Concept

Integral geometry

The study of geometric quantities recovered from averages over random lines, positions or orientations. Buffon's needle is its first result: a probability computed by averaging over positions and orientations turns out to contain the circle constant.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

120 needles on a lined floor. 120 needles dropped at random across evenly spaced lines; 83 of them cross a line.

Getting pi by dropping needles on the floor

Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.

probability · Monte Carlo

A Reuleaux triangle. A curve of constant width on 3 vertices, with 6 pairs of parallel supporting lines drawn across it. Every pair is 180.1 apart.

Round is not the only way to be the same width

A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.

geometry · Constant width

Four shapes of perimeter 300 between their inner and outer circles. A square, an ellipse, a Reuleaux triangle and a stadium, each drawn with the largest circle inside it and the smallest circle around it, the ring between the two shaded, with the ring's width and the widest ring allowed.

Nearly the most means nearly round

A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.

geometry · Isoperimetric

Measuring a closed curve with inlets by throwing lines at it. A curve inside a disc crossed by a sample of random lines with each crossing marked, beside the mean number of crossings and the length it implies against the true length.

A length counted by the lines that cross it

Throw straight lines at random across a curve and count how often they cross it. The average count, times π times the radius of the target, is the curve's length — for a wiggly closed curve, a spiral or a snowflake alike, with no following of the curve and no derivative anywhere. It is Crofton's formula of 1868, and it measures length the way a map-reader's ruled transparency does.

analysis · Arc length

A trefoil, and how far its tangent turns. The (2, 3) torus knot with tube radius 0.45; total curvature 16.8556 = 5.3653π.

A knot must turn twice, and a little more

Follow a closed curve and add up how far its direction turns. Any closed curve turns through at least one full circle; a knotted one turns through more than two. The proof is a count of hilltops — the total turning equals 2π times the number of highest points the curve has, averaged over every direction to look from — and a curve with a single hilltop in any direction cannot be knotted.

topology · Knots

Named alongside it

The objects these essays reach for when they reach for this one.

Pi Convexity Monte Carlo Reuleaux triangle Sampling Arc length Area Barbier's theorem Buffon's needle Circle Constant width Convergence

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