24-cell Honeycomb - Coordinates

Coordinates

The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D₄ root lattice or F4 lattice. Each 24-cell is then centered at a D₄ lattice point, i.e. one of

These points can also be described as Hurwitz quaternions with even square norm.

The vertices of the honeycomb lie at the deep holes of the D₄ lattice. These are the Hurwitz quaternions with odd square norm.

It can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the birectified tesseractic honeycomb vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+½,j+½,k,l), (i+½,j,k+½,l), (i+½,j,k,l+½), (i,j+½,k+½,l), (i,j+½,k,l+½), (i,j,k+½,l+½).