Generator

descent

A generator in the number library, called 4 times across 1 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.

descent 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

`show: "sequence"`

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.

the side of the big square is a whole number between 2 and 200 ×2
the side of the small squares is a whole number between 1 and 200 ×2
and the discrepancy never changes size ×1
and the new pair is strictly smaller ×1
every step is strictly smaller than the one before ×1
the descent carries the discrepancy to minus itself ×1
the descent runs out of room before the discrepancy runs out ×1
the descent takes at least three steps ×1
the pair is one the descent can start from ×1
the two small squares overlap and still reach the corners ×1
the two small squares, less their overlap and plus the corners, are the big one ×1

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

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

The whole library · What the figures prove