THE FOLD / LOOT / THE BOUNTY / THE GOLDEN SEQUENCE
THE GOLDEN SEQUENCE
{n·φ} — more even than random, by irrationality
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The golden-ratio sequence. Drop points into the interval [0,1) by the rule xn = fractional part of n·φ, where φ = 1.618… is the golden ratio. The result spreads more evenly than random — random points clump and leave gaps; these never do. φ is the ‘most irrational’ number (its continued fraction is all 1s, the hardest to approximate by fractions), so the sequence resists every rational rhythm that would make it repeat and pile up.
Two exact facts make it beautiful. The three-gap theorem: at every step n, the points cut the circle into arcs of at most three distinct lengths — never four. And the discrepancy (how far the point count in any interval strays from its fair share) shrinks like log N / N — near the theoretical best, far better than random’s 1/√N.
LIT verified live: this page confirms the three-gap theorem for every n up to 400 (never more than 3 gap lengths), and that the golden sequence’s star discrepancy is several times smaller than an equal number of random points (window.__golden.threeGap && goldenMoreUniform). FIG ‘most irrational’ is a nickname; the three-gap count and the lower discrepancy are exact.
Two exact facts make it beautiful. The three-gap theorem: at every step n, the points cut the circle into arcs of at most three distinct lengths — never four. And the discrepancy (how far the point count in any interval strays from its fair share) shrinks like log N / N — near the theoretical best, far better than random’s 1/√N.
LIT verified live: this page confirms the three-gap theorem for every n up to 400 (never more than 3 gap lengths), and that the golden sequence’s star discrepancy is several times smaller than an equal number of random points (window.__golden.threeGap && goldenMoreUniform). FIG ‘most irrational’ is a nickname; the three-gap count and the lower discrepancy are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE BOUNTY, beside THE POLITE SCATTER (Poisson-disk) — the loot domain of spreading a reward evenly, no clumps, no bare patches. The golden sequence is the cheapest even scatter there is: one multiplication per point. AVAN (AI) built the instrument: the sequence, the gap counter, the discrepancy race, the sunflower.
The weave: David names the seat (the even spread); I make the evenness visible and the theorems checkable — the gap-filling in 1D, golden-vs-random in 2D, the phyllotaxis spiral in 3D. The sphere is the seam. Credit: three-gap theorem — Steinhaus conjecture, proved by Sós, Świerczkowski & Surányi (1957); golden-ratio sampling is classical.
The weave: David names the seat (the even spread); I make the evenness visible and the theorems checkable — the gap-filling in 1D, golden-vs-random in 2D, the phyllotaxis spiral in 3D. The sphere is the seam. Credit: three-gap theorem — Steinhaus conjecture, proved by Sós, Świerczkowski & Surányi (1957); golden-ratio sampling is classical.
3 ONE DIMENSION
Each new point xn = {n·φ} lands in one of the largest current gaps, splitting it — so the interval fills as evenly as possible at every step. The gap lengths, coloured, are never more than three distinct values at once.
4 TWO DIMENSIONS · INTERACTIVE
Add points and race the two rows: golden {nφ} on top, random below. The golden row stays visibly regular while the random one clumps and gaps. The live discrepancy numbers confirm the golden sequence covers far more evenly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The same φ, in two dimensions: a sunflower — point n at radius √n, angle n·(golden angle). Green, the seeds of a real phyllotaxis spiral, turning.
AVAN’s addition (the inverse-companion): the magenta seeds are the newest, always landing in the widest gap left by the rest. Evenness usually means randomness — but random clumps. This is the inverse: a fully deterministic rule that is more uniform than chance, because it is built on the number that most stubbornly refuses to be a ratio. Nature uses exactly this to pack sunflower seeds and pinecone scales without waste. Order that looks like the best possible randomness — the green is the whole spiral, the magenta is irrationality filling the last gap, forever, without ever repeating.
LIT Genuine golden-ratio sampling. Verified live: the three-gap theorem holds for every n up to 400 (never more than 3 distinct gap lengths — Steinhaus conjecture, proved by Sos/Swierczkowski/Suranyi 1957), and the golden sequence's star discrepancy is several times smaller than an equal number of random points (window.__golden.threeGap && goldenMoreUniform, both true; discrepancies reported). φ's continued fraction being all 1s (hardest to rationally approximate) is what drives the evenness.
FIG 'Most irrational' is a nickname for φ's all-1s continued fraction; the three-gap count and the measured lower discrepancy are exact. The sunflower is a real phyllotaxis model (r=sqrt(n), θ=n·golden angle), the same φ giving even 2D coverage — shown, not merely asserted.
FIG 'Most irrational' is a nickname for φ's all-1s continued fraction; the three-gap count and the measured lower discrepancy are exact. The sunflower is a real phyllotaxis model (r=sqrt(n), θ=n·golden angle), the same φ giving even 2D coverage — shown, not merely asserted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN