A Universe in a Spiral of Cells

Part 1 · Where we left off

One row of cells, briefly remembered

If you have not read A Universe in a Row of Cells, start there — this page leans on it. The short version: take one row of cells, each black or white. Each new generation is printed below the last, and every cell's color is decided by exactly three parents in the row above — the cell over it and that cell's two neighbors. Three parents give 8 possible combinations; a rule is an answer sheet for all eight; and 2⁸ makes 256 possible rules, numbered 0 to 255. Some fizzle out, some settle into rhythm, some churn forever, and a precious few come alive with little traveling structures. One of them, Rule 110, was proved capable of any computation a computer can do.

Every picture on that page had the same shape: a rectangle, with time flowing down. Which quietly smuggles in two assumptions — that the world is a straight line with edges, and that all cells change together, generation by generation, like a drumbeat.

This page removes both assumptions, one at a time. The plan: first wind the old pictures into a spiral and see what that buys us. Then make the spiral real — a world that is laid down one cell at a time, from a center outward. Then let that world grow as it goes. And finally, hand the whole experiment over to the plant kingdom's favorite spiral, and see what a sunflower does with Rule 110.

Part 2 · Wind it up

The old universe, on a new axle

Begin gently: change nothing about the math, only the drawing. Join the row's two ends so it becomes a loop — a ring of cells with no edges at all. Run the generations exactly as before. But instead of stacking the generations downward, wrap each one onto the next winding of a spiral: the first generation is the innermost lap, and time flows outward, the way a tree adds rings.

Everything you learned on page one still applies — Rule 30's boiling triangle is right there, just bent around an axle. Two things are new. The world now has no edges: go far enough left and you arrive from the right. And there is a seam — the faint ray pointing up from the center — where each lap closes and the next begins. Remember the seam. It is about to become the whole story.

Pretty — but honestly, this is page one's universe wearing a new coat. The math never noticed the spiral. To go somewhere genuinely new, the spiral has to stop being a way of drawing the world and become the world itself.

Part 3 · Close the loop

A world laid down one cell at a time

Here is the real break with page one. Abolish the drumbeat. In this new universe there are no generations at all — there is only the next cell, laid at the end of the spiral like a bead threaded onto a string, forever counter-clockwise, forever outward.

A new cell still needs parents to decide its color, and it still gets exactly three: the cell laid down just before it, one step back along the spiral; the cell directly beneath it, one full winding inward; and the next cell beneath, just ahead of that one. Three parents, black or white, eight combinations — so the very same answer sheets work here, and every one of page one's 256 rule numbers carries over unchanged. (Fair warning: a rule keeps its number here, not necessarily its personality. The parents sit in new places, and that changes what a rule does — as Rule 90 is about to demonstrate.)

Wolfram himself notes, in the research pages behind his book, that ordinary cellular automata update every cell in parallel purely by convention — and that updating cells one at a time is a studied alternative that usually behaves very differently. Our spiral takes that idea literally: sequence is not a bookkeeping choice here, it is the geometry of the world.

Every world needs a beginning, and this one begins with nine cells: one center cell, and a first ring of eight around it. That seed ring is the innermost winding; the rule takes over from cell ten onward. So the simplest version of this universe keeps eight cells in every winding, forever — a dartboard. Let us see what the rules make of it.

The verdict is swift and a little deflating. A world that is eight cells around is just a loop of eight cells in disguise, and eight cells can only hold 2⁸ = 256 possible states — so every pattern must start repeating within a few laps, and the repetition paints wedges. Wolfram's four fates need room, and this world has none. An honest dead end. But it points directly at the fix: if a constant-width world is too small, let the world grow.

Part 4 · Let it grow

A lattice that expands underneath its own pattern

Keep every cell the same physical size, and something wonderful becomes unavoidable: each lap of the spiral is longer than the last, so each winding holds about six more cells than the one before — 8, then 14, then 20, 27, 33… (Six-and-a-bit, to be exact: 2π.) The world is no longer a loop pretending to be a spiral. It is a lattice that expands as it grows, like a living thing adding cells at its rim.

This growth quietly breaks the neat geometry the rules relied on. "The cell directly beneath" was always the same cell, one winding back, in the dartboard world. Here, the winding beneath has fewer cells than the winding being laid, so beneath-ness drifts steadily out of alignment, and every few cells there is a small seam where a new cell squeezes in above a boundary. No rule was ever designed for this. Which is exactly why it is worth watching.

Growing and shrinking cellular worlds have been studied only rarely — a handful of theoretical papers consider one-dimensional automata that insert new cells as they run. But watching Wolfram's own 256 rules live on an expanding spiral, seams and all, already feels like off-trail territory. And there is one more spiral to visit — the one that has been growing this way, in every meadow, for a hundred million years.

Part 5 · The sunflower

Nature's own expanding helix

A sunflower head builds itself almost exactly the way our Part-3 universe does: one element at a time, from the center outward. Each new floret appears at a fixed angle around from the previous one — and in plant after plant, that angle converges to almost exactly 137.5°, the so-called golden angle, the full circle divided by the golden ratio. In the standard mathematical model of a seed head, floret number n sits at angle n × 137.5°, at a distance proportional to √n — which packs every seed into an equal share of the disk. Wolfram devotes a section of his book to exactly this: he shows that a simple mechanism, where each new element forms wherever the inhibition from recent elements is weakest, settles into the golden angle on its own, no optimization or design required. The spiral is not clever. It is cheap — his favorite kind of answer.

So build the lattice the sunflower way: cells laid one at a time, each 137.5° counter-clockwise from the last, drifting outward. Because the golden angle divides the circle irrationally, no cell ever lands exactly above another — there is no seam anywhere, and "the cell beneath" stops being a matter of bookkeeping and becomes a genuine question of geometry. So let geometry answer it: give each new cell its predecessor as one parent, plus the two nearest cells already laid as the other two. Same eight combinations. Same 256 answer sheets.

And here the lattice hands us a small marvel.

Read that again, because it is the quiet climax of this whole page: on the sunflower lattice, your parents are always a Fibonacci number of cells behind you — and which Fibonacci numbers changes as the world grows. A cellular automaton here is not the fixed machine of page one at all. It is a rule whose very wiring is rewritten, gradually and forever, by the geometry of growth. Now color it in.

Is any of this in the literature? We looked, and each ingredient exists on its own: automata on unusual fixed lattices (Wolfram's book even runs one on a Penrose tiling), automata that update sequentially, automata that insert cells as they run, and a century of beautiful mathematics on phyllotaxis — the science of these spirals, tested even in physics experiments where droplets of magnetic fluid, dripped into a dish, spontaneously arrange themselves at the golden angle. But we could find no published cellular automaton built on a phyllotactic lattice — no one, as far as we can tell, has wired Wolfram's rules into the sunflower's own geometry and pressed play. Perhaps we simply did not find it. Or perhaps you are looking at a small, genuinely new corner of a very large subject.

Part 6 · What the spiral taught us

The rule is only half the universe

Page one's deepest lesson was that complexity is cheap: eight yes-or-no answers can generate endless novelty. This page adds the missing half of that sentence. The same eight answers produced a pinwheel, a wavefront, static, lace, traffic, extinction, and a computing sunflower — without changing a single bit of any rule. All we changed was the shape of space and the order of time. A rule's famous personality — Rule 90 the lace-maker, Rule 184 the traffic warden — turns out to be a partnership between the rule and the world it runs in. Change the world, and the same answers sing a different song.

That is not a defeat for Wolfram's program — it is one of its own predictions, usually left in the footnotes: the choice of a rigid grid and a synchronized clock was always just the simplest laboratory, not a law of nature. His book makes the point in passing, with characteristic confidence:

The sunflower lattice takes that permission further than a tiling — to a space that grows, seamlessly and irrationally, rewiring its own neighborhoods as it goes. And even there, the old miracle held: simple rules, given room, still made worlds worth watching. If nature runs on simple programs, as Wolfram believes, then it runs them on geometries like this one — laid down one cell at a time, at the growing edge of things.

Part 7 · Your turn

Grow your own spiral universe

All four worlds from this page, and all 256 rules, are yours. Pick a geometry, pick a rule by number or flip its eight answers directly, seed the first ring by hand if you like — and watch the world get laid down, cell by cell. The interesting question is the one this page could only begin to answer: which rules keep their personality when the world starts to grow?