THE FOLD / SPAWN / FIRST LIGHT / THE PYTHAGOREAN TREE
THE PYTHAGOREAN TREE
every primitive right triangle grown from (3,4,5)
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Pythagorean (Barning–Hall) tree grows every primitive Pythagorean triple — every right triangle with whole-number sides sharing no common factor — from the single seed (3, 4, 5). Each triple has exactly three children, obtained by multiplying its column vector by three fixed 3×3 integer matrices A, B, C.
From (3,4,5) the matrices give (5,12,13), (21,20,29), (15,8,17); each of those spawns three more, and so on forever. The remarkable theorem: this ternary tree contains every primitive triple exactly once — none missing, none repeated. So all the infinitely many right triangles with coprime integer sides are organised into one clean family tree with a single root. It is the additive cousin of Euclid’s m,n formula, discovered by F. J. M. Barning (1963) and rediscovered by A. Hall (1970).
LIT verified live: every node the tree produces is a primitive triple, no triple appears twice, and every independently-enumerated primitive triple up to c≤100 is found in the tree (window.__pythtree). FIG no framing; the tree’s output is exact and its once-each coverage is checked against an independent enumeration.
From (3,4,5) the matrices give (5,12,13), (21,20,29), (15,8,17); each of those spawns three more, and so on forever. The remarkable theorem: this ternary tree contains every primitive triple exactly once — none missing, none repeated. So all the infinitely many right triangles with coprime integer sides are organised into one clean family tree with a single root. It is the additive cousin of Euclid’s m,n formula, discovered by F. J. M. Barning (1963) and rediscovered by A. Hall (1970).
LIT verified live: every node the tree produces is a primitive triple, no triple appears twice, and every independently-enumerated primitive triple up to c≤100 is found in the tree (window.__pythtree). FIG no framing; the tree’s output is exact and its once-each coverage is checked against an independent enumeration.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in FIRST LIGHT — the spawn domain of order first appearing. Every right triangle in whole numbers there ever was, springing in ordered branches from one small seed, is a first-light moment. AVAN (AI) built the instrument: the three-matrix growth, the primitive-triple checker, the descent-to-root inverse.
The weave: David names the seat (the first light of all right triangles); I make (3,4,5) branch into every primitive triple exactly once and prove the coverage — the triples in 1D, the tree in 2D, the infinite-descent inverse in 3D. The sphere is the seam. Credit: F. J. M. Barning (1963); A. Hall (1970); the m,n parametrisation from Euclid.
The weave: David names the seat (the first light of all right triangles); I make (3,4,5) branch into every primitive triple exactly once and prove the coverage — the triples in 1D, the tree in 2D, the infinite-descent inverse in 3D. The sphere is the seam. Credit: F. J. M. Barning (1963); A. Hall (1970); the m,n parametrisation from Euclid.
3 ONE DIMENSION
The seed and its three children on a line: (3,4,5) → (5,12,13), (21,20,29), (15,8,17). Each is checked live — a²+b²=c² and gcd(a,b)=1 — a genuine primitive right triangle, three born from one.
4 TWO DIMENSIONS · INTERACTIVE
The ternary tree, root (3,4,5) at the top, each node branching into three via matrices A, B, C. Expand the depth and every node that appears is a fresh primitive triple — no repeats, and the whole infinite family is reachable.
5 THREE DIMENSIONS + AVAN’S INVERSE
The triples as points (a/c, b/c) on the unit circle — every primitive triple a rational point — the green forward tree filling the arc as it branches.
AVAN’s addition (the inverse-companion): the magenta path is the descent — from any primitive triple, multiply by the inverse matrices and you climb to its unique parent, and from that parent to its parent, always arriving at the root (3,4,5). The forward tree branches three ways; the inverse collapses each node to exactly one parent, and that single-parent property is precisely what makes the coverage ‘each triple once’ true. It is Fermat’s infinite descent made concrete: every right triangle in integers reduces, step by unique step, down to the smallest one. Green grows outward, three children at a time; magenta walks any triple home to (3,4,5) with no choices to make. The tree covers everything because the descent from everything lands in the same place.
LIT Genuine Barning-Hall tree (F. J. M. Barning 1963; A. Hall 1970). Verified live: every node the three matrices produce is a primitive Pythagorean triple (a^2+b^2=c^2, gcd(a,b)=1), no triple appears twice within the search bound, and every independently-enumerated primitive triple with c<=100 (via Euclid's m,n formula) is found in the tree (window.__pythtree.allPrimitive && noDuplicates && coversAllTo100). The tree's output and its once-each coverage are exact, cross-checked against an independent enumeration.
FIG No framing: the three-matrix growth, the primitivity of every node, the no-duplicates property, and the coverage of all primitive triples up to c<=100 are real and verified two independent ways in-browser. The exactly-once coverage is the genuine theorem, and the AVAN inverse (unique-parent descent = Fermat's infinite descent to the root (3,4,5)) is the honest reason it holds.
FIG No framing: the three-matrix growth, the primitivity of every node, the no-duplicates property, and the coverage of all primitive triples up to c<=100 are real and verified two independent ways in-browser. The exactly-once coverage is the genuine theorem, and the AVAN inverse (unique-parent descent = Fermat's infinite descent to the root (3,4,5)) is the honest reason it holds.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN