Three-dimensional cellular automaton

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A three-dimensional cellular automaton operates in three-dimensional space. Like one- and two-dimensional cellular automata, 3D automata may operate in different neighbourhoods, be totalistic or non-totalistic, isotropic or non-isotropic.

Most commonly, the 3D space is thought of as being divided into a grid of cubic cells. For a 3D version of the Moore neighbourhood, each cell is at the center of a 3 × 3 × 3 neighbourhood, giving it 26 neighbouring cells it touches. For a 3D version of the von Neumann neighbourhood, a cell has 6 neighbours with which it shares a face.

3D Game of Life

In 1987, Carter Bays wrote a paper analyzing what it meant to project the Game of Life into a 3D universe with cubic cells in which a cell has 26 neighbours instead of the 8 neighbours in 2D. Bays proposed two criteria for such a rule having Life-analogous behaviour:

  1. Gliders of some sort must occur naturally from soups
  2. Soups must exhibit bounded growth

Simply applying the B3/S23 rule in 3D space does not meet criterion 2. Having birth on 4 neighbours or fewer results in lightspeed expansion similar to B2 rules in 2D.

One way to allow birth on just a few neighbours without lightspeed expansion is to have birth on three cells as long as the three cells and the cell to be born are all orthogonally coplanar.

In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules.[1] Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S567 produces behaviour similar to Life.[2][note 1]


3D CA in Golly[3]
3D CA in Ready
3D CA in Visions of Chaos

See also


  1. While ash looks similar, there is one major difference, and that's that soups settle much more quickly in the 3D rule than in Life.


  1. Carter Bays (2006). "A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life". Complex Systems.
  2. Carter Bays (1987). "Candidates for the Game of Life in Three Dimensions". Complex Systems.
  3. Andrew Trevorrow (May 15, 2018). 3D.lua (discussion thread) at the forums


Forum threads