Question 1

What is a 3D cellular automaton?

Accepted Answer

A 3D cellular automaton is a grid of cells arranged in three-dimensional space, where each cell is either alive or dead. At each time step, every cell simultaneously examines its neighbors and applies a fixed rule to decide whether it lives, dies, or is born in the next generation. The result is complex, emergent behavior arising from very simple local rules.

Question 2

What is the Moore neighborhood in 3D?

Accepted Answer

In 3D, the Moore neighborhood of a cell consists of all cells within a Chebyshev distance of 1 — that is, all cells that differ by at most 1 in each of the x, y, and z axes. This gives 3³ − 1 = 26 neighbors (excluding the cell itself), compared to 8 neighbors in the 2D Moore neighborhood.

Question 3

How does the survival/birth rule notation work?

Accepted Answer

Rules are written in S/B notation where S is the set of neighbor counts for which a living cell survives, and B is the set of neighbor counts for which a dead cell becomes alive (is born). For example, Crystal Growth uses S=5,6,7 and B=6: an alive cell survives if exactly 5, 6, or 7 of its 26 neighbors are alive; a dead cell becomes alive only if exactly 6 neighbors are alive.

Question 4

What makes the "Crystal Growth" rule form crystal-like patterns?

Accepted Answer

The Crystal Growth rule (S 5,6,7 / B 6) has a very narrow birth condition — cells only become alive when they have exactly 6 living neighbors. This creates stable, faceted structures that grow outward in angular layers similar to real crystal lattices, because new cells can only attach at specific, geometrically constrained positions.

Question 5

How does the Amoeba rule create moving blobs?

Accepted Answer

The Amoeba rule uses a very broad survival set (S 9–26) and a selective birth set (B 5,6,7,12,13,15). Cells in dense regions survive easily, while new cells are born only in moderately populated zones. This creates large amorphous blobs that extend pseudopod-like protrusions into less-populated space, mimicking the motion of single-celled organisms.

Question 6

What is a totalistic cellular automaton?

Accepted Answer

A totalistic cellular automaton is one whose rules depend only on the total count of living neighbors, not their spatial arrangement. The 3D automata simulated here are outer totalistic — the next state depends on the current cell state and the sum of its neighbors. This greatly reduces the rule space: only 27 neighbor counts (0–26) matter, rather than all possible spatial configurations.

Question 7

How do 3D cellular automata relate to Conway's Game of Life?

Accepted Answer

Conway's Game of Life is a 2D cellular automaton with 8 Moore neighbors. 3D cellular automata extend this to a 20×20×20 (or similar) cube with 26 Moore neighbors. While GoL uses a single fixed rule (B3/S23), 3D automata explore an enormous rule space — survival and birth sets each drawn from 0–26 — producing an incredible variety of behaviors from crystal growth to chaotic explosion.

Question 8

What determines whether a 3D CA rule is stable, explosive, or dying?

Accepted Answer

Three factors drive long-term behavior: (1) the width of the birth set — wide birth sets cause explosive growth; (2) the width of the survival set — narrow survival sets cause rapid die-off; (3) their interaction at typical densities. Rules with birth and survival counts clustered near the expected number of alive neighbors at a given density tend to self-regulate into stable or oscillating structures. Rules mismatched to density explode or collapse.

Question 9

Can 3D cellular automata simulate physical phenomena?

Accepted Answer

Yes. Certain 3D CA rules approximate diffusion, crystal growth, dendritic solidification, and even fluid-like flows. The Pyroclastic rule (S 4,5,6,7 / B 6,7,8) produces hot-lava-like spreading structures. Researchers have used related lattice-gas cellular automata to model actual fluid dynamics and reaction-diffusion systems, since local parallel rules can approximate continuous differential equations.

Question 10

How does initial density affect 3D cellular automaton behavior?

Accepted Answer

Initial density strongly influences which rules produce interesting patterns. At low density (5–15%) cells are too sparse for birth conditions requiring many neighbors to fire — the automaton dies out. At high density (40–50%) overcrowding kills cells faster than births can replace them. The sweet spot (20–35%) produces the richest behavior, where local clusters can sustain themselves and propagate. Each preset rule has an ideal density range for interesting dynamics.

Name	Survive	Born	Behavior
Cloud 1	13–26	13,14,17,18,19	Billowing cloud masses
Crystal Growth	5,6,7	6	Angular crystal lattices
Amoeba	9–26	5,6,7,12,13,15	Crawling amoeba blobs
Pyroclastic	4,5,6,7	6,7,8	Lava-like spreading flow
445	4	4,5	Sparse wandering cells

🔳 3D Cellular Automaton — Survival/Birth Rules

About 3D Cellular Automata

S/B Rule Notation

Famous 3D CA Rules

Cell Age Colouring

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