Curves of constant width · Shop-floor demonstration · W = 120.0

Round without being a circle.

Every roller in the mill below gauges exactly 120.0 across — at every angle the calipers can turn. Only one is a circle. The rest have corners, and still the rails sit dead level as they pass. Width is not roundness. Watch the gauge.

Station 01 — The rolling mill 4 lanes · no slip rail gap 120.0 · fixed

Rails fixed 120.0 apart; each roller touches both at every instant. The red trace beneath each is the path of its hub — dead flat for the circle, bobbing for everything else — and the tick on the lower rail is the contact point, which freezes each time a corner pivots. A slab riding on top would glide perfectly level on any lane. This is the machinist's trap: a two-jaw micrometer reads 120.0 on all four, and only one would survive as a shaft.

Station 02 — The square drill guide square 340.0

Wilmerding, Pennsylvania · Pat. 1917

A triangle drills a square hole.

Spin a Reuleaux triangle inside a square guide of the same width and it touches all four walls at every instant — the corners scrape the flats while the arcs brace the span. The Watts Brothers Tool Works patented exactly this in 1917 and has sold square-hole drills ever since.

The catch is the chuck. The bit's centre refuses to stay put: it rides the small red circuit — four elliptical arcs, traced three times per revolution — so the drill needs a floating holder that lets the shank wander while the edge cuts. What it leaves behind is 98.77% of a true square; four blunt slivers survive in the corners.

98.77%

of the square, cleared

1.23%

survives in the corners

±7.7%

of W — centre wander

Definition · two straightedges, any angle

The gauge test.

Constant width is a pact with the calipers: every pair of parallel tangent lines sits exactly W apart, whatever the angle. Trap the shape between two straightedges and spin them — the gap never changes, so the jaws never learn what they are holding.

Build one from any regular polygon with an odd number of corners: swing an arc across each side, centred on the vertex dead opposite. Odd is not optional — every arc needs a single opposite vertex to swing from. Or skip the corners entirely and bend the radius with odd sine waves; lane 04 upstairs is one of those, drawn fresh on every visit.

Station 03 — Roving caliper reading constant

The jaws orbit; the reading doesn't. Red dots mark the true contact points — note they slide off the perpendicular as the jaws pass a corner.

Shop data · four facts off the floor

π·W

Perimeter, always

Barbier, 1860: every curve of width W closes in exactly πW — the same as the circle. The four rollers above finish every revolution in perfect lockstep.

98.77%

of the square

The most a Reuleaux drill can clear. The remaining 1.23% holds on as four blunt slivers, one per corner — a square hole to every gauge but the corner probe.

7 sides

in your pocket

The British 20p and 50p are constant-width heptagons. Coin gauges read them as round; they never jam edge-on, and the flats save the mint metal.

0 axles

ever

The hub of a constant-width roller bobs as it rolls. Under a slab, invisible; on a fixed axle, a hammer blow per turn. Rollers yes, wheels never.