Concept

Dirichlet

Peter Gustav Lejeune Dirichlet, whose name is on both the pigeonhole principle and the theorem that every irrational has unreasonably good rational approximations. Both results carry the same signature: a counting bound forcing something to exist without exhibiting it.

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

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

analysis · The integral

13 into 12. 13 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1.

More things than boxes

If there are more objects than containers, some container holds two. That is the entire principle, it is impossible to disagree with, and it settles questions that look nothing like it.

discrete · Pigeonhole $8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.$

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

number · Pigeonhole

Named alongside it

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

Counting argument Existence proof Nonconstructive Pigeonhole principle Area Collision Continued fraction convergent Continued fractions Convergence Dirichlet's function Fundamental theorem Golden ratio

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