Tetracontoctachoron

Tetracontoctachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Cont
Coxeter diagram	o3x4x3o ()
Elements
Cells	48 truncated cubes
Faces	192 triangles, 144 octagons
Edges	576
Vertices	288
Vertex figure	Tetragonal disphenoid, edge lengths 1 (base) and √2+√2 (sides)
Edge figure	tic 8 tic 8 tic 3
Measures (edge length 1)
Circumradius	$2+\sqrt2 ≈ 3.41421$
Inradius	$\frac{3+2\sqrt2}{2} ≈ 2.91421$
Hypervolume	$14(17+12\sqrt2) ≈ 475.58788$
Dichoral angles	Tic–8–tic: 135°
	Tic–3–tic: 120°
Central density	1
Number of external pieces	48
Level of complexity	3
Related polytopes
Army	Cont
Regiment	Cont
Dual	Bitetracontoctachoron
Conjugate	Great tetracontoctachoron
Abstract & topological properties
Flag count	6912
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The tetracontoctachoron, or cont, also commonly called the 48-cell or bitruncated 24-cell, is a convex noble uniform polychoron that consists of 48 truncated cubes as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular icositetrachoron and itself. Alternatively, it is also the stellation core of the compound of two opposite icositetrachora, the stellated tetracontoctachoron.

It is the second in an infinite family of isochoric cubic swirlchora (the cubiswirlic tetracontoctachoron) and the first in an infinite family of isochoric chiral rhombic dodecahedral swirlchora (the rhombidodecaswirlic tetracontoctachoron). Its cells form 6 rings of 8 truncated cubes.

The solid angle at the vertex is 73/288.

It can form a non-Wythoffian uniform hyperbolic tiling with 64 tetracontoctachora at each vertex with an octagonal duotegum as the vertex figure, called a tetracontoctachoric tetracomb.

Cross-sections

Vertex coordinates

Coordinates for the vertices of a tetracontoctachoron of edge length 1 are all permutations of:

• $\left(±(1+\sqrt2),\,±\frac{2+\sqrt2}{2},\,±\frac{2+\sqrt2}{2},\,0\right),$
• $\left(±\frac{3+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

Representations

A tetracontoctachoron has the following Coxeter diagrams:

• o3x4x3o (full symmetry)
• xo4xw3oo3wx&#zx (BC₄ symmetry)
• xooxwUwxoox4xwwxoooxwwx3ooxwwxwwxoo&#xt (BC₃ axial, cell-first)

Variations

The tetracontoctachoron has a semi-uniform variant with single symmetry called the icositetricositetrachoron, along with isochoric variants with swirlprismatic symmetry.

Related polychora

Uniform polychoron compounds composed of tetracontoctachora include:

• Bitruncated chirohecatonicosachoron (5)
• Bitruncated dodecahedronary hexacosichoron (25)

Isogonal derivatives

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

• Truncated cube (48): Bitetracontoctachoron
• Triangle (192): Biambotetracontoctachoron
• Octagon (144): Small prismatotetracontoctachoron
• Edge (576): Rectified tetracontoctachoron

External links

• Bowers, Jonathan. "Category 7: Bitruncates" (#300).

• Klitzing, Richard. "cont".

• Quickfur. "The Bitruncated 24-cell".

• Wikipedia Contributors. "Bitruncated 24-cell".
• Hi.gher.Space Wiki Contributors. "Xylomesochoron".