Tetracontoctachoron

The tetracontoctachoron, or cont, also commonly called the 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 its dual.

It is the second in a series of isochoric cubic swirlchora (the hexaswirlic tetracontoctachoron) and the first in a series 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 tiling of H4, with 64 tetracontoctachora at each vertex with an octagonal duotegum as the vertex figure.[citation needed]

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.

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