Matpic
mathematics, in pictures
Generator

complex-turn

A generator in the algebra library, called 7 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.

complex-turn 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

Multiplying two complex numbersIn the complex plane, multiplying adds the two angles and multiplies the two lengths.realimaginaryθφzwzw|z| = 1.49|w| = 1.08|zw| = 1.61θ + φ = 76°

show: "powers"

Powers of a complex numberThe first 12 powers of a complex number, each one a further turn and stretch of the last.1z¹z²z³z⁴z¹²

show: "roots"

The 5 5th roots of unity5 points spaced evenly around the unit circle, at the vertices of a regular 5-sided polygon.11/52/53/54/5each is one 5th of a turn from the last

show: "euler"

Euler's formula on the unit circleWalking a distance around the unit circle; at half a turn the point sits exactly at minus one.1i−1−iarc length 3.142e^(iπ) = −1

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.

Algebra

Multiplying is turning

Complex numbers are introduced as an algebraic dodge for square roots of negatives. They are better understood as the arithmetic of rotation, at which point every rule stops needing to be remembered.

The whole library · What the figures prove