Systems · Interactive ← Tools

How far does one small move travel — and what stops it?

Fifty-six magnet-tipped spinners on a sheet of plywood. No motors, no gears, no contact: the only thing between two spinners is a field across an air gap, and the only energy is whatever your hand puts in. Nothing here can see past its neighbours. A city block runs under that constraint. So does a team, and so does a grid. Take hold of one and watch what happens to the rest.

Grab any spinner and turn it. It turns at the speed your hand does. Start a drag on bare plywood to sweep across several.

Turning: 0 / 56
Mean speed: 0.00 rev/s
In step: 0.00
Activity: last 20s

Magnets

Coupling strength 1.00

How hard one magnet can lean on the next.

Bearing drag 0.32

How readily a spinner gives up. Higher settles the board sooner.

Opposed poles

Automaton

Ignition on

Ignition odds 0.45

How easily a still spinner catches. Its neighbour’s speed, the width of the gap and whether a pole happens to be facing it do the rest.

Neighbours needed 1

Recovery 6

How long a spent spinner stays deaf to its neighbours. Set it to zero and waves become a flood.

Overlays

Coupling bonds Cell state Direction

Board

Nothing here touches

Three magnets per spinner, one at the end of each arm, poles facing out. Set two side by side and you have a pair of gears with no teeth: the field does the meshing. Because both rotors are three-fold symmetric, almost all of the interaction between spinner i and spinner j collapses into one term, and that term depends on nothing but the sum of their two angles and the direction of the line between them:

U_ij = −K(g) · cos( 3θ_i + 3θ_j − 6α_ij )
τ_i = −∂U/∂θ_i = −3K(g) · sin( 3θ_i + 3θ_j − 6α_ij )
K(g) = K₀ · (g₀/g)⁵, g_ij = r_ij − 2ℓ

The energy bottoms out wherever 3θ_i + 3θ_j holds still, which leaves a locked pair no choice: ω_i = −ω_j. Turn one clockwise and its neighbour has to go the other way. The board’s only route to order runs through every spinner disagreeing with the ones beside it. When it gets there, the in step reading climbs toward 1.

The g is the air gap between two facing magnets, not the distance between hubs; subtract both arm lengths first. A diagonal neighbour sits only 1.4× further off hub to hub, but its gap is more than twice as wide, so it receives about 2% of a side neighbour’s pull. Two columns over, the magnets are too far apart to feel each other at all.

That weakness is the only reason the board ever holds still. Take any square of four. The four side bonds can be satisfied together, but the two diagonals across that square demand contradictory angles, one insisting on 15° and the other on −45°. Physicists call a lattice like this frustrated: no arrangement satisfies every bond. It settles anyway, because the diagonals are weak and the bearings are stiff.

Physics only goes downhill

Magnets pass energy along. They never make any. Sweep a corner of the board and the disturbance spreads, blurs and dies, and that is the entire repertoire. Travelling waves, fronts that bend, loops that keep re-lighting themselves: all of it needs energy the plywood has no way to supply.

The second layer is not physics. Switch on Ignition and every spinner also becomes a cell in an excitable medium, cycling still → catching → turning → spent → still. A still cell watches its neighbours and catches with probability:

P = 1 − ∏_j ( 1 − p_j ), p_j = c · (|ω_j| / ω_ref) · (g₀ / g_ij)³ · A_ij
A_ij = ½ · [ 1 + cos( 3θ_i + 3θ_j − 6α_ij ) ]

Three things move those odds: how fast the neighbour is going, how wide the gap is, and whether a pole happens to be pointed your way at that instant. The last is the term A. It falls to nothing when the arms are aimed wrong, so a fast neighbour caught mid-phase still fails to light its partner.

Recovery is what turns spreading into travelling. Without it, activity floods outward and fills the board. With it you get fronts that move, bend around the gap, and now and then close into a loop that keeps re-lighting itself. Ask for three live neighbours instead of one and most fronts starve before they get across.

What’s real here

Derived The coupling term and the counter-rotation it forces. The fifth-power falloff across the gap. Equal and opposite torque at both ends of a bond. The frustration in the diagonals. Viscous and dry friction in the bearing. All of it integrated at 120 Hz.

Tuned by hand Coupling strength, drag, inertia, and how much your hand counts for. All set by feel until it moved the way the board on the table moves. No datasheet, no gauss meter. The rev/s figures are approximate.

Invented Ignition, recovery and the neighbour threshold are automata theory bolted on. They put energy into the board that neither the plywood nor your hand supplied, which is the one claim at the top of this page they break. That is why it opens with ignition off. Turn it on and the board does things the plywood never could.

The same board, at other scales

A lattice is what you get when the parts of a system can only reach the parts immediately beside them. Plenty of things are built that way.

Influence is short-sighted. A diagonal neighbour gets two percent of what a side neighbour gets; two columns over, nothing at all: the gap is past the reach of the magnets. The bare stretch down the middle of this board stops a front for the same reason London’s firebreaks did after 1666. A park, a rail cut, a six-lane arterial. Org charts produce them by accident: the reporting boundary nobody designed as a firebreak turns out to be one.
Order shows up as disagreement. No spinner here is told what to do. Each pair is told only to differ from the one beside it, and that is enough to produce a pattern across the board that nobody specified. Schelling made the same move with segregation. An alternating one-way grid moves traffic that no single intersection understands.
Frustration is the normal condition. A zoning map, a seating chart, a platform team boundary: every real adjacency problem carries more constraints than it can satisfy at once. Frustrated systems do not fail. They fall into one of many nearly equal compromises and stay there, because leaving costs more than the improvement is worth.
Recovery is what makes a wave instead of a flood. Remove it and activity fills the board at once and is finished. Leave it in and you get a travelling front: an epidemic curve, a stop-and-go traffic wave, a rumour crossing a company one department at a time. Heart tissue runs this rule, and when the front closes into a loop and keeps re-lighting itself, the clinical name is fibrillation.
Nothing sustains itself. Switch ignition off and every disturbance dies, because magnets pass energy along and never make any. The efforts in a neighbourhood or a company that are still moving a year later are the ones somebody is still feeding.

Firefly Field and Zone Territories work the same seam from the other direction, structure arriving out of local rules with nobody in charge, and the 3D graph automaton runs it on a network instead of a grid.

Try this

Sweep hard across the top-left corner with ignition off. Count the columns it reaches before it gives out. That is the range of a magnet.
Switch ignition on and do it again. The bare stretch in the middle still stops most fronts. Distance is doing that, not the rule.
Hold one spinner and turn it slowly, arm by arm. Its neighbours pick the motion up backwards, but only at about half your rate, slipping a notch whenever friction and their own neighbours outvote you.
Push coupling to 3 and turn at the same speed. The slipping stops: neighbours now track at 0.93 against a perfect 1.00. That is the difference between a magnetic gear that holds and one that skips.
Set recovery to zero and ignition to 1. Everything turns and nothing travels: a wave needs the board to remember it has just fired.
Flip to opposed poles. Pairs still counter-rotate, but they settle 60° away from where they did before, and the in-step reading tells you the moment they find it.

The board is real: 56 spinners, four colours, in the layout you see here. Everything past that is arithmetic. The comparisons to cities and teams are analogies; nothing here proves anything about either.