Verification Venue · geometry
The Wheel That Isn't Round
Why are manhole covers round? So they can't fall through the hole, goes the famous answer. It's true — but it quietly implies the circle is the only shape that can't fall in. It isn't. A whole family of shapes has the same width in every direction, rolls under a board that never bobs, and drills a square hole. You have one in your pocket.
The board rests on the shape's highest point. Watch it as the shape turns: on all three it never moves. That is constant width. (The blue trail is the centre — flat on the circle, wavy on the others: the catch, below.)
The width of a shape, in some direction, is the gap between the two parallel lines that just touch it on either side — the reading you'd get from a pair of calipers. A circle gives the same reading whichever way you hold the calipers; that's what makes it roll. But so does the Reuleaux triangle: take an equilateral triangle and replace each straight side with a circular arc drawn from the opposite corner. Every arc has the same radius, and that radius is the width. Turn it any way you like — the calipers never budge.
Spin any of them between the jaws. The gap is pinned at 1.000 the whole way round — the definition, made visible.
Two consequences fall out, and both are exact. First, every curve of constant width w has the same perimeter, πw — the same as a circle of diameter w. That's Barbier's theorem (1860), and it's why a coin-sorting machine can gauge a 50p by width alone. Second, of all of them the Reuleaux triangle is the skinniest: it encloses the least area for its width (the Blaschke–Lebesgue theorem), about a tenth less than the circle.
Now the party trick. Because a Reuleaux triangle of width w fits snugly inside a square of side w at every orientation, you can spin it inside that square — and its corners scrape out very nearly the whole square. Put a cutting edge on it and a full-floating chuck to let its centre wander, and you have a drill that bores a (nearly) square hole. Harry Watts patented exactly this in 1917.
The gold shape is the drill bit; the darkening region is the hole it has carved. It stalls at 98.77% — the four corners stay slightly rounded. That number is exactly 2√3 + π/6 − 3.
The one you carry, and one you don't
This isn't only a curiosity. The British 50p (1969, the world's first seven-sided coin) and 20p (1982) are genuine curves of constant width — equilateral-curve heptagons, real circular arcs — so a vending machine can measure them without caring which way they went in. The Canadian loonie is an eleven-sided one. Reach into your pocket and you're holding a theorem.
One famous example, though, is softer than it's told. The Wankel rotary engine's rotor is often called a Reuleaux triangle — but it isn't quite: its sides are flattened to match the housing (itself an epitrochoid, unrelated to width), so the rotor does not have constant width. It's Reuleaux-inspired, not the real thing — and this page won't pretend otherwise.
The check
Every figure here is recomputed offline in research/reuleaux-constant-width/verify.mjs (15/15 checks), and the browser rebuilds the same geometry live:
· Constant width: the triangle, pentagon, and heptagon each measure 1.000 w in all 360 directions (deviation <10⁻³) — so the board on top never moves.
· Barbier's theorem: each has perimeter π·w = 3.14159, recovered by summing the arcs.
· Blaschke–Lebesgue: areas rise triangle 0.7048 < pentagon <
heptagon < circle 0.7854 (all at width w=1); the triangle
is the proven minimum, ½(π−√3), about 10.3% under
the circle.
· The square-hole drill: a union-of-rotations simulation (with exact registration) climbs to the closed form 2√3 + π/6 − 3 = 0.98770 (OEIS A066666); the four corners are provably never reached, so the hole is rounded, not perfect.
Idealisations, stated plainly: the shapes are exact circular-arc curves; "the board stays level" is the geometric width, not a friction-and-wobble physics of a real cart; the drill coverage is the mathematical swept area, and a real bit adds tool clearance.