3D Life is a three-dimensional extension and exploration in the variants of Conway's Game of Life. It was first discovered Carter Bays. A number of different semitotalistic rules for the 3D rectangular Moore neighborhood were investigated. It was popularized by A. K. Dewdney in his "Computer Recreations" column in Scientific American magazine.

Background

Recontextualizing Conway's Game of Life

In every step of Conway's Game of Life follows four rules: These events may be simplified in a way where we emphasize a given cells next state based on the range of the number of that cell's neighbors This reworking of the rules allows us to formalize them. We define E as the number of living neighbors required to prevent a currently living cell from dying, which upper and lower limits E_l and E_u respectively. Similarly, we define F as the number of living neighbors required to create a new living cell, with upper and lower limits F_l and F_u respectively. We define a transition rule R that states the following: This transition rule may further be defined as the 4-tuple. For example, Conway's Game of Life has a transition rule. We can use this transition rule to create different variants of the Game of Life. For instance, an automaton with results in an explosive variant of the Game of Life called 3-4 Life, and one of the earliest studied variants of the famous automaton.

The third dimension and defining a Game of Life

Because of the transition rule's complete independence from the number of dimensions, we may translate the transition rule into the third dimension. Each cell can have from zero to 26 living neighbors, greatly increasing the number of possible rules from 1296 in two dimensions to 123,201 in three dimensions. Unsurprisingly, many of the transition rules either decays too quickly, expands too quickly, or doesn't create anything interesting in a given "primordial soup" seed (ex: gliders, spaceships, blinkers, etc.) We loosely formalize a Game of life like so: An automaton with a transition rule is a Game of Life if and only if both of the following are true.