About 3D Cellular Automata
A 3D cellular automaton (CA) is a discrete dynamical system defined on a three-dimensional lattice of cells. Each cell is either alive (1) or dead (0). At every generation, every cell simultaneously inspects its neighbors and applies a simple local rule to determine its next state. Despite the simplicity of the rule, emergent global structures — crystals, amoebae, clouds, volcanic flows — appear spontaneously.
The most studied neighborhood in 3D is the Moore neighborhood: all cells within Chebyshev distance 1 in x, y, and z. That gives 3³ − 1 = 26 neighbors. A cell's fate depends only on how many of those 26 neighbors are currently alive — making this an outer totalistic rule.
S/B Rule Notation
Rules are written as S n₁,n₂,… / B m₁,m₂,…:
- S (Survive): a living cell with exactly n ∈ S living neighbors stays alive next generation.
- B (Born): a dead cell with exactly m ∈ B living neighbors becomes alive next generation.
- Any other cell dies (or stays dead).
Because neighbor counts range from 0 to 26, the theoretical rule space is enormous: 2²⁷ × 2²⁷ ≈ 3.6 × 10¹⁶ possible S/B combinations. Only a tiny fraction have been studied — most either die instantly, explode to fill the grid, or converge to random static noise. The presets below are among the most visually rich.
Famous 3D CA Rules
| 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 |
Cell Age Colouring
Cells are coloured by how many generations they have been alive. Newborn cells glow bright amber/yellow (#fbbf24). As cells age through 5 generations they shift to deep orange (#f97316), and cells older than 15 generations burn deep red (#dc2626). This reveals internal structure — new growth is bright at the surface while ancient cores are dark red.
Frequently Asked Questions
What is a 3D cellular automaton?
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.
What is the Moore neighborhood in 3D?
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. Each cell must therefore count up to 26 neighbors before deciding its fate.
How does the survival/birth rule notation work?
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. All other cells die or stay dead.
What makes the "Crystal Growth" rule form crystal-like patterns?
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 on the surface of existing structures.
How does the Amoeba rule create moving blobs?
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 such as Amoeba proteus.
What is a totalistic cellular automaton?
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 2²⁶ possible spatial configurations of those neighbors.
How do 3D cellular automata relate to Conway's Game of Life?
Conway's Game of Life is a 2D cellular automaton with 8 Moore neighbors and a single fixed rule (B3/S23). 3D cellular automata extend this concept to a cubic lattice with 26 Moore neighbors. While GoL's rule space is tiny, 3D CA rules have 27 possible birth counts and 27 possible survival counts — producing an enormous variety of behaviors from crystal growth and cloud formation to chaotic explosion and rapid extinction.
What determines whether a 3D CA rule is stable, explosive, or dying?
Three factors drive long-term behavior: (1) the width of the birth set — wide birth sets cause explosive growth that fills the grid; (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 within a few generations.
Can 3D cellular automata simulate physical phenomena?
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 spreading structures reminiscent of volcanic lava flows. 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 when averaged over many cells.
How does initial density affect 3D cellular automaton behavior?
Initial density strongly influences which rules produce interesting patterns. At low density (5–15%) cells are too sparse for birth conditions requiring many neighbors — 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, which is why the density slider is provided.