Still Lifes: Patterns That Never Change

The word “still” is almost misleading here. A still life in Conway’s Game of Life is not still in any passive or accidental sense. It is still in the way a suspension bridge is still — an object under constant stress, held motionless by forces that exactly cancel each other. Every live cell in a still life is simultaneously constrained by two requirements: it must have two or three neighbors (or it dies), and every dead cell at its boundary must have too few or too many live neighbors (or it springs to life and destroys the equilibrium). The whole structure is a negotiation. The wonder is not that still lifes exist but that the four rules of Life permit any configuration to be so precisely balanced at all.

That they are common — the Block appears roughly once per 473 cells in any large random soup — is stranger still. The rules were not designed to favor stability. Conway was searching for rules that would produce interesting, unpredictable behavior. He got that, but he also got, as a side effect, an enormous catalog of configurations that are permanently, unshakeably inert.

The Mathematics of Standing Still

A still life satisfies exactly two conditions simultaneously, for every cell in and around the pattern:

Every live cell has exactly 2 or 3 live neighbors (survival condition).
Every dead cell adjacent to the pattern has either 1, 2, 4, 5, 6, 7, or 8 live neighbors — anything except 3 (the birth condition cannot fire).

Both conditions must hold for every cell, every generation, forever. This is a constraint satisfaction problem, and the constraints are stringent. Most small configurations fail: they either have a live cell with only 1 neighbor (which dies), or a dead cell at the edge with exactly 3 live neighbors (which is born, immediately perturbing everything adjacent to it).

The minimum cell count for a still life is 4. You cannot make a stable pattern with 1, 2, or 3 cells: a single live cell has zero neighbors and dies; two live cells each have one neighbor and die; three cells in any arrangement either have an exposed cell with one neighbor, or create a dead cell with exactly three live neighbors. Four cells is the minimum, and only two distinct 4-cell still lifes exist.

Strict vs. Pseudo Still Lifes

The catalog distinguishes strict still lifes — connected configurations (or multi-island patterns where no island can be removed without destabilizing the rest) — from pseudo still lifes, which consist of two or more stable islands that happen to interact in a way that keeps a dead cell between them with exactly the wrong number of neighbors to be born.

The simplest pseudo still life is two Blocks placed adjacent but not touching, in a position where a dead cell between them has exactly four live neighbors (above the birth threshold). Each Block is stable by itself; together, they form a new stability. This distinction matters for enumeration: at low cell counts, virtually all still lifes are strict. As cell count grows, pseudo still lifes proliferate faster.

The Block

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Four cells. A 2×2 square. Every live cell has exactly three live neighbors; every dead cell immediately adjacent to the Block has either two live neighbors (the cells on the edges) or one live neighbor (the cells at the corners). None has three. Nothing is born. Nothing dies. The Block is perfectly, permanently stable.

The Block is the most common object in Life — the most common still life, the most common object of any kind. In the statistics compiled by the Catagolue project (which has processed over a trillion random soups), the Block dwarfs every other pattern in raw frequency. It forms because the Block is an attractor: many chaotic dynamics happen to produce a 2×2 square as a byproduct of their final stabilization. Glider collisions often leave Blocks. Active regions cool into Blocks. The four-cell configuration is so stable, and so easy to produce, that it is effectively the resting state of the Life universe.

The Block also serves as a functional component in larger constructions. The two stabilizing anchors in the Gosper Glider Gun are Blocks. The “eater” pattern — a still life that can absorb an incoming glider without being destroyed — is built around a Block-like structure. The most common still life in Life is also one of its most useful tools.

The Beehive

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Six cells, forming a ring with two live cells on each of three rows. The Beehive is named for its hexagonal appearance — it really does look like a section of honeycomb. It is the second most common still life in Life (after the Block) and the most common 6-cell still life.

The Beehive’s stability arises from its ring structure: each live cell has exactly two neighbors (the cells on either side of it in the ring). Every dead cell adjacent to the Beehive has either 1 neighbor (the cells above and below the top and bottom pairs) or 2 neighbors (the cells between the pairs). None has 3.

The Beehive is interesting not just as a stable object but as evidence that Life’s equilibrium conditions admit multiple essentially different solutions at low cell counts. The Block’s stability comes from density — four cells packed together. The Beehive’s stability comes from connectivity — six cells arranged so that the ring topology prevents both underpopulation and unwanted births.

The Loaf

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Seven cells. Like the Beehive, but with a “corner” removed from one end, creating an asymmetric but stable configuration. The Loaf is the third most common still life in natural soups. Its name is informal but apt — it has the shape of a rectangular bread loaf viewed from above, cut at one corner.

Each live cell in the Loaf has exactly 2 or 3 live neighbors; no dead adjacent cell has exactly 3 live neighbors. The asymmetry of the Loaf compared to the Beehive is instructive: still life equilibrium does not require symmetry. Many still lifes have no symmetry at all. The rules care only about neighbor counts; the arrangement can be arbitrarily lopsided as long as those counts come out right.

The Boat

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Five cells. The Boat is the only 5-cell strict still life (a strict still life because removing any cell would destabilize it). It looks like a small sailing vessel with a prow, which is exactly why Conway or his colleagues named it as they did. It is the fourth most common still life in soups.

The Boat is notable for being the minimum-size example of a “tail”: the single cell at the bottom-left of the structure is held in place not by the cells directly adjacent to it but by the geometry of the whole. Move any one of the five cells and the constraint conditions fail somewhere.

The Tub

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Four cells, forming a diamond. Like the Block, the Tub is a 4-cell still life — but where the Block is compact (2×2), the Tub is open (2×3). The Tub’s four cells form a hollow square; each has exactly two neighbors. Every dead cell in the pattern’s “interior” and border has either 0, 1, 2, or 4 neighbors — never 3.

The Tub and Block are the only two 4-cell strict still lifes. Together, they form a complete census of the minimum-cell boundary.

The Enumeration Effort

How many still lifes exist? The question is well-posed because Life’s grid is infinite but still lifes are finite configurations. For each cell count n, we can ask: how many distinct still lifes with exactly n cells exist?

The enumeration has been performed computationally by a succession of researchers. The sequence of strict still life counts by cell number is: 0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, and it grows roughly as O(2.46^n). The work was divided among John Conway (4–7 cells, by hand), Robert Wainwright (8–10 cells), David Buckingham (11–13 cells), Peter Raynham (14 cells), and Mark Niemiec, Simon Ekström, and Nathaniel Johnston, who extended the enumeration through 34 cells — at which point there are tens of billions of distinct strict still lifes.

The enumeration is computationally intensive because checking stability requires verifying conditions for every cell in and adjacent to the pattern. But it is also conceptually clean: the constraints are precise, there is no ambiguity, and “complete” means something definite. The catalog of still lifes is, in this sense, a more mathematically finished enterprise than the catalog of oscillators or spaceships, where the search continues indefinitely.

Still Lifes as Attractors and Tools

The preponderance of still lifes in the “ash” of Life dynamics is not an accident. Life’s rules implement a specific form of dissipation: chaotic regions lose energy (in the informal sense of cell-count turnover per generation) as they interact with each other and with stable structures, and eventually the turbulence resolves. The attractors — the configurations where the system comes to rest — are dominated by still lifes, supplemented by low-period oscillators.

The first five still lifes — Block, Beehive, Loaf, Boat, Tub — account for 96% of the still life instances in natural soups. This extreme concentration at the low end of the complexity scale is striking. The Life universe is rich and unpredictable in its dynamics, but it converges overwhelmingly to a tiny set of stable forms.

This is the deeper non-obvious truth about still lifes: they are not the boring configurations, the things you see when nothing interesting is happening. They are the vocabulary of Life’s final state, the grammar that all dynamics ultimately write in. Understanding which still lifes exist, and why, is understanding the ground state of the Life universe — the configurations that are, in a real sense, the system’s resting temperature.

They are also, as mentioned, components. Every functional device built inside Life — every glider gun, every logic gate, every memory cell — is assembled largely from still lifes interacting with moving patterns. The eater, the block, the pond, the carrier: still lifes are the resistors and capacitors of Life’s electronics. They don’t just survive — they do work.