Beyond Conway: The Universe of Life-Like Rules

Conway’s Game of Life is not a discovery. It is a choice.

Someone had to decide: born with exactly 3 neighbors, survives with 2 or 3. Those numbers were not derived from first principles. Conway found them by trial and observation, sitting on the floor of the Cambridge mathematics common room, pushing Go stones around a board. He tried hundreds of rules before settling on B3/S23. Most of them were uninteresting. This one was extraordinary.

The question that follows immediately — and which the cellular automaton research community has spent fifty years exploring — is: how extraordinary is it? Is Conway’s Life a unique gem in a featureless desert of boring rules, or is it one rich region in a landscape full of rich regions? And what does the answer tell us about the nature of complexity itself?

The answer turns out to be: somewhere between those two pictures. Conway’s Life is not the only interesting rule. But interesting rules are genuinely rare. And their distribution across the rule space — where they cluster, how they relate to each other, what they have in common — is one of the most revealing things we know about the mathematics of emergence.

The Space of All Life-Like Rules

The framework that makes this question tractable is B/S notation — the language for describing any Conway-style cellular automaton. Every cell has two states (alive or dead). Every cell has eight neighbors (the Moore neighborhood). At each tick, every cell updates based on exactly two facts: whether it is currently alive or dead, and how many of its eight neighbors are alive.

That’s all the information available. The rule must decide, for each combination of (current state, neighbor count), what the next state will be.

This reduces to two lists:

B (birth): which neighbor counts (0 through 8) cause a dead cell to become alive
S (survival): which neighbor counts (0 through 8) cause a live cell to stay alive

Each list is a subset of {0, 1, 2, 3, 4, 5, 6, 7, 8}. A set with nine elements has 2⁹ = 512 possible subsets. So the total number of possible rules is 512 × 512 = 262,144.

Conway’s Life — B3/S23 — is one point in this space of a quarter-million universes.

Read the full introduction to B/S notation →

The Distribution Problem: Most Rules Are Boring

Run any random rule on a random initial configuration and you will almost certainly see one of two things:

Everything dies. The grid rapidly empties and stays empty. This happens when survival conditions are too tight — live cells can’t find enough neighbors to cling to, or birth conditions are too weak to replace the ones that die. Rules like B/S (no births, no survival) or B123456789/S (births always, survival never) die off almost instantly.

Everything solidifies. The grid rapidly fills with live cells and stays full. This happens when birth conditions are too aggressive or survival conditions are too permissive. The grid reaches a static or near-static maximum density from which it rarely recovers.

Between these two attractors — pure death and pure life — lies a narrow band of rules where something interesting happens. The balance between birth and death is close enough to produce dynamics that neither explode nor collapse, and complex enough to support objects with internal structure.

Christopher Langton, studying cellular automata in the early 1990s, developed a parameter he called lambda (λ): the fraction of a rule’s transitions that result in a live cell. Lambda = 0 means everything dies; lambda = 1 means everything lives. Conway’s Life has λ ≈ 0.273 — well below the midpoint, slightly on the ordered side. Langton found that the rules with the richest behavior clustered near a critical value of lambda, a phase transition between order and chaos that he called the “edge of chaos.”

The phrasing is evocative, and later researchers found it somewhat oversimplified — lambda alone doesn’t fully predict complexity, and not all “interesting” rules cluster neatly at the same lambda value. But the core insight survives scrutiny: the interesting rules are not uniformly distributed across rule space. They cluster near a phase transition, and that transition is what Conway found by hand in 1968.

The Neighborhood of Conway’s Life

One of the most illuminating ways to explore rule space is to look at Conway’s Life’s immediate neighbors — rules that differ from B3/S23 by exactly one element.

Add 6 to the birth list: B36/S23 (HighLife). Almost identical behavior to Life, with the same gliders, the same still lifes, the same overall texture. But with one crucial difference: HighLife supports a natural replicator — a pattern that produces exact copies of itself every 12 generations without any external machinery. This pattern doesn’t exist in Conway’s Life.

Add 5 to the birth list: B35/S23. The rule becomes somewhat more chaotic; patterns tend to grow faster and stabilize less reliably.

Remove 3 from survival: B3/S2. The rule becomes much more volatile; most configurations rapidly explode or die.

Add 4 to survival: B3/S234. More stable than Life; configurations tend to solidify into dense still-life fields.

What this neighborhood tour shows is that Life sits in a region of rule space where small perturbations produce qualitatively similar but measurably different dynamics. It is not a knife-edge singularity — the interesting behavior is somewhat robust to small rule changes. But the neighborhood is not large. Wander too far and the behavior degrades rapidly.

This structure — a small interesting region surrounded by boring rules in most directions — is consistent with the edge-of-chaos picture. And it explains why the search for interesting rules proceeds mostly by exploring the neighborhoods of already-known interesting rules.

The Major Variants

Over fifty years of exploration, a handful of rules have been studied enough to constitute genuine fields of knowledge. Each illuminates something different about the relationship between local rules and global behavior.

HighLife (B36/S23)

Discovered by Nathan Thompson in 1994, HighLife is Conway’s closest studied relative. It shares all of Life’s most common objects — the block, the blinker, the glider, the LWSS — but it supports something Life doesn’t: a replicator.

The HighLife replicator is a compact pattern of about 22 cells that, after exactly 12 generations, has produced a second copy of itself displaced four cells diagonally. From two copies it makes four, from four it makes eight, doubling on a geometric schedule at positions forming a diagonal line. It spreads at speed c/6. It occurs naturally from random initial configurations — frequently enough that watching a random soup in HighLife, you will eventually see replication start spontaneously.

The replicator is why HighLife is the most studied Conway variant. It is also why it poses one of the deepest questions in the field: if self-replication arises this easily from a minor rule change, what does that say about the difficulty of self-replication in general? Read the full HighLife page →

Day & Night (B3678/S34678)

Invented by Nathan Thompson in 1997 and explored in depth by David I. Bell the following year, Day & Night has a structural property that no other well-known rule possesses: perfect alive/dead symmetry.

In B3678/S34678, the fate of a cell depends on its neighbor count in a way that is invariant under the exchange of alive and dead. If you take any pattern in Day & Night and flip every cell — making every live cell dead and every dead cell alive — the result evolves exactly as the complement of the original pattern would evolve. Every still life has a “negative” that is also a still life. Every glider has a mirror-image twin that moves in the same direction at the same speed.

The rule is called Day & Night because the “daytime” world (sparse live cells on a dead background) and the “nighttime” world (sparse dead cells on a live background) are mathematically identical. Read the full Day & Night page →

Seeds (B2/S)

Seeds is the extreme case on one end of the birth-survival balance. The survival list is completely empty: every live cell dies every generation, no exceptions. The birth list has exactly one entry: 2. A dead cell with exactly two live neighbors is born.

No cell ever survives. Yet Seeds is not trivial — far from it. The birth rule applied to its own products propagates patterns explosively across the grid. Even a single pair of live cells expands into a rapidly growing diagonal structure. Most Seeds patterns grow forever without bound.

What Seeds lacks is any form of memory. Because no cell ever survives, Seeds has no still lifes (by definition) and no traditional oscillators. The concept of stability barely applies. Seeds is the cellular automaton equivalent of a reaction with no inhibitor — pure activation, no brake. Read the full Seeds page →

Replicator (B1357/S1357)

B1357/S1357 — known as the Replicator rule — has a property that sounds impossible until you understand why it’s true: every pattern, without exception, replicates. A single cell replicates. A random scribble replicates. A pattern that spells your name in binary replicates.

The rule was studied by Edward Fredkin and is sometimes called Fredkin’s rule. Its universal replication property follows from the mathematics of XOR: since the rule fires on odd neighbor counts, the state of any cell after n generations is the XOR (parity) of the contributions from each initial live cell independently. At generation 2ⁿ, a single initial cell produces eight copies of itself at positions ±2ⁿ along each axis. By superposition, any initial pattern produces eight copies of itself at those same displacements. The copies then copy themselves. The result looks like a Sierpinski triangle growing across the grid. Read the full Replicator page →

SmoothLife

In 2011, Stephan Rafler submitted a paper to arXiv (arXiv:1111.1567) proposing a generalization of Conway’s Life that dissolves the grid entirely. The square lattice becomes a continuous 2D field. Alive and dead become real-valued field intensities between 0 and 1. The step-function transitions of Life become smooth sigmoid functions. The 8-cell Moore neighborhood becomes an annular integration region with an inner disk (the “cell”) and an outer ring (the “neighborhood”).

The result — called SmoothLife — looks nothing like Conway’s Life visually: patterns move fluidly, without the pixelated snapping of discrete grids. It looks organic. But the dynamical structure is recognizably similar: there are persistent objects, there are moving objects (smooth gliders), there are objects that grow and objects that shrink. The behaviors that define Life’s richness survive the move to continuous space. Read the full SmoothLife page →

Generations Rules (Brian’s Brain, Star Wars)

Generations rules add a third state to the two-state Life framework: dying. When a live cell fails its survival condition, instead of immediately dying, it enters a dying state — or a sequence of dying states — before finally expiring. The dying cells don’t interact directly with birth conditions; only fully live cells count as neighbors for birth purposes.

The visual effect is trails: patterns leave colored ghosts behind them, fading over successive generations. The dynamical effect is more significant: dying states act as a kind of refractory period, preventing live cells from being born back into the space they just vacated. This refractory effect radically changes what patterns can exist.

Brian’s Brain (B2/S/3 — born at 2, never survives, 3 total states) was devised by computer scientist Brian Silverman and first described in print in Toffoli and Margolus’s 1987 book Cellular Automata Machines. In Brian’s Brain, almost every pattern is a spaceship. The rule produces an explosion of gliders from almost any starting configuration.

Star Wars (B2/S345/4), devised by Mirek Wójtowicz in March 1999, produces light-speed spaceships (“fireballs”) and the visual impression of a dogfight, with glowing trails behind every moving object. Read the full Generations page →

2×2 and Morley

A few other rules are worth naming.

2×2 (B36/S125), studied extensively by Nathaniel Johnston in a 2012 paper, gets its name from the fact that patterns made of 2×2 blocks continue to evolve as patterns made of 2×2 blocks. It has a natural diagonal glider, rich oscillator families, and chaotic-but-eventually-stable behavior similar to Life — but the objects look nothing like their Life equivalents. Johnston proved the existence of oscillators with period 2^ℓ(2^k − 1) for any integers k, ℓ ≥ 1.

Move (Morley, B368/S245) was proven Turing-complete in August 2020 by Layz Boi, who constructed a Rule 110 unit cell within it — meaning Move can simulate any Rule 110 computation, which is itself Turing-complete. Move supports 16 known gliders and a rich ecosystem of slow-moving spaceships.

Extensions: Beyond the 262,144

The 262,144 rules above all share the same fundamental structure: two states, square grid, eight-cell Moore neighborhood. Relax any of these constraints and you get a different family of rules entirely.

Generations rules add dying states, expanding the rule space dramatically. Larger Than Life rules expand the neighborhood radius — examining cells two, three, or more steps away — enabling very large patterns to interact as if they were single objects. Non-totalistic rules distinguish between different arrangements of the same number of neighbors, giving the rule access to much finer geometric information.

And then there’s SmoothLife, which dissolves the grid altogether — a different kind of generalization that asks what Life-like behavior looks like when the discreteness of the substrate is removed entirely.

Each extension is a different way of asking: what features of Conway’s Life are essential, and which are accidents of its particular formulation? The answer, consistently, is that fewer features are essential than you might expect. The emergence of complex behavior from local rules is robust across a surprising range of substrates.

What the Variants Tell Us About Life

The most important thing the variants reveal is not any individual pattern or behavior — it is the shape of the landscape they collectively define.

Conway’s Life is not an isolated miracle. It is one high point in a rough terrain that has multiple high points, each approached through nearby regions of interesting-but-lesser behavior. HighLife is one such high point, reached by moving one step from Life in the birth direction. Day & Night is another, reached by moving far across the space toward the symmetry axis. Brian’s Brain is reached by crossing the boundary into multi-state rules entirely.

The high points are rare — the vast majority of rule space is dead or frozen. But they are not vanishingly rare. They cluster near phase transitions. They share structural properties: balance between birth and death, sensitive dependence on initial conditions, support for persistent moving objects.

The picture this paints is that emergence — the spontaneous appearance of complex organized behavior from simple local rules — is not a miraculous accident that happened to occur in one particular universe. It is a feature of a certain region of the space of possible rule systems, a region that can be approached from multiple directions and that has internal structure.

Conway found one path into that region by trial and error in the late 1960s. The variants show us the region itself.