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Concept

Geometric mean

The nth root of the product of n positive numbers, which is the average of their logarithms exponentiated. It is never larger than the arithmetic mean, with equality only when every number is the same, and the gap between the two is the logarithm's concavity.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

computation · Constructible numbers A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

analysis · Convexity Five rules, one dial. Seats for each of 5 regions at 21 settings of the rounding threshold, with the three settings that are the named methods marked; the largest region gains and the smallest loses as the threshold rises.

Five rules and one dial

Adams, Webster and Jefferson are usually taught as three rules for rounding a share. They are one rule with a number in it, and turning that number from nought to one moves seats from the smallest region to the largest, one at a time.

applied · Apportionment Which regions each rule favours. Average seats above or below exact quota for the largest and the smallest region, under each of the five methods, over 400 generated instances.

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

applied · Apportionment What each rule is answering. A table of five apportionments against three measures of inequality between two regions, with a tick where no transfer of a seat reduces the measure; each measure certifies exactly one of the five.

Choosing what unfair means

Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.

applied · Apportionment

Named alongside it

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

ApportionmentDivisor methodQuotaCounting argumentMonotonicityRoundingArithmetic meanBiasCentroidChordCircleClosure

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