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Your Universe of Digital Possibilities
A plank rolls perfectly level on rollers that are not round. Curves of constant width — the Reuleaux triangle and its family — measure the same between parallel jaws in every direction, so the plank on top never rises or falls. Every one of them has perimeter π times its width, the triangle has the least area any of them can, and one of them, turned in a square, drills a hole with almost-square corners. Three rooms, one support function, and an auditor counting what is left over.
h(θ) is the distance from the centre to the tangent line facing direction θ; the caliper width is h(θ) + h(θ + π). It is the same in every direction exactly when h is w/2 plus odd harmonics only. The radius of curvature h + h″ must stay non-negative, or the curve folds over itself and is no roller at all.
Every curve of constant width w — circle, Reuleaux polygon or anything stranger — has perimeter exactly π times its width. The perimeter is the integral of the support function, and only its constant term, w/2, survives a full turn.
Of all curves with the same constant width, the Reuleaux triangle encloses the least area; the circle, π w²/4, encloses the most. Same width, same perimeter — about an 11% spread in area.
A Reuleaux triangle of width w can turn inside a square of side w touching all four sides at once. One full turn sweeps everything but four thin slivers at the corners — a hole that is square to within 1.23% of its area.
The second Heron machine trades bars for outlines: the program is the curve itself. One function — the support h(θ) — places the plank, closes the jaws and seats the drill, and the auditor checks each against its closed form: the width spread and the plank’s rise sit at the float floor, the trundle lands on π to its step size, the drill’s mask on 98.77% to its grain. Barbier proved his theorem with Buffon’s needle; the straight line Peaucellier drew in The Straight is the degenerate roller’s path, the area a wheel can read is The Planimeter’s, and how well curved tiles fill a plane is The Mosaic’s question.