Icositetrachoric tetracomb

The icositetrachoric tetracomb or icot, also known as the 24-cell tetracomb or 24-cell honeycomb, is one of three regular tetracombs or tessellations of 4D Euclidean space. 3 icositetrachora join at each face, and 8 join at each vertex of this honeycomb.

It can be obtained from the tesseractic tetracomb by decomposing alternate tesseracts into 8 cubic pyramids and attaching those to the neighbouring tesseract, making it an icositetrachoron. It can also be obtained as a birectified tesseractic tetracomb, or a rectified hexadecachoric tetracomb.

Vertex coordinates
The vertices of an icositetrachoric tetracomb of edge length 1 are given by all permutations of: where i, j, k, and l range over the integers.
 * $$\left(\sqrt2i,\,\sqrt2j,\,±\frac{\sqrt2}{2}+\sqrt2k,\,±\frac{\sqrt2}{2}+\sqrt2l\right),$$

An alternate set of coordinates can be given by: where i, j, k, l are integers.
 * $$\left(i,\,j,\,k,\,l\right),$$
 * $$\left(i+\frac12,\,j+\frac12,\,k+\frac12,\,l+\frac12\right)$$ for i+j+k+l even

Representations
An icositetrachoric tetracomb has the following Coxeter diagrams:


 * (full symmetry)
 * (U5 symmetry, as rectified hexadecachoric tetracomb)
 * (R5 symmetry, birectified tesseractic tetracomb)
 * (S5 symmetry, rectified demitesseractic tetracomb)
 * (Q5 symmetry, cells of four different types)