Generator

continued

A generator in the number library, called 9 times across 5 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.

continued 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: "convergents"`

`show: "error"`

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 quotients rebuild 34/13 ×3
the number of terms is a whole number between 2 and 12 ×2
√2 has a convergent above √5, as Hurwitz says every irrational must ×1
and the last of them has closed on √5 ×1
between two and five constants ×1
e has a convergent above √5, as Hurwitz says every irrational must ×1
each convergent is closer than the one before ×1
neighbouring convergents differ by a determinant of one ×1
the constant is one this figure knows ×1
the convergents alternate above and below the value ×1
the denominator is a whole number between 1 and 10000000 ×1
the golden ratio's convergents never leave the neighbourhood of √5 ×1
the numerator is a whole number between 1 and 10000000 ×1
the tower's last convergent is a number ×1
there are enough convergents to draw ×1
π has a convergent above √5, as Hurwitz says every irrational must ×1
π has a convergent that beats √5 by two orders of magnitude ×1
φ has a convergent above √5, as Hurwitz says every irrational must ×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

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.

Number

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

Number

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

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.

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