Matpic
mathematics, in pictures
Generator

lattice-circle

A generator in the number library, called 10 times across 3 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.

lattice-circle 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

The circle of radius √25 on the integer latticeA circle drawn on the whole-number grid, with the lattice points it passes through marked.12 lattice points sit on the circle — and 4 × (3 − 0) = 12divisors of 25: 1, 5, 25 are 1 mod 4, none are 3 mod 4

show: "rational"

Rational points on the unit circleLines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.(−1, 0)3/4 → 7, 24, 252/3 → 5, 12, 131/2 → 3, 4, 52/5 → 21, 20, 291/3 → 8, 6, 101/4 → 15, 8, 17each line of rational slope meets the circle a second time at a rational pointclearing the denominators turns that point into a Pythagorean triple, and every triple arises this way

show: "gaussian"

Primes below 40 as sums of two squaresEach prime with its remainder on division by four, and the two squares that add to it where they exist.primemod 4two squares221² + 1² = 233— no way at all —512² + 1² = 573— no way at all —113— no way at all —1313² + 2² = 131714² + 1² = 17193— no way at all —233— no way at all —2915² + 2² = 29313— no way at all —3716² + 1² = 37every prime one more than a multiple of four is a sum of two squares, and no other odd prime istwo is the exception that proves nothing: 1² + 1², and it is the only even prime

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

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

One way to factor, and no other

Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

 Number

Two squares, and a lattice

Whether a prime is the sum of two squares is decided entirely by its remainder on division by four. A fact about circles is settled by a fact about remainders, and neither statement contains any hint of the other.

The whole library · What the figures prove