Thought Toys · Shape & space · Exhibit 126
A wheel needs an axle, and an axle needs a circle. A roller needs neither. A roller only has to be the same height however it happens to be lying, and the circle is not the only shape with that property; geometers call the others curves of constant width. Take a triangle, bulge each side out into an arc drawn from the opposite corner, and roll it under a plank. The plank never bumps. The roller's centre, meanwhile, bobs up and down three times a turn, because the thing under the plank is not a circle and never was.
Two rollers, one plank each, turning together ● the arc-sided roller ● the same corners, straight-sided — where the plank would be at one width
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The amber roller is a triangle whose three sides have been swapped for arcs, each arc drawn with its compass point on the opposite corner. It rolls along the floor, and the plank on top rides on it. Watch the plank against the dashed line. It never leaves it. However the roller is lying, the distance from the floor to the plank is the same, so this shape is called a curve of constant width. Now watch the small trail behind the roller's centre. It bobs. The centre is highest when a corner points straight down and lowest when a corner points straight up, three times every turn. The outside of the roller keeps its height while the middle of it does not, and that is the whole point.
The cyan roller in the lower scene has the same three corners with straight sides, the plain triangle. Its plank bumps. It sits at the dashed height only while one side stands upright, and it sags by about an eighth of the roller's width whenever a corner points straight up or straight down, six times a turn. The only difference between the two rollers is the arcs, and the arcs are what make the width constant. Whichever corner is touching the floor, the arc opposite it is exactly one width away, because that arc was drawn from that corner.
Now drag How many corners up. Seven corners makes a Reuleaux heptagon, the shape of a British twenty-pence coin, which is why a coin machine can gauge it in any orientation. With every pair of corners you add, the centre bobs less, from 15% of the width down to half a percent at fifteen corners, and the shape looks more like a circle. The plank, meanwhile, does not care. It rode level on three corners and it rides level on fifteen. Only an odd number of corners works: each arc needs a single corner opposite it to be drawn from.
Look at the tick marks sliding past on the floor. Every roller here advances exactly π widths per turn, the same as a circle of that diameter would, whatever its corner count. That is Barbier's theorem: every curve of constant width has the same perimeter as the circle, π times the width. The straight-sided triangle in the lower scene rolls less far per turn, three of the arc-sided roller's widths against 3.14, because its perimeter is not that of any circle.
None of these shapes would make a wheel. Fix an axle through the centre and the axle bobs with it, once for every corner. That is why the Wankel engine, whose rotor is close to a Reuleaux triangle, lets the rotor orbit on an eccentric shaft instead of spinning on a fixed one. A roller does not need a fixed centre. It only needs a constant height, and that is a weaker thing to ask of a shape than roundness. A manhole cover does not need to be round either: it only needs to be unable to fall through its own hole, and any of these shapes manages that.
The rule, exactly. A shape's support function h(θ) is how far it reaches from its centre in direction θ. Its width in that direction is h(θ) + h(θ + π), the gap between two parallel rails squeezing it from either side. A curve of constant width w has that sum equal to w for every θ. Rolling to the right on a floor, turned clockwise by α, the roller's centre sits at height h(α − π/2) and advances at that same rate per radian, and the plank rests at
h(α − π/2) + h(α + π/2) = wA Reuleaux polygon on n corners (n odd) sets them on a circle of radius R = w / 2 sin(π(n − 1)/2n); for three corners that is w/√3, about 0.577w, the corner-to-centre distance of a triangle with sides w. Each arc has radius w and angle π/n, so the perimeter is n · (π/n) · w = πw: Barbier's theorem (1860) by direct count. The centre bobs between w − R and R, so the bob is 2R − w: 15.5% of w for three corners, 5.2% for five, 0.55% for fifteen. Cauchy's formula gives any convex shape's perimeter as the integral of h over a full turn, which is how the page's distance-per-turn readout is computed, and why every constant-width roller shows π.
Verified in node (improve/verify/126-constant-width.js, 7 checks, all before this
page existed):
Leonhard Euler studied these curves in 1778 and Franz Reuleaux catalogued them in 1876 as parts of machines. Joseph-Émile Barbier proved the perimeter theorem in 1860. The Reuleaux triangle has the sharpest corners any curve of constant width can have, 120 degrees, and, by a theorem of Blaschke and Lebesgue, it encloses the least area for its width.
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