euclid
euclid 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: "irrational"
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 smallest square has side gcd(34, 13) ×4
- the squares tile the 34 by 13 rectangle ×4
- the leftover strip is still there after every pass ×1
- the squares peeled at each pass are the continued fraction of the ratio ×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.
A fraction that never closes
Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.
GeometryThe oldest algorithm, drawn as a tiling
Euclid's method for finding a greatest common divisor is usually presented as a loop. It is also a way of tiling a rectangle with squares, and the tiling explains why it works.
GeometryThe rectangle that eats itself
Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.
NumberThe 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.