SERIES C // RECURSIVE RULES

Recursive geometry posters

Sound that builds. Thousands of blocks of light grow by a set of recursive rules - branching, bending and repeating until the signal fades. The architecture of a single moment.

Read the guide: Perlin noise explained →
All 58 Attractors 12 Mandelbrot 6 Flame 4 Structure 6 Flux 6 City 6 Ridges 6 Orbit 6 Hopf 4 Objects 2
Structure - Transmitter Transmitter Structure from €39 Structure - Galaxy Galaxy Structure from €39 Structure - Signal Tree Signal Tree Structure from €39 Structure - Binary Star Binary Star Structure from €39 Structure - Dyson Sphere Dyson Sphere Structure from €39 Structure - Supernova Supernova Structure from €39

MATHEMATICS

The maths behind the picture

  1. 01

    A grammar instead of a drawing

    No hand draws the picture, a set of rules does: "a ray is a block, then a new ray a little further and a little bigger". The program applies the rules over and over, like a sentence that writes itself. The idea comes from Context Free Art and Structure Synth.

    ray → block · ray(x + 2, s × 1.045)
  2. 02

    Chance with rules

    Each rule has a probability: most of the time a ray simply extends, sometimes it forks in two, sometimes it places a node that grows arcs. The same seed always makes the same choices, so every piece is exactly reproducible: every print is identical to the original.

    p(extend) ≈ 84% · p(fork) ≈ 9% · p(node) ≈ 5%
  3. 03

    Sound sets the reach

    How many steps each ray lives depends on a sound signal s(t): a sum of sine waves under swells of loudness. Loud parts of the signal become long rays, quiet parts stay short.

    s(t) = Σ aₖ·sin(2π·fₖ·t + φₖ) · envelope(t)
  4. 04

    Perlin noise

    Nature is neither perfectly smooth nor completely random: clouds, mountains and waves are somewhere in between. In 1983 Ken Perlin invented a noise that looks like nature: a smooth function that gives random but gentle changes. The program adds four layers (octaves) of this noise, each half as strong and twice as fine as the last. That is what gives the lines life, so they don't look like they were drawn with a ruler.

    noise(x) = Σₖ 0.5ᵏ · perlin(2ᵏ·x), k = 0…3

HISTORY

Where this idea comes from

  1. 1956

    Noam Chomsky describes formal grammars: a finite set of rules that can generate infinitely many sentences.

  2. 1968

    Biologist Aristid Lindenmayer applies the same idea to plants. His L-systems describe how branches grow using rewriting rules.

  3. 2005

    Context Free Art brings grammars into art: a picture is a program of a few rules that call themselves.

  4. 2007

    Mikael Hvidtfeldt Christensen releases Structure Synth, extending grammars into three dimensions. This series grows from there.

Perlin noise

  1. 1982

    Ken Perlin works on the effects for the film Tron and is frustrated that computer images look too "machine-like".

  2. 1983

    Perlin invents his noise and publishes it in 1985 at SIGGRAPH, in the paper "An Image Synthesizer".

  3. 1997

    Perlin receives an Academy Award for Technical Achievement: his noise becomes the basis of smoke, fire and cloud textures in film.

  4. 2001

    Perlin publishes an improved version and "simplex" noise. Today games, films and programmer art all over the world use it.

In nature

Recursive rules for plant growth were first written down in 1968 by biologist Aristid Lindenmayer, watching algae: a cell divides, and each new branch repeats the old one.