Three-dimensional cellular automaton

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.

In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules. Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S57 produces behaviour similar to Life.

See also

• One-dimensional cellular automaton

External links

• WOLFRAM Demonstrations Project - 3D Totalistic Cellular Automata
• Stephan Wolfram, A New Kind of Science, Chapter 5: Two Dimensions and Beyond
• 3D Cellular Automata on Softology's Blog