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mathematics, in pictures
Concept

Non-measurable set

A set of points to which no length, area or volume can be assigned consistently with translation and countable addition. Such sets exist because of the axiom of choice, and a ball can be cut into five of them and reassembled into two balls.

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

Every way of choosing one thing from each of 4 pairs. A table with one row per choice function on a small family of pairs, each row giving what it takes from each pair, with the row a stated rule names picked out.

The choice nobody can write down

Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.

logic · Axiom of choice The share of a ball that is its own edge. A plot of the proportion of each ball formed by its outermost shell against the radius, one line per group — falling towards nothing for the lattice groups and holding steady for the free group.

The edge that is as big as the ball

In a lattice the boundary of a large ball is a negligible fraction of it. In a tree it is two thirds of it at every size — and that single ratio, not the group's size, is what decides whether a set can be cut into pieces and reassembled into two copies of itself.

algebra · Cayley graph

Named alongside it

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

Axiom of choiceCayley graphChoice functionFree groupGroup actionGrowth rateIndependenceInvariantMaximal elementTransfinite recursionWell-orderingWord metric

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