MATHEMATICS
The maths behind the picture
Every piece is computed, not drawn. These are the real equations and numbers from the program that made this exact piece.
H(i, j) = (0.75·s(i/G)·s(j/G) + 0.25·n(i, j))^2.1 · c(i, j)i = frequency band, j = time slices(t) = Σₖ aₖ·(½ + ½·sin(2π·fₖ·t + φₖ)) · Σⱼ bⱼ·e^(−((t − cⱼ)/wⱼ)²)- Grid
- 44 × 44
- Towers
- 1,936
- Signal frequencies
- 5.814 · 16.239 · 16.879 · 16.824
- Swells
- 0.596±0.133 · 0.627±0.136
- Perlin noise
- 4 octaves · falloff 0.5
- Tallest tower
- 26, 10
- Mean height
- 0.031
The form is driven by a sound signal made of the frequencies 5.814 · 16.239 · 16.879 · 16.824. Where the signal is louder, the structure grows further.
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.
HISTORY
Where this idea comes from
City
- 1822
Joseph Fourier publishes that any periodic signal can be broken into a sum of sine waves. It is the basis of sound analysis.
- 1947
Bell Labs presents "visible speech": a spectrograph that turns sound into a picture of frequencies over time.
- 1965
Cooley and Tukey publish the Fast Fourier Transform (FFT), putting spectrograms within reach of every computer.
Perlin noise
- 1982
Ken Perlin works on the effects for the film Tron and is frustrated that computer images look too "machine-like".
- 1983
Perlin invents his noise and publishes it in 1985 at SIGGRAPH, in the paper "An Image Synthesizer".
- 1997
Perlin receives an Academy Award for Technical Achievement: his noise becomes the basis of smoke, fire and cloud textures in film.
- 2001
Perlin publishes an improved version and "simplex" noise. Today games, films and code art all over the world use it.
