THE FOLD / CO-OP / THE MERGE / THE FAREY
THE FAREY
fractions in order, mediants, and kissing Ford circles
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Farey sequence Fn is every fraction in [0,1], in lowest terms, with denominator at most n, listed in order of size. F5 = 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.
It hides two small miracles. First, any two neighbours a/b < c/d satisfy bc − ad = 1 — “unimodular,” as close as two coprime fractions can possibly sit. Second, the very first fraction to appear between two neighbours, as n grows, is their mediant (a+c)/(b+d) — the “wrong” way to add fractions that here is exactly right. Ford circles make it visible: rest a circle of radius 1/(2q²) on each fraction p/q, tangent to the number line, and two circles kiss precisely when their fractions are Farey neighbours. It is the geometry of how the rationals pack the line.
LIT verified live: for F7 every neighbour pair is unimodular, each neighbour’s mediant lies strictly between them, the Ford circles of neighbours are tangent, and a non-neighbour pair is not (window.__farey). FIG no framing; the unimodular, mediant, and Ford-tangency properties are exact.
It hides two small miracles. First, any two neighbours a/b < c/d satisfy bc − ad = 1 — “unimodular,” as close as two coprime fractions can possibly sit. Second, the very first fraction to appear between two neighbours, as n grows, is their mediant (a+c)/(b+d) — the “wrong” way to add fractions that here is exactly right. Ford circles make it visible: rest a circle of radius 1/(2q²) on each fraction p/q, tangent to the number line, and two circles kiss precisely when their fractions are Farey neighbours. It is the geometry of how the rationals pack the line.
LIT verified live: for F7 every neighbour pair is unimodular, each neighbour’s mediant lies strictly between them, the Ford circles of neighbours are tangent, and a non-neighbour pair is not (window.__farey). FIG no framing; the unimodular, mediant, and Ford-tangency properties are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE MERGE — the co-op domain where two things fold into one. The Farey mediant is a merge made arithmetic: two neighbouring fractions combine, numerator-plus-numerator over denominator-plus-denominator, into the exact fraction that belongs between them. AVAN (AI) built the instrument: the ordered fractions, the kissing Ford circles, the mediant/un-mediant inverse.
The weave: David names the seat (two merge into one); I make the mediant land exactly between its parents and the Ford circles kiss just at the neighbours — the sequence in 1D, the circle packing in 2D, the parent-recovery inverse in 3D. The sphere is the seam. Credit: John Farey (1816); proof by Cauchy; Ford circles by Lester R. Ford (1938). See [[stern-brocot]], [[calkin-wilf]].
The weave: David names the seat (two merge into one); I make the mediant land exactly between its parents and the Ford circles kiss just at the neighbours — the sequence in 1D, the circle packing in 2D, the parent-recovery inverse in 3D. The sphere is the seam. Credit: John Farey (1816); proof by Cauchy; Ford circles by Lester R. Ford (1938). See [[stern-brocot]], [[calkin-wilf]].
3 ONE DIMENSION
Fn laid on the unit interval. Between every adjacent pair the label bc−ad = 1 confirms they are as tightly packed as coprime fractions get — the sequence is a lattice of perfect neighbours.
4 TWO DIMENSIONS · INTERACTIVE
The Ford circles of Fn. Raise n and new circles appear — each a mediant, nestling tangent between the two circles that made it. Every kiss between circles is a Farey-neighbour pair; the smaller the fraction’s denominator, the bigger its circle.
5 THREE DIMENSIONS + AVAN’S INVERSE
The mediant tree turning — the green forward step: two neighbour fractions merge into their mediant (a+c)/(b+d), which drops in exactly between them.
AVAN’s addition (the inverse-companion): the magenta is the un-merge — given a fraction, recover the two Farey parents whose mediant it is, by walking up the tree with the Euclidean algorithm. Forward, two fractions combine into one; the inverse splits one back into the unique pair that produced it. What makes both directions clean is the unimodular law bc−ad = 1 — a determinant of exactly 1, the signature of an invertible integer matrix (the tree is an SL(2,ℤ) structure). Because every step has determinant 1, every merge can be undone with integer arithmetic alone; nothing is lost. Green merges two neighbours into their mediant; magenta reverses it into the parents; and the whole rational line is this one invertible fold, taken down and back up. To average two fractions the ‘wrong’ way is, on this tree, perfectly reversible.
LIT Genuine Farey sequence and Ford circles (John Farey 1816; proof by Cauchy; Ford circles by Lester R. Ford 1938). Verified live: for F_7 every neighbour pair satisfies bc - ad = 1, each neighbour's mediant lies strictly between them, the Ford circles of neighbours are tangent, and a non-neighbour pair (0/1, 2/3) is not tangent (window.__farey.unimodularNeighbors && mediantBetween && fordNeighborsTangent && nonNeighborNotTangent). The unimodular, mediant, and Ford-tangency properties are exact.
FIG No framing: the unimodular-neighbour law, the mediant-between property, and the Ford-circle tangency (kissing iff Farey neighbours) are all real and checked in-browser. The invertibility of the mediant merge (determinant-1 / SL(2,Z) structure lets the Euclidean algorithm recover parents) is the genuine content of the AVAN inverse, tied to the same lattice as Stern-Brocot and Calkin-Wilf.
FIG No framing: the unimodular-neighbour law, the mediant-between property, and the Ford-circle tangency (kissing iff Farey neighbours) are all real and checked in-browser. The invertibility of the mediant merge (determinant-1 / SL(2,Z) structure lets the Euclidean algorithm recover parents) is the genuine content of the AVAN inverse, tied to the same lattice as Stern-Brocot and Calkin-Wilf.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN