Antiot's Field

The rules

Each square of the grid holds a fertility value between zero and a capacity E. On top of that, cells live and die by ordinary Life-like neighbour counts. Every generation, simultaneously:

A dead cell is born if it has the right number of live neighbours and the fertility beneath it is at least T.
A live cell survives if it has the right number of live neighbours and the fertility beneath it is at least G. If it is not, it starves regardless of its neighbours.
A square with something living on it loses G fertility.
A square with nothing on it recovers R fertility, up to the capacity E.

Set E high and G to zero and you get Conway's Life back exactly - the ground never runs out, so the fertility term never bites. Everything interesting happens when grazing is faster than regrowth.

Rule notation

B3/S23/E16/G1/R1/T4 — born on 3, survives on 2 or 3, ground holds 16, a cell grazes 1 per generation, empty ground recovers 1 per generation, and a birth needs 4 in the ground. The B and S parts are standard Life notation, so any Life-like rule can be used as the kernel.

Why standing still is fatal

In Conway's Life a random soup burns itself out within a few hundred generations and leaves ash - a static field of blocks, beehives and blinkers that sits there forever. That ash is the reason Life eventually stops being interesting to watch.

Here it cannot happen. A block sitting in one place grazes the same four squares every generation. Sooner or later those squares reach zero, and the block starves. The same is true of every still life and every oscillator, because an oscillator returns to the same squares over and over.

A glider, on the other hand, is somewhere new every four generations. It grazes fresh ground continuously and leaves a trail of exhausted black squares behind it, which recover long after it has gone. Under this rule a glider is not merely one pattern among many - it is the only kind of thing that can survive indefinitely.

So the rule performs a kind of selection. It takes a soup containing every sort of object Life produces, and removes all of them except the ones that travel.

Waves, and why they appear

Raise T until a birth needs nearly full fertility and the field stops behaving like Life altogether. It becomes an excitable medium - the same class of system as a nerve fibre or a Belousov-Zhabotinsky reaction, and the reason waves travel through heart muscle in one direction only.

The three ingredients are all present, and none of them was designed in on purpose:

Excitation — a cell is born on fertile ground.
A refractory period — it grazes that ground to nothing, and nothing can be born there again until it recovers.
Recovery — the ground slowly refills.

Drop a small blob onto rested ground under the Wavefront preset and within eighty generations it is a clean expanding ring with a hollow centre. The front cannot burn backwards, because behind it the ground is exhausted; it can only advance into ground that has not yet been touched. Two blobs produce fronts that annihilate where they meet, exactly as chemical waves do. A broken ring has two free ends, and a free end curls back on itself, so you get rotating scrolls instead of a single clean ring.

Try this

Let the default run until the field is busy. Set the tool to Ellipse and the action to Scorch, drag a large circle across the middle, and press Play. The burnt ground is impassable, so the fronts flow around it like water round a stone - and then, as the soil recovers from the edges inward, the hole closes and life pours back in. Nothing in Conway's Life can do that, because in Life the board has no state of its own.

What actually happens

These are measured results, not predictions. Each was run for 10,000 generations on a 200×120 torus starting from a 30% random soup - about 7,200 live cells. Moving is the fraction of live cells occupying a square they were not in four generations earlier; a value near 1.0 means essentially everything alive is travelling.

Preset	Gen 1,000	Gen 3,000	Ground	Moving	Outcome
Wavefront	4,807	5,290	72%	0.45	Permanent; travelling fronts rather than uniform churn
Pasture	7,050	6,832	52%	0.31	Permanent; half the field grazed at any moment
Grassland	10,906	10,862	67%	0.31	Permanent, ~1,000 births a generation
Bloom	10,297	10,249	83%	0.57	Holds its population while replacing over half of it every generation
Maze	7,172	7,132	51%	0.31	Permanent; grows corridors rather than colonies
Foraging	846	437	99%	0.72	A sparse wandering population, sustained indefinitely
Selection	607	78	100%	0.69	10 cells by generation 10,000 — two gliders, nothing else
Drought	—	0	—	—	Extinct at generation 258
Overgrazed	—	0	—	—	Extinct at generation 26

Selection is the striking one. A population of 7,200 cells containing hundreds of distinct objects collapses over ten thousand generations to a population of exactly ten — two gliders — which then circle the torus forever, grazing a thin trail that has always recovered by the time they come back round. Nothing else survives, because nothing else moves.

It is worth sitting with that for a moment. Nothing in the rule mentions spaceships. The selection happens because travelling is the only strategy that does not exhaust its own food supply, which is not a fact about Life at all.

Three regimes

Starvation: Grazing outruns regrowth badly, or the ground is too shallow to begin with. Everything dies within a few hundred generations. Try Drought or Overgrazed.
Selection: The balance is tight enough that static objects starve but travellers survive. The population falls for thousands of generations and then holds steady at a handful of spaceships. Try Selection or Foraging.
Ecology: A more generous survival kernel keeps the field permanently occupied. Fronts sweep across the ground, exhaust it, die back, and are replaced from the flanks once the soil recovers - a boom-and-bust cycle that never converges. Try Grassland, Bloom or Thicket.

Where this sits

This rule was designed for this site, and the parameters were chosen by running the search rather than by guesswork. It is not a famous automaton and it is not claiming to be a discovery in the scientific sense - resource-limited variants of Life are a natural idea and others will have built things in the same family. What can be said honestly is that it is ours, that it is not one of the standard named rules, and that the behaviour described above is real and reproducible: set the same preset, press play, and watch it happen.

If you find a set of parameters that does something the presets do not, the rule string in the panel is the whole specification. It is worth writing down.

The refractory idea - a square that cannot immediately be reused - is what Brian's Brain does with cell states rather than with terrain, and it is why both produce travelling waves instead of static debris. For the wider family of rule variations see variants and other worlds, and for ordinary Conway's Life with no fertility limit at all, the Lab.