16-cell honeycomb
The 16-cell honeycomb (Schläfli symbol {3,3,4,3}), whose vertices correspond to the point of the D4≃F4 lattice, is one of three proper regular tilings of four-dimensional space, alongside the tesseractic honeycomb and 24-cell honeycomb, which is the dual of the 16-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 16-cell honeycomb perfectly. Also like the triangular tiling, and unlike the square, hexagonal, cubic, tesseractic, and 24-cell tilings, 16-cells do not tesselate Euclidean space through translation only, there in fact three "orientations" of 16-cells, meaning that if emulated on a linear (tesseractic) grid the different orientations would require the use of three different types on neighborhoods.
In one straightforward coordinate system the cells correspond to points in 4-space with coordinates given by integers (x,y,z,w) with x+y+z+w even or (x/2,y/2,z/2,w/2) with each of x, y, z, and w odd. Note that this is similar but not the same as the F4 lattice points, this is probably the source of the misconception that the 16-cell honeycomb corresponds to the F4 lattice. In this coordinate system 16 cells adjacent to the origin are given by the points (±1/2,±1/2,±1/2,±1/2).
Symmetric directions
The best way to define the symmetric directions on the 16-cell honeycomb is probably based on the corresponding directions on the dual 24-cell honeycomb.
Neighbourhoods on the 16-cell honeycomb
A von Neumann-style neighbourhood can be constructed by considering the 16 adjacent cells to a chosen central cell. A Moore-style neighbourhood can be constructed by combining the set of all of the cells sharing each of the eight vertices of a chosen cell.
Just as in the triangular tiling a Moore-style neighbourhood is built up from a 3 cell von Neumann-style neighbourhood by first adding the 6 cells of the outer neighbourhood followed by the 3 cells of the radiation neighbourhood, so in the 16-cell honeycomb the Moore-style neighbourhood is built up from a 16 cell von Neumann-style neighbourhood by first adding 24 cells of a level two neighbourhood followed by 64 cells of a level three neighbourhood and finally 8 cells of a level four neighbourhood.
In the coordinate system described above the cells of the level two neighbourhood of the origin are given by permutations of (±1,±1,0,0), the cells of the level three neighbourhood are given by permuations of (±3/2,±1/2,±1/2,±1/2), and the cells of the level four neighbourhood are given by permutations of (±2,0,0,0).
Symmetries
The symmetries of the 16-cell honeycomb are the same as those of its dual tiling, the 24-cell honeycomb, which have been enumerated here.
Software support
As of September 2025, there are no known attempts to implement cellular automata on this tiling.