240 pentacontahexahecatonicosihexaexa, with 8 of each joining at each vertex forming a birectified octaexon as the vertex figure.

The diacositetraconta-dischiliahecatonhexaconta-zetton contains the vertices and edges of an inscribed small cellated diacosipentacontahexazetton and small cellidemiocteract.

Vertex coordinates

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

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

all permutations and even sign changes of

• $\left({\frac {5{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)$,

and all permutations and odd sign changes of

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

Related polyzetta

A diacositetracontadischiliahecatonhexacontazetton contains the vertices and edges of a small cellated diacosipentacontahexazetton, small cellidemiocteract, octadecazetton, teritruncated enneazetton, petiprismated enneazetton, birectified hecatonicosihexapentacosiheptacontihexaexic prism, hexagonal-rectified pentacontatetrapetic duoprism, triangular-small demiprismated icosiheptaheptacontidipetic duoprism, decachoric duoprism, rectified pentachoric-great rhombated pentachoric duoprism, prismatorhombated pentachoric-small rhombated pentachoric duoprism, square-small prismated hexacontatetrapetic duoprism, octahedral-small prismated triacontiditeric duoprism, cuboctahedral-small rhombated triacontiditeric duoprism, truncated octahedral-penteractitriacontiditeric duoprism, icositetrachoric-rectified icositetrachoric duoprism, tesseractic-rectified icositetrachoric duoprism, rectified tesseractic-truncated hexadecachoric duoprism, truncated tetrahedral-small rhombated demipenteractic duoprism, tetrahedral-great rhombated demipenteractic duoprism, hexagonal-cellirhombated dodecateric duoprismatic prism, triangular-prismatotruncated duoprismatic prism, and hexagonal tetraprism.

External links

• Klitzing, Richard. "bif".
• Wikipedia contributors. "1₄₂ polytope".