24-cell honeycomb
The 24-cell honeycomb (Schläfli symbol {3,4,3,3}), whose (hyper)cells correspond to the points of the D4≃F4 lattice, is one of three proper regular tilings of four-dimensional space, alongside the tesseractic honeycomb and 16-cell honeycomb, which is the dual of the 24-cell honeycomb. It is constructed with three 16-cells being placed around each face.
Due to the exotic nature of this tiling, no implementations of cellular automata are known to have been attempted. It is, however, listed in the Golly rule table repository roadmap, implying that such rules have at least been considered.
Coordinates, directions and displacements
Like with the triangular tiling and hexagonal tiling, the standard Cartesian coordinate system does not generalise to the 24-cell honeycomb perfectly. In fact the 24-cell honeycomb has so many directions that no coordinate system can preserve all of them. Nevertheless, there are three basic coordinate systems that can be used, we will call them the D4 coordinate system, the F4 coordinate system, and the lattice basis system.
D4 coordinates
The cells correspond to points in the D4 lattice, that is points in 4-space with coordinates given by integers (x,y,z,w) with x+y+z+w even. In this coordinate system 24 cells adjacent to the origin are given by the points (±1,±1,0,0) and permutations thereof.
F4 coordinates
The cells correspond to points in the F4 lattice, that is points in 4-space with coordinates given by integers (x,y,z,w) or (x/2,y/2,z/2,w/2) with each of x, y, z, and w odd in the latter case. This is the same as the noncommutative ring of Hurwitz quaternions. In this coordinate system 24 cells adjacent to the origin are given by the points (±1,0,0,0) and (±1/2,±1/2,±1/2,±1/2) and permutations thereof.
Basis coordinates
Using a basis has the advantage that of giving a one-to-one correspondence between 4-tuples of integers and cells at the expense of preserving even fewer of the symmetries. A basis for the lattice is given in D4 coordinates by:
( 1, 1, 0, 0) (-1, 1, 0, 0) ( 0, 0, 1, 1) ( 0, 1, 1, 0)
and in F4 coordinates by:
( 1, 0, 0, 0) ( 0, 1, 0, 0) ( 0, 0, 1, 0) (1/2,1/2,1/2,1/2).
Symmetric directions
A 24-cell has 24 orthogonal neighbors, each touching one of its constituent cells; these may be considered as the orthogonal directions. We can take the diagonal directions to be those corresponding to the 96 (2-)faces, the paragonal directions to be those corresponding to the 96 edges, and metagonal directions to be those corresponding to the 24 vertices.
Neighbourhoods on the 24-cell honeycomb
A von Neumann-style neighbourhood can be constructed by considering the 24 adjacent cells to a chosen central cell. A Moore-style neighbourhood can be constructed by combining von Neumann-style neighbourhood with the 24 additional cells that share one of the twenty-four vertices but aren't adjacent.
Symmetries
The 501 static symmetries and 1864 additional kinetic symmetries have been enumerated here.
Software support
As of September 2025, there are no known attempts to implement cellular automata on this tiling. However, the D4 coordinate system described above imply that a 24-cell honeycomb cellular automata can be simulated using tesseractic cellular automata via appropriate choice of neighbourhood, namely a diagonal one. However, it's not clear to what extent the existing 4d cellular automata programs support this type of neighbourhood.