Python on FoundingFuture

Numbers and sets

Mon, 31 Aug 2026 20:10:00 +0200

Each kind of number was invented because an operation ran out of answers. Subtraction ran out of the counting numbers, division ran out of the whole numbers, and roots ran out of the fractions. Complex numbers are the last of those widenings, and they arrive the same way the earlier ones did.

Anatomy of the Mandelbrot set

Sun, 30 Aug 2026 12:54:33 +0200

The pictures that carry the name Mandelbrot show a black bulb with spirals and curls along its edge. They are coloured in bands, and magnifying the edge reveals more detail without end. Those pictures are portraits of a set of complex numbers. One short calculation, repeated, decides which numbers belong. This page defines the set, works that calculation, and names the shapes the picture turns out to hold.

Angles and polar form

Mon, 31 Aug 2026 21:03:19 +0200

A point of the plane can be named by two coordinates, or by one distance and one turn. The second naming is the polar form, and it is the one under which multiplication of complex numbers becomes easy to read. Reaching it takes an angle measure and two ratios.

Matrices and vectors

Mon, 31 Aug 2026 22:25:15 +0200

A vector is a column of numbers. A matrix is a rectangle of them, so a vector is the case of a single column. Adding either works one position at a time. Multiplying two matrices builds each entry of the answer from a whole row and a whole column. That one rule is what the rest of the subject rests on.

Iteration and orbits

Mon, 31 Aug 2026 22:05:53 +0200

Applying one rule repeatedly, with each answer becoming the next question, produces a sequence. Which sequence depends entirely on where the process starts. Two starting points a hair apart can produce sequences that end in different places.

One-pole filters, shelves and all-passes

Sat, 05 Sep 2026 00:35:37 +0200

A filter is a rule that treats fast changes in a signal differently from slow ones. The simplest rule that does so keeps one number in memory and moves it a fixed fraction of the way towards each new sample. Everything on this page is that rule and its mirror image. The last of them rearranges a delay so that it moves a click in time and leaves every steady tone at its level.

Linear, cubic and spherical interpolation

Sat, 05 Sep 2026 00:35:37 +0200

Given two values and a fraction, interpolation names a value in between. Which value depends on what the two are: numbers on a line, samples of a curve, or directions. Each case has its own rule, and the wrong rule gives an answer that lies between the inputs and is still wrong.

Random numbers and noise

Sat, 05 Sep 2026 00:35:37 +0200

A computer has no dice. What it has is a rule that scrambles a number so thoroughly that the sequence it produces passes for chance. It also has a way of turning that sequence into a signal with a chosen colour.

Colour spaces and blending

Mon, 07 Sep 2026 23:58:15 +0200

Three ways to take a colour halfway between two others, and what each one costs.

A colour halfway between two others depends on the space the halving happens in. Averaging the bytes a file stores gives one colour. Averaging the light those bytes stand for gives another, and averaging in a space built to match perception gives a third. This page defines the three spaces, gives the transforms between them, and measures the midpoint each produces for one pair of colours.

Cutting a small-capitals face

Mon, 24 Aug 2026 00:00:00 +0000

Fake small capitals are shrunken capitals. It shows.

Set a word in them and it comes apart. The menu on this site sets anything that opens in small capitals. Ask a browser for that and you write:

font-variant-caps: small-caps;

If the font has small-capital glyphs, you get them. If it does not, the browser makes some by shrinking the capitals, and that is where it goes wrong.