Repelling clocks make 333

A clock that pushes its neighbours’ hands away cannot sit opposite all six of them at once. On a triangular lattice the best compromise is a third of a turn, repeated everywhere — and that compromise is the 333 pattern.

Turn the finished picture about one of its clocks and it matches itself at once. Turn it about a triangle centre and it matches only after you wind the film on a third of a turn, or two thirds. Those two waits are the little numbers in r33₁3₂. Everything below is you doing those turns and paying those waits.

The symbol has four parts, and every part is a move you make and a wait you then pay. Here they are up front, in Symmetries of Things notation; the rest of the page is you earning them.

The shaded part of each little dial is the wait. A plain 3 is a point you can turn the picture about with no wait at all.

1 · One rule, and what it forces

Four small pictures, in order. Each one adds a clock.

A · one clock

hand: 0°

Every clock on this page has one hand. Its angle is everything that clock is doing, and every hand turns at the same steady speed: one turn in six seconds. Drag the hand — it carries on from where you let go.

B · two clocks

apart: —

One rule, the whole page: push your hand away from your neighbours’. Two clocks that push each other apart end up opposite — half a turn, every time, from any start.

C · three on a triangle

gaps, going round the way it came out: —

Three cannot all be opposite — someone has to give. The best they manage is a third of a turn each, and no pair gets what it wanted. There are two ways round the triangle, and scrambling picks one.

D · the fourth clock has no choice

fourth hand: — ·

settings that keep both triangles happy: —

Drag the fourth hand all the way round. Exactly one setting keeps both triangles happy, and it is the same angle as the corner across from it. So only three settings ever appear anywhere in the picture — and once the first triangle has picked its way round, every clock after it is forced.

2 · 144 clocks from a scramble

spin off · arrangement settling

one way round ◴ the other way ◷ a stuck triangle

how slowly the shaking fades: none

worst triangle: — from even · stuck triangles: — · triangles winding each way: — · your scrambles: 0 clean of 0

144 clocks on a lattice that wraps round at every edge. The spin is off, so the only motion is hands moving relative to each other.

Press Scramble. Every clock starts at a random angle and then obeys that same one rule, with the overall turning switched off, so the only thing moving is hands finding their places.

It usually gets stuck. About one random start in ten settles into a single clean arrangement — — in 100 over — tries, — in 100 allowing for luck. The other nine in ten nearly always freeze with a seam: one patch went one way round the triangles, its neighbour the other, and the two can never be glued. That is the hatched scar, and it is why the shading has two kinds.

Clean runs in 100, measured before this page was written, with 95% intervals. The marker rides wherever your slider sits.

Shake the clocks and let the shaking fade away slowly, and most runs come out clean — — in 100 at the slowest setting, against — in 100 with no shaking at all. What matters is how slowly you stop: a quick shake barely helps (— in 100), and shaking twice as hard while still stopping quickly is no better than not shaking (— in 100).

That costs time; it does not save it. The slow fade shakes for — time units of its own before the clocks are left alone, and by then the ones that come out clean are already in place — they stop within — more, against — for an unshaken run that finds its way. End to end that is — units against —: — times longer, at the same speed on screen. Shaking buys patience, not haste.

Tidy it up is different — it averages the picture over its own symmetries and lands clean — times in —, from a stuck picture or a settled one. That is arithmetic imposed from outside, not the clocks working it out.

3 · Switch the spin on, then move it and wait

spin off · arrangement fixed

Now let every hand turn, all at the same speed. The arrangement never changes again — the three families stay a third of a turn apart for ever. That turning picture is the film, and from here on the only thing that ever happens is waiting.

hands disagree by —
wait so far: 0 of a turn · no move yet

Drag the wedge dial, or press Wait a third. The small grid is the difference between the moved picture and the live film, one square a clock: it goes black when they agree.

move	wedge	wait

Tap a move. Every clock lands exactly where another clock was, so there is nothing to look at but the hands. The number tells you how far out they are; wind the dial until the picture goes black, and the wedge you swept is the wait.

Turn about a clock and it matches at once — no wait at all. Turn about a triangle centre and it only matches once you have wound the film on a third of a turn; the other kind of triangle needs two thirds. The two kinds point opposite ways, which is why one costs twice what the other does.

Even a plain sideways slide costs a third. (One step the other way costs two thirds, and three steps cost nothing.) A clock keeps its own reading when it is carried: a hand angle says where that clock is in its cycle, not which way it points on the page.

4 · Four moves, four waits — and that is the symbol

Disagreement against wait, one full turn across. A crosshair rides the wait you wound in section 3. ○ a clock · △ an up-triangle · ▽ a down-triangle · → a slide, whose tent sits exactly on the up-triangle’s because both cost a third.

Each move has exactly one wait that works. How far out the hands are is 360° times how far you are from that wait, so each move is a tent with a single zero: nothing, a third, two thirds, a third.

r33₁3₂

A big 3 means: turn the picture a third of the way round this kind of point and it repeats. The small number under it is how many thirds of a turn you must then wait — so 3 on its own is a point that needs no wait, 3₁ needs one third, 3₂ needs two. The r in front says even a plain sideways slide costs a wait.

Read it back: r33₁3₂. That is Symmetries of Things notation, after the book by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. There is a published film with exactly this symmetry, built from a chemical reaction instead of clocks.

5 · One frame says 333, the film says r33₁3₂

Tap a clock, or a triangle centre.

Frozen, the three families take three shapes — dot, ring, bar — so you can see them without colour.

Stop the film and the only points you can turn the picture about are the clocks themselves, which fall into three families — three kinds of three-fold point, and that is exactly what 333 says.

Start the film and the three families become one kind, because the turning carries each into the next; in exchange the two triangle centres become turning points too, at the price of a third and two thirds of a turn. 333 becomes r33₁3₂.

	one frame	the film
turning points	the clocks, in three families	the clocks, all one family, plus both triangle centres
symbol	333	r33₁3₂

6 · Where this sits

This is a repelling clock rule on a triangular lattice: every clock turns at the same steady speed and pushes its hand away from its six nearest neighbours’. What it settles into, when it settles cleanly — about one random start in ten, or most of them if you let the shaking fade slowly first — is the 333 pattern, turning: as a film, r33₁3₂ in Symmetries of Things notation.

There is a published film with exactly this symmetry built from a chemical reaction instead of clocks, but there the clockwork is the fragile state and unison is the stable one; here it is the other way round, which is why a scramble lands on the clockwork at all. Finding a reaction that makes this film stable is the open question.

The published film with the same symmetry, from a chemical reaction.
The catalogue row: r33₁3₂.
Symmetric animation analogs, the Mon 26 Oct 2026 (week 44) page this one grew out of.

For Chaim Goodman-Strauss and Scott Vorthmann

In Symmetries of Things notation the film is r33₁3₂; its kernel is 333 (p3) on the √3 × √3 superlattice. The rule is θ̇_i = −(K/6) ∑_j ∼ i sin(θ_j − θ_i) with K = +1 (repulsive), identical natural frequency taken out into the separately-switched spin, six nearest neighbours on a periodic 12 × 12 triangular lattice, RK4 at dt = 0.2. The attractor is the three-sublattice 120° state.

Convention: (g, τ) is a symmetry iff q(g·x, t + τT) = q(x, t). Operationally, freeze the moved picture and let the live film run forward; the other sign is 120° out everywhere. A move permutes sites and carries each phase unchanged — the hand is a clock reading, not a vector, so it does not co-rotate with the picture.

The nine below are the three rotations of the kernel times the three translations of the superlattice coset: α the lattice site, β the up-triangle centroid (1/3, 1/3), γ the down-triangle centroid (2/3, 2/3), in the index coordinates of a₁, a₂. The Greek letters are used here and nowhere above. A centre therefore appears twice, once for each sense: the 240° turn about β is the inverse of the 120° one, so its τ is the complement — which is the 3₁ of the symbol read the other way round, not a second cost.

operation	centre	τ catalogue	τ measured	site residual

Three questions for Vladimir Bulatov

Does the forced fourth clock in section 1 land for you as the reason thirds appear, or would a different small piece do it better?
Is winding the dial until the picture goes black the right way to feel a time cost, or would you rather read it off a ruler?
Would SymHub’s notation want the two triangle centres named, or does 3₁ and 3₂ carry it?

Wed 28 Oct 2026 (week 44). Every number on this page was measured by src/measure.mjs and src/measure-stats.mjs with fixed seeds, and the simulation core is tested in tests/clocks333.test.mjs.