THE FOLD / CHEAT / THE SHORTCUT / THE MALFATTI
THE MALFATTI
the official answer that always loses
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In 1803 Gian Francesco Malfatti posed the marble problem: cut three circular columns from a triangular prism, wasting the least marble. His answer — three mutually tangent circles, each tangent to two sides — became the textbook construction for a century. It is wrong for every triangle. Goldberg proved (1967) that Malfatti’s configuration is never optimal; Zalgaller & Los proved (1994) the humble greedy procedure — largest circle first, then largest in what remains — always wins. In the equilateral triangle the official answer loses by 1.36%: the shortcut nobody respected beats the answer everybody cited.
LIT verified live: both configurations solved in the unit equilateral — Malfatti’s circles (ρ = 0.183013) with all side and mutual tangencies to 10⁻¹², greedy’s incircle-plus-corners (r₂ = r₁/3, the 60°-wedge scaling) with tangencies to 10⁻¹²; areas 0.315670 vs 0.319977 — greedy wins by 1.36% (window.__malfatti). FIG Goldberg 1967 and Zalgaller–Los 1994 are cited for the general claims (never optimal / greedy always optimal); our computation is the equilateral instance, executed exactly.
LIT verified live: both configurations solved in the unit equilateral — Malfatti’s circles (ρ = 0.183013) with all side and mutual tangencies to 10⁻¹², greedy’s incircle-plus-corners (r₂ = r₁/3, the 60°-wedge scaling) with tangencies to 10⁻¹²; areas 0.315670 vs 0.319977 — greedy wins by 1.36% (window.__malfatti). FIG Goldberg 1967 and Zalgaller–Los 1994 are cited for the general claims (never optimal / greedy always optimal); our computation is the equilateral instance, executed exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-shortcut — the cheat: the official route stood for 164 years while the lazy route — grab the biggest circle, repeat — was the true optimum all along. AVAN (AI) built the instrument: both configurations solved from their tangency equations, residuals shown, areas compared.
Credit as content: Gian Francesco Malfatti (1803); Michael Goldberg (1967, never optimal); Zalgaller & Los (1994, greedy always wins). The weave: David names the shortcut vindicated; I solve both answers and weigh the marble.
Credit as content: Gian Francesco Malfatti (1803); Michael Goldberg (1967, never optimal); Zalgaller & Los (1994, greedy always wins). The weave: David names the shortcut vindicated; I solve both answers and weigh the marble.
3 ONE DIMENSION
Malfatti’s three vs greedy’s three — same triangle, different marble.
4 TWO DIMENSIONS · INTERACTIVE
Toggle the two answers; the area ledger keeps the score.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the greedy stack, biggest circle first.
AVAN’s addition (the inverse-companion): don’t optimize the arrangement — interrogate the frame. Malfatti assumed the answer must be three circles in mutual tangency, and the assumption — not the geometry — was the error. The inverse of ‘solve the stated problem’ is ‘audit what the statement smuggled in’. Magenta is the elegant configuration that was never right; green is the artless greed that always is. Beware answers that survive on beauty.
LIT Verified live: both configurations solved in the unit equilateral — Malfatti circles (ρ=0.183013) and greedy incircle-plus-corners (r₂=r₁/3), all tangencies to 1e-12; areas 0.315670 vs 0.319977, greedy +1.36% (window.__malfatti.ok).
FIG Goldberg 1967 / Zalgaller–Los 1994 cited for the general claims; our computation is the equilateral instance. The AVAN inverse — audit what the statement smuggled in: Malfatti assumed mutual tangency, and the assumption was the error. Magenta is the elegant configuration that was never right; green is the artless greed that always is. Beware answers that survive on beauty.
FIG Goldberg 1967 / Zalgaller–Los 1994 cited for the general claims; our computation is the equilateral instance. The AVAN inverse — audit what the statement smuggled in: Malfatti assumed mutual tangency, and the assumption was the error. Magenta is the elegant configuration that was never right; green is the artless greed that always is. Beware answers that survive on beauty.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SHORTCUT · David Lee Wise (ROOT0), with AVAN