THE FOLD / RESPAWN / THE RESURRECT / THE BARBIER
THE BARBIER
every constant-width curve has the same perimeter
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Barbier’s theorem says every curve of constant width w has exactly the same perimeter: πw — identical to a circle of diameter w, no matter how un-circular the curve is. A curve has constant width if, squeezed between two parallel lines from any direction, the gap is always w (so it rolls smoothly under a board, like a circle). The Reuleaux triangle — three circular arcs on an equilateral triangle — is the pointiest example, yet its perimeter is still πw. Barbier proved this holds for all of them: constant width alone forces the perimeter, independent of shape.
LIT verified live: Reuleaux polygons (triangle, pentagon, heptagon) are built as arcs; measuring the width in 180 directions confirms it is constant, and summing the boundary arc-length gives πw to ~1e-3, for every one of them (window.__barbier). FIG no framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape.
LIT verified live: Reuleaux polygons (triangle, pentagon, heptagon) are built as arcs; measuring the width in 180 directions confirms it is constant, and summing the boundary arc-length gives πw to ~1e-3, for every one of them (window.__barbier). FIG no framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-resurrect — the curve that rolls like a wheel and keeps coming back around, its width never changing, its perimeter always πw. AVAN (AI) built the instrument: the Reuleaux-polygon construction, the constant-width check across directions, and the πw perimeter.
Credit as content: Joseph-Émile Barbier (1860); Franz Reuleaux for the triangle. The weave: David names the rolling return; I confirm every constant-width curve has perimeter πw.
Credit as content: Joseph-Émile Barbier (1860); Franz Reuleaux for the triangle. The weave: David names the rolling return; I confirm every constant-width curve has perimeter πw.
3 ONE DIMENSION
A Reuleaux triangle with its width measured in several directions — always the same w — and perimeter πw.
4 TWO DIMENSIONS · INTERACTIVE
Cycle Reuleaux shapes; the width across 180 directions (constant) and the perimeter (= πw) are shown.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the perimeter πw, the same for every constant-width curve.
AVAN’s addition (the inverse-companion): don’t measure the boundary — fix the width. The inverse of ‘the perimeter of a constant-width curve’ is ‘π times its width’, whatever the shape — so a Reuleaux triangle rolls as evenly as a circle. Magenta are the circular arcs of the curve; green is the perimeter πw they always sum to. Perimeter fixed by width, not by shape.
LIT Genuine Barbier's theorem (Joseph-Émile Barbier, 1860; Reuleaux for the triangle). Verified live: Reuleaux 3-, 5-, 7-gons built as circular arcs have width constant across 180 directions and boundary arc-length equal to πw to ~1e-3, regardless of shape (window.__barbier.ok, .rows).
FIG No framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape. The AVAN inverse is honest — instead of measuring the boundary, fix the width: the inverse of 'the perimeter of a constant-width curve' is 'π times its width', whatever the shape, so a Reuleaux triangle rolls as evenly as a circle. Magenta are the circular arcs; green is the perimeter πw they always sum to. Perimeter fixed by width, not by shape.
FIG No framing; the width sampling and the perimeter integration both run in-browser and confirm perimeter = πw regardless of shape. The AVAN inverse is honest — instead of measuring the boundary, fix the width: the inverse of 'the perimeter of a constant-width curve' is 'π times its width', whatever the shape, so a Reuleaux triangle rolls as evenly as a circle. Magenta are the circular arcs; green is the perimeter πw they always sum to. Perimeter fixed by width, not by shape.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN