Bitetracontoctachoron

Bitetracontoctachoron
(OFF file)
Rank	4
Type	Noble
Notation
Bowers style acronym	Bicont
Coxeter diagram	o3m4m3o ()
Elements
Cells	288 tetragonal disphenoids
Faces	576 isosceles triangles
Edges	144+192
Vertices	48
Vertex figure	Triakis octahedron
Measures (based on two icositetrachora of edge length 1)
Edge lengths	Lacing edges (144): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	Edges of icositetrachora (192): 1
Circumradius	1
Inradius	${\frac {2+{\sqrt {2}}}{4}}\approx 0.85355$
Dichoral angle	$\arccos \left(-{\frac {1+2{\sqrt {2}}}{4}}\right)\approx 163.15788^{\circ }$
Central density	1
Related polytopes
Army	Bicont
Regiment	Bicont
Dual	Tetracontoctachoron
Conjugate	Great bitetracontoctachoron
Abstract & topological properties
Flag count	6912
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The bitetracontoctachoron or bicont, also known as the tetradisphenoidal diacosioctacontoctachoron or octafold octaswirlchoron, is a convex noble polychoron with 288 tetragonal disphenoids as cells. 24 cells join at each vertex, with the vertex figure being a triakis octahedron. It can be constructed as the convex hull of 2 dual icositetrachora.

It is the second in an infinite family of isogonal octahedral swirlchora (the octafold octaswirlchoron) and the first in an infinite family of isogonal chiral cuboctahedral swirlchora.

The ratio between the longest and shortest edges is 1:${\sqrt {\frac {2+{\sqrt {2}}}{2}}}$ ≈ 1:1.30656.

The tetragonal disphenoid cells of this polychoron are similar to those used as the vertex figure of the great tetracontoctachoron.

Vertex coordinates

Coordinates for the vertices of a bitetracontoctachoron of circumradius 1, centered at the origin, are given by all permutations of:

• $\left(\pm 1,\,0,\,0,\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,0\right).$

Variations

The bitetracontoctachoron has a number of isogonal or isotopic variations:

• Disphenoidal diacosioctacontoctachoron (digonal disphenoid cells, isotopic)
• Octafold octaswirlchoron (96 tetragonal and 192 phyllic disphenoids, swirlprism)

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Tetragonal disphenoid (288): Tetracontoctachoron
• Isosceles triangle (576): Rectified tetracontoctachoron
• Edge (144): Small prismatotetracontoctachoron
• Edge (192): Biambotetracontoctachoron

External links

• Klitzing, Richard. "bicont".
• Wikipedia contributors. "Disphenoidal 288-cell".