Tesseractic honeycomb

The tesseractic honeycomb (Schläfli symbol {4,3,3,4}) is one of three proper regular tilings of four-dimensional space, alongside the 16-cell honeycomb and 24-cell honeycomb. It is constructed with four tesseracts being placed around each face.

Rules on the tesseractic grid appear to be the only four-dimensional cellular automata actually investigated, likely due to its ease of implementation compared to other candidate grids, due to it being the higher-dimensional generalization of the square tiling and cubic honeycomb.

Coordinates, directions and displacements

Cartesian coordinate system

Much like the square tiling in 2D, the tesseractic honeycomb can be very easily described with Cartesian coordinates, as any combination of four integers will correspond to a unique grid cell.

Notating displacement

Cartesian coordinates allow for the very easy definition of directions and displacements of moving objects; one need only consider said object's displacement in the x, y, z and w directions in order to notate its overall displacement. As such, displacement can be notated as

(x,y,z,w)

where:

• x is the object's displacement in the x-direction (usually the highest value)
• y is the object's displacement in the y-direction (usually the higher of the middle two values)
• z is the object's displacement in the z-direction (usually the lower of the middle two values)
• w is the object's displacement in the w-direction (usually the lowest value)

How many directions an "asymmetric displacement" can correspond to in 4D is yet to be determined.

Displacement notation can also be used to notate directions in general, in which case all four numbers are usually positive and in descending order.

Notating velocity

For periodic moving objects such as spaceships, puffers, breeders and replicators, displacement can be used to notate the velocity of an object. The x-, y-, z- and w-displacement, and period of the object are then combined into a single string which describe the object's motion:

(x,y,z,w)c/p

where:

• x is the object's displacement in the x-direction (usually the highest value)
• y is the object's displacement in the y-direction (usually the higher of the middle two values)
• z is the object's displacement in the z-direction (usually the lower of the middle two values)
• w is the object's displacement in the w-direction (usually the lowest value)
• p is the object's period

c is not a variable, and instead is used to represent the object's velocity in relation to the speed of light.

Symmetric directions

There are four "symmetric" directions of travel which arise on the tesseractic honeycomb:

• orthogonal, in which movement happens only on one axis,
• diagonal, in which movement happens to the same extent on two axes, with no movement on the other two,
• paragonal, in which movement happens to the same extent on three axes, with no movement on the fourth,
• metagonal, in which movement happens to the same extent on all four axes simultaneously

Neighbourhoods on the square tiling

The two most commonly investigated neighbourhoods on the tesseractic honeycomb are higher-dimensional analogues of 2D square grid neighbourhoods:

• the von Neumann neighbourhood is the set of eight tesseracts orthogonally adjacent to the central tesseract;
• the Moore neigbourhood is the set of eighty tesseracts orthogonally, diagonally, paragonally or metagonally adjacent to the central tesseract.

Given that the Moore neighbourhood in 4D is far larger than in 2D with respect to the number of cells, higher ranges do not appear to have been investigated so far.

Symmetries

Static

The static symmetries of the tesseractic honeycomb include the many inherited from the cubic honeycomb.

It is not known if all static symmetries have been enumerated so far, or if they have, if they have been assigned names.

Kinetic

Kinetic symmetries on the tesseractic honeycomb affect oscillators and spaceships. It is not known if these have been properly enumerated either.

Software support

Software support for four-dimensional rules is exceedingly rare, and appears to be almost exclusively relegated to purpose-written programs.

Visions of Chaos is a notable example of a program that supports cellular automata on a tesseractic grid, although without editing capabilities.