Pure randomness draws one exact shape, forever.

A point sits inside a triangle. Every step: jump partway toward a corner picked by a coin no one controls. Repeat forever. There is no plan, no memory, no target — and yet the cloud of everywhere it has ever been locks onto one unmistakable shape, the same one every time.

What you're seeing

One point, one rule, applied forever: pick one of the triangle's three corners at random, then move the point partway there — by default, exactly halfway. Nothing is being drawn on purpose. Nothing remembers where the point has been. Every single choice is a fair coin (well, a fair three-sided one) with no plan behind it at all.

And yet, drop enough points and a shape appears — the same shape, every time you run it: three smaller triangles of dots, each holding three smaller still, each of those holding three smaller still, forever. And at the center of the big triangle, and at the center of every one of those sub-triangles, no matter how long you wait: nothing. A perfectly empty hole. Toggle "show the forbidden middle" to outline exactly where that hole sits — then drag the ratio and watch the dots respect that boundary precisely at 0.50, and violate it the moment you drop below.

Here is why the middle is untouchable, exactly at 0.50 and not a hair below: describe any point inside the triangle by three numbers — how much it "belongs" to each corner (its barycentric coordinates, always summing to 1). Jumping halfway toward a corner can never leave that corner's own share below one half, no matter where the point started. So the instant every corner's own coordinate is guaranteed to reach at least 0.50 on arrival, it becomes impossible for all three coordinates to be under 0.50 at once — which is precisely the definition of the empty middle. Lower the ratio and that guarantee simply isn't strong enough anymore, and the middle stops being safe.

The rule, exactly. A point P with barycentric coordinates (a, b, c), a+b+c = 1 jumps toward a uniformly random corner k by P′ = (1−r)P + r·V_k, which transforms coordinate k to (1−r)·a_k + r ≥ r since a_k ≥ 0 always. At r = 0.5 that is enough to prove the empty middle (all three coordinates below 0.5) can never be reached. Verified in node (improve/verify/90-chaos-game.js): zero violations of that guarantee across hundreds of thousands of genuinely random jumps at five ratios from 0.50 to 0.85. Negative controls: drop the ratio below 0.50 and the same code measurably starts landing in the middle — up to roughly a third of the time at r = 0.30, falling smoothly to about 14% right at the edge, r = 0.48, and to exactly zero from r = 0.50 on; and removing the randomness entirely (always choosing the same corner, at the same r = 0.5) collapses the point onto that one corner — a single point, not a fractal — at precisely the geometric rate (1−r)ⁿ the algebra predicts.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 90.