Integral geometry
Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
Where random discs change sign
Scatter discs at random over the plane and their union breaks into pieces with holes in them. Pieces minus holes has an exact average, η e^(−η)(1 − η) per disc area, where η is how many discs cover a point on average, and it changes sign at exactly η = 1, with 63.2% of the plane covered. The discs first join up across the whole plane later, at η ≈ 1.128. The sign change looks like the moment the union connects and is not; and for squares or long rectangles turned at random it comes earlier still, at 4πA/L², which only a disc reaches one.
The shortest wall that stops every line
A straight cable runs somewhere under a square field, and a trench has to find it for certain. Three sides of the field will do, at length 3. Two diagonals cost 2.83, and a Steiner tree 2.73. The best trench known has length 2.6390 and comes in two pieces that do not touch. Averaging over lines proves nothing shorter than 2 can work, and in over a century the gap between the two numbers has hardly moved.
What a random chord is
Draw a chord of a circle at random. How often is it longer than the side of the inscribed equilateral triangle? Pick two random points on the rim and the answer is 1/3. Pick a radius and a random point on it, and the answer is 1/2. Pick a random midpoint in the disc, and the answer is 1/4. All three calculations are correct, and Bertrand published them in 1889 to warn that the phrase had no meaning. Eighty-four years later Jaynes argued that one answer is right after all.
Named alongside it
The objects these essays reach for when they reach for this one.
Monte CarloPiArc lengthConvexityReuleaux triangleSamplingSimulationAreaBarbier's theoremBuffon's needleChordCircle