Every number here is measured from the shape on screen, not typed in.
Where's the maths? This is geometry — the geometry of circles, arcs and angles. A shape has constant width if, whichever way you turn it, it just fits between the same pair of parallel lines: it is exactly as "thick" in every direction. A circle obviously does that, and almost everyone assumes circles are the only shapes that do. They are not. The Reuleaux triangle on screen is drawn with nothing but a compass: take an equilateral triangle, and from each corner sweep an arc joining the other two. Because every arc has the same radius as the triangle's side, the shape is as wide in one direction as in any other, so it rolls under the plank without the plank rising or falling at all — even though its own centre bobs up and down. The only mathematics you need to check this is the arc: a radius, an angle at the centre measured in radians, and the arc length r θ. Add the arcs up and something startling appears — every shape of constant width w has exactly the same perimeter, πw.
Beyond the syllabus: curves of constant width in general are enrichment.
Building it. Draw an equilateral triangle of side w. Put the compass point on one vertex, open it to w, and swing the arc between the other two vertices. Do that three times. Each arc is part of a circle of radius w, and because the triangle's angles are 60° = π/3 radians, each arc subtends π/3 at its centre.
Why the width never changes. Take any direction and squeeze the shape between two parallel lines. One line always touches a corner, and the opposite line touches the arc drawn from that very corner. The gap between them is therefore the radius of that arc — which is w, for every direction. That is a symmetry property of circles at work: every point of a circle is the same distance from its centre.
Barbier's theorem — the perimeter. Arc length is s = rθ with θ in radians. Three arcs, each with r = w and θ = π/3, give 3 × w × π/3 = πw. A Reuleaux n-gon (n odd) has n arcs of π/n, so n × w × π/n = πw again. A circle of width w has diameter w and circumference πw. So:
Every curve of constant width w has perimeter πw.
Check it in the panel on the left: switch between the triangle, the pentagon, the heptagon and the circle and the "perimeter ÷ w" readout stays at 3.1416. A square of side w does not — it reads 4, because its width swings from w to w√2.
Same perimeter, different area. The circle is the fattest curve of constant width, with area πw²/4 ≈ 0.785w². The Reuleaux triangle is the thinnest possible, at (π − √3)w²/2 ≈ 0.705w² — sector areas minus the triangle, which is a Sec 3/4 area-of-a-segment calculation.
Rolling. Under the plank the shape is a perfect roller: the plank height equals the width, so it is dead level. The centre is a different story. For the triangle the circumradius is R = w/√3 ≈ 0.577w, so the centre climbs to R and drops to w − R ≈ 0.423w — a bob of about 0.155w, which is exactly the gold wave in the graph. That is why cars have round wheels and not Reuleaux ones: the axle would shake even though the load would not.
Drilling a square hole. Constant width means the shape's bounding box is a w × w square at every angle, so a Reuleaux triangle can always be turned inside a square of side w while touching all four sides. Spin it and it sweeps out about 98.8% of the square, leaving four very slightly rounded corners. Watch the pale line inside the bit: that is the path of its centre, and it is not a circle but four arcs — which is why a real square-hole drill needs a floating chuck. Harry Watts patented a bit on exactly this idea in 1914.
In your pocket. UK 50p and 20p coins are Reuleaux heptagons. A slot machine measures a coin by rolling it and reading its width, so a constant-width coin passes as easily as a round one but is easy to tell apart by touch. The same idea keeps manhole covers safe: a cover of constant width cannot be turned to fall down its own hole.
Try this: choose Square and watch the plank rock and the crate lurch; then switch to the Reuleaux triangle without pausing. The blue plank trail snaps to a dead-straight line while the gold centre trail keeps waving. Now read the left panel: perimeter ÷ w is 3.1416 for the triangle, the pentagon, the heptagon and the circle — four different shapes, one perimeter. Then switch to Drill a square hole.
Controls: choose a scene and shape on the left, drag the width and speed sliders, and use Pause / Step / Reset — or press space to pause, ← → to turn the shape by hand and R to reset.