Generations Rules: Introducing Cell Age

In Conway’s Life, death is instantaneous. A cell that fails its survival condition is alive one tick and completely gone the next. There is no transition, no trace, no lingering. The cell contributes to one generation’s neighbor counts, and then it vanishes.

Generations rules change this. When a live cell fails its survival condition, instead of dying immediately, it enters a dying state — a kind of cellular afterlife in which it is no longer counted as a neighbor by birth rules, but it still exists, still has a color, still occupies space. After one generation in the dying state, the cell enters the next dying state, if there is one. After exhausting all dying states, it finally expires and the space goes blank.

The visual effect is immediate and striking: patterns leave colored trails, ghosts of their previous positions fading over successive generations. A spaceship in a Generations rule looks like it has a comet tail. An oscillator leaves a pulsing halo. The grid becomes a time-lapse image of itself, with recent history always visible.

The dynamical effect is more significant. Those dying cells are not merely decorative. They are absent from the birth calculation — a dying cell is not counted as a live neighbor — which means they create a refractory zone: a region where new births cannot be triggered until the dying cells expire. This refractory effect changes what patterns can exist, how they move, and how they interact. The same birth condition that produces explosively unstable behavior in two-state rules can, with dying states, produce stable, directed, glider-rich dynamics.

The Generations Notation

Generations rules are specified with three parameters: B (birth conditions), S (survival conditions), and G (the total number of states including alive and dead).

The notation is B[birth]/S[survival]/G[number of states], where:

State 0 is dead (blank)
State 1 is fully alive (counts as a neighbor for birth/survival purposes)
States 2, 3, …, G-1 are dying states (do not count as neighbors)
A state-(G-1) cell transitions to state 0 (dead) on the next tick

G=2 recovers ordinary Life-like rules: alive or dead, no intermediate states. G=3 gives one dying state — the simplest non-trivial Generations rule. G=4 gives two dying states. G can be any integer.

The key constraint is that only state-1 cells (fully alive) count as neighbors for both birth and survival. Dying cells (states 2 through G-1) are invisible to the rule — they don’t help trigger births or count toward survival. This is what makes them “refractory” rather than merely “decaying alive.”

Brian’s Brain

The most celebrated Generations rule is Brian’s Brain, specified as B2/S/3:

A dead cell is born if it has exactly 2 live (state-1) neighbors.
A live cell never survives (the survival list is empty).
G = 3: one dying state between alive and dead.

The first thing to notice is the survival rule: empty, just like Seeds (B2/S). Every live cell becomes a dying cell on the next tick, no exceptions. But unlike Seeds, dying cells don’t disappear immediately — they linger for one generation in the dying state, during which they are invisible to the birth rule.

The second thing to notice is that this dying-state invisibility transforms the dynamics completely. In Seeds, a live cell and a dying cell have the same effect on their neighbors (they are both present and both die next tick). In Brian’s Brain, a dying cell has no effect on births. The refractory period means that the wake behind a moving pattern cannot immediately refill — there’s a one-generation delay.

This delay changes everything.

Brian’s Brain was devised by computer scientist Brian Silverman, whose earlier Seeds rule investigation gave him the intuition that birth-at-2 with no survival could produce rich dynamics if the dying cells could be made refractory. The rule was first described in print by Tommaso Toffoli and Norman Margolus in their landmark 1987 book Cellular Automata Machines, which remains one of the foundational texts of the field.

The glider explosion

Run Brian’s Brain from almost any starting configuration, and within a few dozen generations you will see something extraordinary: the pattern explodes into a chaotic mess that quickly organizes itself into dozens — sometimes hundreds — of independently moving spaceships, all traveling diagonally at various speeds.

The reason is the refractory period. In Brian’s Brain, the dying state creates a kind of directionality: a cluster of live cells, surrounded by dying cells behind it and empty space ahead, is more likely to trigger births in the empty-space direction than in the dying-space direction. The dying cells prevent backfilling. The result is a self-reinforcing asymmetry: live cells propagate forward, and the refractory zone behind them prevents the pattern from collapsing back on itself.

Almost every Brian’s Brain pattern is a spaceship. The rule produces far more gliders per unit area of initial configuration than any two-state rule. This was the key insight Silverman was after: the refractory period, by preventing backward propagation, channels the system’s activity into persistent forward motion.

A 2×2 block of live cells in Brian’s Brain becomes an ever-expanding diamond: four diagonal wavefronts traveling outward at the rule’s speed of light, leaving a complex interior of smaller patterns behind.

Star Wars

Star Wars, specified as B2/S345/4, was devised by Mirek Wójtowicz in March 1999:

A dead cell is born if it has exactly 2 live (state-1) neighbors.
A live cell survives if it has 3, 4, or 5 live neighbors.
G = 4: two dying states.

The birth condition is the same as Brian’s Brain: birth at 2. But Star Wars adds a non-trivial survival condition: cells survive with 3, 4, or 5 neighbors. This means that dense clusters of live cells can maintain themselves — stable or semi-stable structures can exist alongside the refractory zones created by the two dying states.

The visual character of Star Wars lives up to its name. Random starting configurations produce explosions of light-speed spaceships — small, very fast objects moving in all directions — that collide, combine, and scatter. The dying cells create luminous trails behind every object. Collisions produce flashes of color as dying cells from different patterns interfere.

The most characteristic Star Wars pattern is the fireball: a tiny structure (typically just a few live cells) that moves at c (one cell per generation, the rule’s theoretical maximum) leaving a trail of two-generation-long colored traces. Fireballs appear spontaneously and are extraordinarily abundant. They are the “photons” of the Star Wars universe — small, fast, indestructible, everywhere.

The survival conditions (S345) allow denser structures to persist alongside the fireballs: slow-moving clumps that periodically shed fireballs, or stable islands that deflect fireballs without being destroyed. The combination of fast (c) and slow patterns gives the rule its dogfight character.

What Dying States Add: Memory Without Storage

The dying states in Generations rules have a specific dynamical function that is worth making explicit: they create local temporal memory.

In a two-state rule, the grid at time t contains no information about time t-1 except what is implicit in the birth-and-survival dynamics of the current configuration. A live cell “remembers” nothing of its past; it responds only to its current neighbors.

In a Generations rule, the dying cells at time t are literally the cells that were alive at time t-1, t-2, etc. They are the grid’s explicit recent history, encoded in the dying-state color. A cell looking at its neighborhood can “see” not just the current live cells, but the recent past — the positions where live cells were in the previous one or two generations.

This is not true memory in the computational sense — the dying cells don’t influence births — but it creates a geometric record of recent motion that shapes the environment into which the next generation is born. The refractory zone that prevents backfilling is precisely this: the past positions of live cells, lingering just long enough to block their immediate replacement.

This one-or-two-generation memory is enough to create qualitatively new dynamics. Directed motion (Brian’s Brain) and persistent fast objects (Star Wars) are both consequences of the refractory zone — patterns that cannot exist in two-state rules with the same birth and survival conditions.

Cyclic Cellular Automata

Cyclic CA are a related multi-state family with a different structure from Generations rules, but worth discussing here because they share the key feature of sequential state transitions and produce some of the most visually striking dynamics in cellular automaton theory.

In a Cyclic CA with k states (labeled 0, 1, 2, …, k-1), a cell advances from state n to state n+1 (mod k) if any of its neighbors is in state n+1. Otherwise it stays at state n. Every cell is always “trying” to advance to the next state, and it succeeds whenever a suitable neighbor is present.

The result is rotating spiral waves that propagate across the grid. Any initial configuration with more than k/2 different states quickly organizes into stable spiral structures: target patterns, rotating pinwheels, and complex interactions between them. The spirals expand at a fixed speed, collide, and annihilate, leaving behind new spirals.

The spirals in Cyclic CA strongly resemble two biological phenomena: cardiac arrhythmia (rotating electrical waves in heart muscle, which underlies ventricular fibrillation) and Belousov-Zhabotinsky reactions (the chemical oscillating reactions that produce spiral waves in a shallow dish of reagents). The mathematical reason for the resemblance is genuine: all three systems — Cyclic CA, cardiac muscle, and BZ reactions — are examples of excitable media: systems where a stable state can be triggered into activity, which then propagates to neighboring elements, followed by a refractory period.

The biological connections of Generations rules and Cyclic CA are not superficial. The refractory period in Brian’s Brain and Star Wars is the same mechanism that makes cardiac tissue directional: once a cell has fired, it cannot fire again immediately, and this forces electrical waves in the heart to propagate in one direction rather than returning on themselves. Understanding the cellular automaton version is a direct route to intuition about the biological version.

The Space of Generations Rules

The space of Generations rules is much larger than the space of two-state Life-like rules.

For a rule with G states, the birth and survival conditions are each subsets of {0, 1, …, 8} (still 512 × 512 = 262,144 combinations), but now combined with a choice of G from 2 to any value. The total space is infinite if G is unbounded — but for practical exploration, researchers typically focus on G = 3 (one dying state) through G = 6 or 7.

At G = 3, the space is exactly 262,144 rules, each with one dying state. Most are as boring as most two-state rules — they explode or die immediately. A small fraction produce interesting behavior.

Brian Silverman found Brian’s Brain by focusing on the B2/S condition (which he knew from Seeds was interesting at two states) and adding the dying state. Wójtowicz found Star Wars by systematic exploration. The broader Generations space has been explored heuristically — by tweaking known interesting rules and seeing what changes — rather than systematically mapped.

The general principle that has emerged from this exploration is: dying states are most interesting when combined with birth conditions that produce near-critical behavior in the two-state version. B2/S (Seeds) at two states produces explosive growth; at three states (Brian’s Brain) the dying state stabilizes the explosiveness into directed motion. The dying state acts as a regulator, dampening the most extreme tendencies of the birth rule into something more sustained.

This suggests a general design intuition: if you want a rich Generations rule, start with a two-state rule that is too explosive to be interesting on its own, and add dying states to tame the explosion. The dying states’ refractory effect channels the surplus energy into persistent moving structures rather than undirected growth.