Penteractic honeycomb
The penteractic honeycomb (Schläfli symbol {4,3,3,3,4}) is the only proper regular tiling of five-dimensional space. It is constructed with four penteracts being placed at each cell.
Rules on the penteractic grid appear to be the only five-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 and tesseractic honeycomb.
Coordinates, directions and displacements
Cartesian coordinate system
Much like lower n-cubic honeycombs, the penteractic honeycomb can be very easily described with Cartesian coordinates, as any combination of five 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 and v 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 in the middle)
- w is the object's displacement in the w-direction (usually the second lowest value)
- v is the object's displacement in the v-direction (usually the lowest value)
How many directions an "asymmetric displacement" can correspond to in 5D 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- and v-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 in the middle)
- w is the object's displacement in the w-direction (usually the second lowest value)
- v is the object's displacement in the v-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 five "symmetric" directions of travel which arise on the penteractic honeycomb, corresponding to the five diagonal types of a penteract: (1,0,0,0,0), (1,1,0,0,0), (1,1,1,0,0), (1,1,1,1,0) and (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 would be called,
Neighbourhoods on the penteractic honeycomb
The two most commonly investigated neighbourhoods on the penteractic honeycomb are higher-dimensional analogues of 2D square grid neighbourhoods:
- the von Neumann neighbourhood is the set of ten penteracts orthogonally adjacent to the central penteract;
- the Moore neigbourhood is the set of two hundred and forty-two penteracts in any way touching the central penteract.
Given that the Moore neighbourhood in 5D 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 penteractic honeycomb include the 686 inherited from the tesseractic honeycomb. However, due to the extremely large predicted quantity, there have been no enumeration attempts.
Kinetic
Kinetic symmetries on the penteractic honeycomb affect oscillators and spaceships. There are presumably analogues of the 3542 oscillator symmetries and unknown number of spaceship symmetries of the tesseractic honeycomb, however there have been no attempts thus far to enumerate these either.
Software support
As of October 2025, the only programs known to support rules on the penteractic honeycomb are Visions of Chaos, which permits the 242-cell Moore and 10-cell von Neumann neighbourhoods (patterns cannot be edited, however), as well as NDCell.