Oscillators: Patterns with a Heartbeat

Life’s four rules produced, almost immediately, something unexpected: rhythm.

Not movement — that came later, with the Glider. Not explosion. Not death. Rhythm: patterns that change, that transform, that generate and consume cells in a regular cycle — and then return, exactly, to where they started. These are oscillators, and they occupy the strange middle ground between the dead and the mobile. A still life is frozen; a spaceship is fleeing; an oscillator is neither. It pulses. It breathes. It is, in the most suggestive sense, alive.

The simplest oscillator — the Blinker, three cells that flip between horizontal and vertical every generation — was noticed by Conway himself in 1969, while he was still developing the rules. He watched three cells become five become three become five, in perpetual alternation, and he understood at once that something interesting was possible.

He did not know, in 1969, that it would take until 2023 for the full catalog of oscillation periods to be completed.

What Makes a Pattern Oscillate

A still life achieves equilibrium: every cell is always satisfied. An oscillator achieves something more subtle — periodic satisfaction. The conditions are not met every generation; they are met on a schedule. The pattern passes through states that are locally unstable, sustained by the momentum of the cycle itself.

Consider the Blinker. In its horizontal state, the three cells form a line. The middle cell has two neighbors (satisfied). The two end cells each have one neighbor — which under Life’s rules means they die. But the three dead cells above and below the middle cell each have exactly three live neighbors — so they are born. The pattern has destroyed itself and replaced itself simultaneously, one tick at a time.

This precarious cycling is what defines an oscillator. It’s a sustained tension between death and birth — and it continues only because the pattern it produces in each generation is the exact predecessor of the pattern in the next.

The minimum cell count for an oscillator is 3 (the Blinker). The minimum period is 2.

Period-2 Oscillators: Blinker, Toad, and Beacon

The Blinker (3 cells, discovered by Conway in 1969) is the simplest oscillator and the most common by far. It flips between two states each generation. In the full statistics of random-soup evolution, the Blinker appears more frequently than any other oscillator — roughly one per every hundred or so live cells in the ash of a large soup.

Generation 0    Generation 1
. . .           . # .
# # #           . # .
. . .           . # .

The Toad (6 cells, period 2) is a more complex period-2 oscillator with a 4×4 bounding box. Its two states look like overlapping 3-cell rows — one shifted by one cell from the other. The Toad was discovered in 1970 by Simon Norton at Cambridge, working with Conway.

The Beacon (6 cells, period 2) consists of two diagonal Blocks that interact at one corner. The corner cells alternate: in even generations, they are both alive; in odd generations, both dead. The Beacon is notable for being, in one sense, two still lifes that are just barely unstable by proximity — place two Blocks one cell apart diagonally and the corner interaction drives a period-2 oscillation.

These three — Blinker, Toad, Beacon — are all period-2 oscillators that fit in small bounding boxes. They are also the only period-2 oscillators commonly encountered in natural soups; higher-period oscillators appear rarely without deliberate construction.

Period-3: The Pulsar

Conway found the Pulsar in early 1970, while studying the evolution of rows of cells. It is 48 cells, has period 3, and is one of the most visually striking objects in Life.

. . # # # . . . # # # . .
. . . . . . . . . . . . .
# . . . . # . # . . . . #
# . . . . # . # . . . . #
# . . . . # . # . . . . #
. . # # # . . . # # # . .
. . . . . . . . . . . . .
. . # # # . . . # # # . .
# . . . . # . # . . . . #
# . . . . # . # . . . . #
# . . . . # . # . . . . #
. . . . . . . . . . . . .
. . # # # . . . # # # . .

The Pulsar cycles through three states before returning to its starting configuration. Its four-fold symmetry — it looks the same rotated by 90 degrees — and its 12-fold overall symmetry (reflecting across both axes and both diagonals) make it one of the most symmetric patterns in Life.

Despite its size, the Pulsar is the fourth most common oscillator overall, and the most common oscillator of period greater than 2. It appears in natural soups with surprising regularity. Conway was reportedly delighted by it, and its elegance is partly responsible for the early enthusiasm that spread Life research beyond Cambridge.

The Pulsar exists because a specific arrangement of 12 “prongs” — three cells each — forms a mutually sustaining structure. Each prong is simultaneously a rotor (the part that changes) and a stator (the stabilizing frame). The interaction among the four quadrants drives the three-state cycle.

Period-15: The Pentadecathlon

The Pentadecathlon (10 cells, period 15) is one of the most elegant high-period oscillators, and also one of the most unusual: it begins as a line of 10 cells and evolves through a remarkable expansion and contraction before returning to its starting state 15 generations later.

Conway discovered it in 1970 while systematically tracking the fates of short rows of cells — asking, essentially, what happens if I start with 10 cells in a line? Most lengths stabilize quickly or die. 10 cells, it turns out, produce an oscillator of period 15.

. . # . .
# # . # #
. . # . .
. . # . .
. . # . .
. . # . .
. . # . .
. . # . .
# # . # #
. . # . .

(shown in one of its intermediate states)

The Pentadecathlon expands dramatically over its 15-generation cycle — at its widest, it occupies a much larger bounding box than its 10-cell starting state suggests. This expansion-and-contraction rhythm is what gives it its name: pentadeca- from the Greek for 15, -thlon from the word for contest, because the 15-step cycle was seen as a kind of endurance competition.

The Pentadecathlon is notable for having a period that is genuinely not obvious from its size. Period 15 is not a small prime or a power of 2; it has no obvious relationship to period-2 or period-3 oscillators. It emerges from the specific interaction geometry of 10 cells, and it took direct observation to find it.

The Quest for All Periods: 53 Years of Searching

From 1970 onward, the question of oscillator periods became one of the great open problems in Life research: for which periods n does there exist an oscillator?

The small periods filled in quickly. Period 2, 3, 4, 5, 6, 8, 14, 15 were all found in the first few years. But some periods proved extraordinarily elusive. Period 19 was unresolved for decades. Period 41 was never found, despite searches.

The field’s answer was not always “search harder.” In 1996, David Buckingham showed that a construction method using Herschel conduits — sequences of still lifes and eaters that route a specific active pattern — could generate oscillators of any period ≥ 58. These were not naturally occurring oscillators; they were engineered constructions of considerable complexity. But they proved the periods existed.

In April 2013, Mike Playle discovered the Snark — a small, reflector-based pattern that bounces a glider using a stable catalyst. Snarks enabled the construction of oscillators of all periods ≥ 43 by creating efficient period-multiplying loops.

The final gaps were periods 19 and 41. They fell in the summer of 2023. The period-19 oscillator required a new catalyst interaction that had eluded search algorithms by being simultaneously simple and geometrically unusual — as one researcher noted, “People had been trying all kinds of really complicated searches with lots of catalysts and lots of rare active things in the middle, but all that was necessary was finding this new chunky catalyst.” Period 41 was found by Nico Brown, then an undergraduate mathematics student at UC Santa Cruz, using the novel approach of using gliders themselves as catalysts in the oscillator.

In December 2023, Brown and six co-authors — Carson Cheng, Tanner Jacobi, Maia Karpovich, Matthias Merzenich, David Raucci, and Mitchell Riley — published “Conway’s Game of Life is Omniperiodic” on arXiv (arXiv:2312.02799), formally proving that oscillators of every positive integer period exist in Conway’s Game of Life.

The proof took 53 years.

What the Omniperiodicity Result Means

Omniperiodicity is a statement about the expressive power of Life’s rules. It says that the four rules do not privilege any particular rhythm over any other. Any clock rate you want — period 2, period 41, period 1000003 — is achievable. The rules are, in this sense, rhythmically universal.

This matters computationally. A Life-based computer can have a clock at any desired frequency; it is not limited to specific periodicities. More abstractly, omniperiodicity is a completeness result — it closes a question that had been open almost from the beginning of the field.

The fact that periods 19 and 41 were the last to fall is itself interesting. There is no obvious mathematical reason why these two periods should be harder than, say, period 23 or period 37 — and in retrospect, the oscillators for those periods are not especially complex. The difficulty was that the specific geometric configurations required — particular catalyst interactions at those periods — simply were not in the existing vocabulary of known Life constructions. New ideas were needed. The gap between “we believe this exists” and “we have actually found it” remained open for five decades.

Oscillators as Components

Oscillators are not just objects of study. They are the timing elements of Life’s computational machinery.

Any periodic signal in a Life-based computer — a clock pulse, a data stream, a synchronization signal — is, fundamentally, an oscillator or a product of one. The Pulsar and Pentadecathlon are sometimes used directly in constructions that require specific periods. The Blinker appears in hundreds of larger patterns as an incidental component.

More importantly, glider guns are oscillators: they return to their initial state every 30 generations (in the case of the Gosper Gun), with the side effect of emitting a glider on each cycle. The gun is an oscillator that produces output. This is the bridge between oscillators and computation: an oscillator that emits spaceships is a signal generator, and signal generators are where Life’s computational architecture begins.