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Concept

Measure

An assignment of size to the parts of a set, adding up whenever disjoint parts are put together.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

0123456740 steps of a rotation by √2 − 1 of a turnthe gaps between neighbouring points take 3 distinct values — never more than three, at any number of steps

The orbit that must come back

A system with finitely many states has to repeat itself. Poincaré showed the same thing holds when the states are a continuum — almost every starting point returns arbitrarily close to where it began, however complicated the rule, and the argument is the pigeonhole principle with volume in place of counting.

dynamics · pigeonhole the cutter's measure, segment by segmentthe chooser's measure, segment by segment52530201553010552030cut at 4/9the left piecethe right piecethe cutterthe chooser5050130/3≈ 43.33170/3≈ 56.67the cutter's piecethe chooser's piecethe cutter's running total crosses 50 inside segment 3, 2/3 of the way through it, so the cut is at 4/9both pieces are worth exactly 50 to the cutter; the chooser takes the right one at 170/3 ≈ 56.67 andgains 20/3 ≈ 6.67 over half

One cuts and the other chooses

The oldest rule in fair division promises each of two people at least half the cake by their own measure, and it keeps that promise exactly. It does not promise what the word "fair" is usually asked to carry, and the gap opens the moment the two measures disagree across the cut.

applied · fair division

Named alongside it

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

Counting argumentDivide and chooseEntropyEnvy freenessFair divisionIrrational rotationIterationOrbitPigeonholeProportionalityRecurrenceReversibility

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