Matpic
mathematics, in pictures
Generator

pigeonhole

A generator in the discrete library, called 9 times across 4 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

pigeonhole is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

13 into 1213 items spread as evenly as 12 boxes allow. Even at their most even, some box holds 2, because 13 is more than 12 × 1.211111111111spread as evenly as possible, the fullest box still holds 2

show: "scale"

The same counting argument, four timesFour statements of the pigeonhole principle. In each, the number of items exceeds the number of boxes, so some box is forced to hold more than one.items exceed boxes, so some box is forcedpeople / possible birthdays367 into 366forces 2socks / colours8 into 7forces 2points in a square / quadrants5 into 4forces 2Londoners / possible hair counts1,000,000 into 150,000forces 7

What it checks while it draws

Collected by running the family and listening to lib/verify.js, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Number

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.

 Discrete

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

Six people at a party

Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.

 Probability

Twenty-three people

A room needs 253 people before someone probably shares a birthday with you. It needs 23 before two of them probably share one with each other. The gap between those numbers is the whole problem.

The whole library · What the figures prove