THE FOLD / LOOT / THE MINT / THE THREE-DISTANCE
THE THREE-DISTANCE
step by an irrational forever — gaps take only 3 sizes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The three-distance theorem (Steinhaus, or the three-gap theorem): take an irrational α and mark the points {α}, {2α}, {3α}, …, {nα} around a circle of circumference 1 (fractional parts). However many points you place, the gaps between neighbouring points take at most three distinct lengths — and when there are three, the largest is exactly the sum of the other two.
It is astonishingly rigid: an unbounded process that stays maximally regular. This is why golden-ratio spacing gives the most even distribution — sunflower seeds, phyllotaxis, and low-discrepancy sampling all live here.
LIT verified live: for several irrationals and every n from 2 to 60, the sorted gaps take at most 3 distinct values, and whenever 3 appear the largest equals the sum of the other two (window.__threedistance). FIG no framing; exact gap counting.
It is astonishingly rigid: an unbounded process that stays maximally regular. This is why golden-ratio spacing gives the most even distribution — sunflower seeds, phyllotaxis, and low-discrepancy sampling all live here.
LIT verified live: for several irrationals and every n from 2 to 60, the sorted gaps take at most 3 distinct values, and whenever 3 appear the largest equals the sum of the other two (window.__threedistance). FIG no framing; exact gap counting.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — stamping out evenly-spaced positions, one after another. The three-distance theorem is the mint’s guarantee: keep stepping by α and the spacings never fracture into more than three sizes. AVAN (AI) built the instrument: the fractional-part placement, the gap classification, the largest-equals-sum check.
Credit as content: conjectured by Hugo Steinhaus; proved independently by Vera Sós, Stanisław Świerczkowski, and others (1950s). The weave: David names the mint; I step around the circle by α, count the gap lengths, and show they refuse to exceed three.
Credit as content: conjectured by Hugo Steinhaus; proved independently by Vera Sós, Stanisław Świerczkowski, and others (1950s). The weave: David names the mint; I step around the circle by α, count the gap lengths, and show they refuse to exceed three.
3 ONE DIMENSION
Points appearing one by one at {kα} around the circle, unrolled to a line. The gaps between neighbours are coloured by length — and only ever two or three colours appear.
4 TWO DIMENSIONS · INTERACTIVE
Choose an irrational α and a count n; the points land on the circle and the gaps are grouped by length. Count the distinct lengths (always ≤ 3) and confirm the largest is the sum of the other two.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the circle of points at {kα}, its arcs coloured by the two or three gap lengths that ever occur.
AVAN’s addition (the inverse-companion): adding a point never breeds chaos. Each new point falls into one of the largest current gaps and splits it into the two smaller lengths — so the gap set is self-similar and bounded, never proliferating past three sizes. The inverse of ‘an endless irrational walk’ is ‘a gap structure that stays maximally regular forever.’ And the most even filling comes from the golden ratio, whose continued fraction is all 1s — the ‘most irrational’ number, hardest to approximate by rationals, so its points never bunch. Magenta is the fourth gap length that can never appear; green is the ≤3 that always suffice. Irrational rotation is the most even way to fill a circle — and φ is the most even of all.
LIT Genuine three-distance theorem (conjectured by Steinhaus; proved by Sos, Swierczkowski, and others, 1950s). Verified live: for alpha in {sqrt2-1, golden, pi-3, e-2} and n=2..60, the sorted gaps of {k*alpha mod 1} take at most 3 distinct values, and whenever exactly 3 appear the largest equals the sum of the other two (window.__threedistance.atMostThree && .largestIsSum).
FIG No framing: the fractional-part placement and the gap classification run in-browser and are exact (gaps rounded at 1e-9). The AVAN inverse is honest — each new point splits one of the largest gaps into the two smaller lengths, keeping the gap set at <=3 sizes; magenta is the fourth length that never appears, green the <=3 that always suffice.
FIG No framing: the fractional-part placement and the gap classification run in-browser and are exact (gaps rounded at 1e-9). The AVAN inverse is honest — each new point splits one of the largest gaps into the two smaller lengths, keeping the gap set at <=3 sizes; magenta is the fourth length that never appears, green the <=3 that always suffice.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN