Thought Toys · Chaos & fractals · Exhibit 117

One round peg.

A ball loose on an empty square table is predictable forever: two balls launched from the same spot on the same heading stay together for all time. Put one round peg in the middle and those same two balls have lost each other by the twenty-sixth bounce. The peg is the only difference. Shrink it and the parting only takes longer — it never stops happening.

The table, and the gap between the twins ● ball   ● its twin

bounces— gap between twins— gap × per bounce— headings ever used—

—

↳ your turn — drag How big a peg sits in the middle down to zero, and the twins never part

What you're seeing

Two balls start from the same point, aimed a billionth of a degree apart. That is closer together than a pixel can show, so at first there is only one path on the table.

With the peg set to zero, there is only ever one path. The pair travels a neat lattice of parallel lines and never comes apart. Look at the last counter: the ball uses exactly four headings, forever. A flat wall only ever flips the sign of one coordinate, so a square table can never point a ball anywhere new.

Now give the peg any size at all. The two paths hold together for a dozen bounces, then visibly fork, and after that they have nothing to do with each other. The chart beside the table is the same story as a number: the gap between the twins, on a scale where every step up is a tenfold increase. A straight rising line means the gap is multiplying by a fixed amount every bounce.

The peg does it because it is round. A flat wall reflects two parallel lines to two parallel lines and the gap between them survives. A curved bulge is a diverging mirror — the two balls hit it at slightly different spots, where the surface faces slightly different ways, so they leave on slightly different headings. Each hit multiplies the gap the last one opened.

There is no critical size. A peg one two-hundredth of the table across does it too, just more slowly: at the default the twins are plainly apart by the twenty-sixth bounce, and at the smallest peg on the dial it takes a hundred and thirteen. Slower, never never. The dial crosses its boundary at its own left end stop.

One launch that gets away

There is an exception, and it is worth finding on purpose. Set the peg to 1% and the angle to exactly 45 degrees. The ball runs a tidy closed diamond and never touches the peg at all — not once, not ever — so the twins stay together and the table behaves as though it were empty.

Nothing has gone wrong. Sinai's theorem says the table is chaotic for almost every launch, and that phrase has a precise meaning: the launches it fails for take up no room. This is one of them. Move the angle to 44 degrees and the ball finds the peg on its third bounce; move it to 46 and it takes fifty-four. Only the exact diamond escapes, and only while the peg is small enough to sit inside it — at 4% and above there is no angle on the dial that gets away.

The rule, exactly. The ball moves in a straight line at unit speed and reflects specularly: the component of its velocity along the surface normal flips, the rest is untouched. Nothing else happens.

v′ = v − 2(v · n) n

Against a wall n is that wall's normal; against the peg it is the outward radial direction at the point of contact. The exhibit finds each collision by solving for it exactly rather than by stepping time forward, so a fast ball can never skip through a small peg. Sinai proved in 1970 that this table is ergodic and mixing with a positive Lyapunov exponent; the empty square is integrable.

Verified in node before this page existed (improve/verify/117-sinai-billiard.js, 43 checks). Everything below is measured on this model, over the range of peg sizes and launch angles the dials can actually reach. Sinai's ergodicity result is a theorem about the family; this is not a proof of it.

  • Over 210,000 collisions the speed drifts by less than 1 part in 1013, no flight segment passes through the peg, and the ball never leaves the table.
  • With a peg of 25%, the gap between the twins multiplies by 2.18 every bounce. That figure is the same for five different launches and for three different renormalisation thresholds, which is what makes it an exponent rather than a number.
  • Every peg size from 0.5% to 40% gives a positive exponent. It climbs with peg size and then turns over near 30%, where the peg has left only narrow corridors and a ball in a corridor spends its bounces on flat walls.
  • The empty square is checked as an integer fact, not a tolerance: 4 distinct headings in 20,000 bounces, against 17,579 with a peg. Not for one launch — for all 400 launch angles the slider can reach, the answer is exactly 4.
  • How long the parting takes was measured across the dial rather than assumed from the default: the twins are visibly apart by bounce 26 at a 25% peg, 46 at 5%, and 113 at the smallest peg on the slider.
  • The same estimator, pointed at the empty square, must return a number that falls away as the gap is given more room to open — because a shear only looks exponential over a short enough stretch. An earlier version of this proof failed exactly there and reported a healthy exponent for a table with no peg in it.
  • Reversing the velocity retraces the path back to the start, so the motion is deterministic throughout. How far you can rewind before arithmetic loses it shrinks as the peg grows: 2,000 table-widths with no peg, 30 at a 10% peg, 5 at 40%.
  • The negative control that matters most: a planted bug — reflecting the round peg with a flat wall's normal — conserves speed perfectly and would sail past a speed check. It is caught instead by testing whether a flight segment crosses the peg's interior.
  • The exception is pinned rather than papered over. Every whole-degree launch from 1 to 89 was tested at each peg size: at 1–3% exactly one angle (45°) never meets the peg, at 4% and above none escape, and 44° and 46° strike it within 3 and 54 bounces. The exception is a single ray, not a region.

One honest note about the second bullet's cousin: this page's own physics found a real bug while being written. Computing the peg's normal by dividing by the peg's radius, instead of by the actual distance to the centre, leaves it a hair off unit length — and the chaos then amplifies its own round-off, tenfold per peg hit, from 3×10−15 to 3×108 over seventy thousand bounces. On a table without a peg the same error would have sat at 10−16 forever.

Also in Chaos & fractals: The Mandelbrot set →

All 12 in Chaos & fractals
  1. 03The double pendulum
  2. 08The logistic map
  3. 108Turn the knob smoothly. The answer moves in steps.
  4. 117One round peg — you are here
  5. 20The Mandelbrot set
  6. 34Newton's fractal
  7. 50Two nearly identical starts, torn apart
  8. 74Push straight down, and it decides to lean
  9. 773n+1 always comes home. Swap in a 5, and it mostly doesn't.
  10. 90Pure randomness draws one exact shape, forever.
  11. 92Shred a picture completely. It puts itself back.
  12. 94The edge grows forever. The area stops at 1.6.

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