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mathematics, in pictures
Concept

Pigeonhole

The observation that distributing more things than there are places forces some place to take two.

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 every node has at most 3 children and the tree reaches level 416 nodes on the bottom row; the marked walk takes a surviving child at every step

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

logic · models

Named alongside it

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

CompactnessEntropyFinite branchingGraph colouringInfinite pathIrrational rotationIterationKonig lemmaMeasureOrbitRecurrenceReversibility

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