Tetracontoctachoron

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.

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