Hexeractic honeycomb
The hexeractic honeycomb (Schläfli symbol {4,3,3,3,3,4}) is the only proper regular tiling of six-dimensional space. It is constructed with four hexeracts being placed at each cell.
Rules on the hexeractic grid appear to be the only six-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, cubic honeycomb, tesseractic honeycomb and penteractic honeycomb.
Coordinates, directions and displacements
Cartesian coordinate system
Much like lower n-cubic honeycombs, the hexeractic honeycomb can be very easily described with Cartesian coordinates, as any combination of six 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, w, v and u directions in order to notate its overall displacement. As such, displacement can be notated as
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 second highest value)
- z is the object's displacement in the z-direction (usually the higher of the middle two values)
- w is the object's displacement in the w-direction (usually the lower of the middle two values)
- v is the object's displacement in the v-direction (usually the second lowest value)
- u is the object's displacement in the u-direction (usually the lowest value)
How many directions an "asymmetric displacement" can correspond to in 6D is yet to be determined - the lower bound is 384, though is presumably much higher.
Displacement notation can also be used to notate directions in general, in which case all five 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-, w-, v- and u-displacement, and period of the object are then combined into a single string which describe the object's motion:
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 second highest value)
- z is the object's displacement in the z-direction (usually the higher of the middle two values)
- w is the object's displacement in the w-direction (usually the lower of the middle two values)
- v is the object's displacement in the v-direction (usually the second lowest value)
- u is the object's displacement in the u-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 six "symmetric" directions of travel which arise on the hexeractic honeycomb, corresponding to the five diagonal types of a hexeract: (1,0,0,0,0,0), (1,1,0,0,0,0), (1,1,1,0,0,0), (1,1,1,1,0,0), (1,1,1,1,1,0) and (1,1,1,1,1,1). While, in analogy to the tesseractic honeycomb and below, the first four may be called orthogonal, diagonal, paragonal and metagonal, it is not immediately obvious what the fifth and sixth would be called.
Neighbourhoods on the hexeractic honeycomb
The two most commonly investigated neighbourhoods on the hexeractic honeycomb are higher-dimensional analogues of 2D square grid neighbourhoods:
- the von Neumann neighbourhood is the set of twelve hexeracts orthogonally adjacent to the central hexeract;
- the Moore neigbourhood is the set of seven hundred and twenty-eight hexeracts in any way touching the central hexract.
Given that the Moore neighbourhood in 6D 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 hexeractic honeycomb include the many inherited from the penteractic honeycomb. However, due to the extremely large predicted quantity, there have been no enumeration attempts.
Of interest is the fact that the hexeractic honeycomb has areas of icosahedral symmetry, as its vertices are 6-orthoplexes, which have 12 vertices which are analogous to those of the icosahedron; that is, it implements the icosahedron's orthoplex realization.
Kinetic
Kinetic symmetries on the hexeractic honeycomb affect oscillators and spaceships. There are presumably analogues of the oscillator and spaceship symmetries of the penteractic honeycomb, however there have been no attempts thus far to enumerate these either.
Software support
As of October 2025, the only program known to support rules on the hexeractic honeycomb is NDCell.