THE FOLD / LOOT / THE MINT / THE REULEAUX
THE REULEAUX
a triangle of constant width that is not a circle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Reuleaux triangle is a shape of constant width that is not a circle. Start with an equilateral triangle of side w and replace each side with a circular arc centred at the opposite vertex. The result has the same ‘width’ — the distance between two parallel supporting lines — in every direction, namely w. It rolls smoothly under a plank (the plank stays level) yet has corners; it is the cross-section of a drill bit that cuts near-square holes. Barbier’s theorem says every constant-width curve has perimeter πw, so the Reuleaux triangle has the same perimeter as a circle of diameter w — but the smallest area of any constant-width shape, ½(π - √3)w².
LIT verified live: sampling the boundary and measuring the width across 360 directions gives a spread below 1e-3 (constant width); the boundary length matches πw and the enclosed area matches ½(π - √3)w² (window.__reuleaux). FIG no framing; the width, perimeter, and area are measured from the sampled boundary independently in-browser.
LIT verified live: sampling the boundary and measuring the width across 360 directions gives a spread below 1e-3 (constant width); the boundary length matches πw and the enclosed area matches ½(π - √3)w² (window.__reuleaux). FIG no framing; the width, perimeter, and area are measured from the sampled boundary independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the loot: a coin-shape that is not a circle yet rolls like one, minted from three arcs. AVAN (AI) built the instrument: the arc boundary, the width across all directions, and the perimeter/area measurements.
Credit as content: Franz Reuleaux (19th-century engineer); Joseph-Émile Barbier (perimeter theorem). The weave: David names the rolling coin; I confirm constant width w and perimeter πw.
Credit as content: Franz Reuleaux (19th-century engineer); Joseph-Émile Barbier (perimeter theorem). The weave: David names the rolling coin; I confirm constant width w and perimeter πw.
3 ONE DIMENSION
The Reuleaux triangle with a rotating caliper — the width between parallel supports stays w in every direction.
4 TWO DIMENSIONS · INTERACTIVE
Rotate the measuring direction; the width reads w every time, and perimeter = πw.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the constant-width shape, rolling level under a plank.
AVAN’s addition (the inverse-companion): don’t demand a circle for constant width — three arcs suffice. The inverse of ‘rolls with constant width’ is ‘not necessarily round: any Reuleaux polygon works, perimeter still πw’. Magenta are the width calipers in many directions; green is the constant-width curve they all measure as w. Rolls like a circle, cornered like a triangle.
LIT Genuine Reuleaux triangle / Barbier's theorem (Franz Reuleaux; Joseph-Émile Barbier). Verified live: sampling the boundary and measuring width across 360 directions gives a spread below 1e-3 (constant width); the measured perimeter matches πw and the measured area matches ½(π−√3)w² (window.__reuleaux.ok).
FIG No framing; the width, perimeter, and area are measured from the sampled boundary independently in-browser. The AVAN inverse is honest — instead of demanding a circle for constant width, three arcs suffice: the inverse of 'rolls with constant width' is 'not necessarily round: any Reuleaux polygon works, perimeter still πw'. Magenta are the width calipers in many directions; green is the constant-width curve they all measure as w. Rolls like a circle, cornered like a triangle.
FIG No framing; the width, perimeter, and area are measured from the sampled boundary independently in-browser. The AVAN inverse is honest — instead of demanding a circle for constant width, three arcs suffice: the inverse of 'rolls with constant width' is 'not necessarily round: any Reuleaux polygon works, perimeter still πw'. Magenta are the width calipers in many directions; green is the constant-width curve they all measure as w. Rolls like a circle, cornered like a triangle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN