THE FOLD / CO-OP / THE MERGE / THE JOHNSON CIRCLES
THE JOHNSON CIRCLES
three circles hand off to a fourth of equal size
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Johnson’s circles: take three circles of the same radius ρ that all pass through one common point H. Each pair meets again at a second point; call the three second points P₁₂, P₁₃, P₂₃. The theorem: those three points lie on a fourth circle of exactly the same radius ρ (the Johnson circle). Even better, its centre is C = O₁+O₂+O₃−2H, and each second point sits at distance |Oₖ| = ρ from it — a clean vector identity, since Pₖₗ = Oₖ+Oₗ−H.
LIT verified live: for 5,000 random configurations of three equal-radius circles through a common point, the three second intersections are all at distance ρ from C = O₁+O₂+O₃−2H (window.__johnson) — a same-radius circle every time. FIG no framing; the second points and their common radius are computed in-browser.
LIT verified live: for 5,000 random configurations of three equal-radius circles through a common point, the three second intersections are all at distance ρ from C = O₁+O₂+O₃−2H (window.__johnson) — a same-radius circle every time. FIG no framing; the second points and their common radius are computed in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-merge — three circles meet at one shared point and their pairwise merges hand off to a fourth of equal size, a clean four-way symmetry. AVAN (AI) built the instrument: place three equal circles through H, compute each pair’s second intersection by the vector identity, and confirm all three are radius ρ from the Johnson centre.
Credit as content: Roger Arthur Johnson (1916). The weave: David names the merge; I show the three second points ride a circle of the very same radius, with centre O₁+O₂+O₃−2H.
Credit as content: Roger Arthur Johnson (1916). The weave: David names the merge; I show the three second points ride a circle of the very same radius, with centre O₁+O₂+O₃−2H.
3 ONE DIMENSION
Three equal circles through a common point H, their three second intersections, and the equal-radius Johnson circle threading those three.
4 TWO DIMENSIONS · INTERACTIVE
Randomize the three equal circles; the Johnson circle through the second points always has the same radius ρ.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fourth circle, same radius, through the three second points.
AVAN’s addition (the inverse-companion): don’t intersect circles pair by pair — add the centres. The inverse of ‘find each pairwise second point’ is ‘Pₖₗ = Oₖ+Oₗ−H, so the whole figure is one vector sum, symmetric in H and the fourth centre.’ Magenta are the three second points; green is the equal circle they share. Four circles, one radius.
LIT Genuine Johnson's theorem (Roger Arthur Johnson, 1916): three equal-radius circles through a common point have their three other pairwise intersections on a fourth circle of the same radius. Verified live: over 5000 random configs, the three second points Pᵢⱼ=Oᵢ+Oⱼ−H are all at distance ρ from C=O₁+O₂+O₃−2H to <1e-9 (window.__johnson.equalRadius).
FIG No framing: the second points and their common radius are computed in-browser. The AVAN inverse is honest — the vector identity Pᵢⱼ=Oᵢ+Oⱼ−H makes the whole figure one symmetric sum in H and the fourth centre, which is why the fourth radius equals ρ; magenta are the three second points, green the equal circle they share. Four circles, one radius.
FIG No framing: the second points and their common radius are computed in-browser. The AVAN inverse is honest — the vector identity Pᵢⱼ=Oᵢ+Oⱼ−H makes the whole figure one symmetric sum in H and the fourth centre, which is why the fourth radius equals ρ; magenta are the three second points, green the equal circle they share. Four circles, one radius.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN