Technique · Simulations
Elementary automaton
One row of cells computing the next by a rule numbered 0 to 255 (30, 90, 110 ...), scrolling as it grows.
Elementary automaton is a simulation that evolves step by step. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Rule 90 draws the Sierpinski triangle: each cell becomes black when exactly one of its two neighbours was black, which is the same as Pascal's triangle with the even numbers coloured white.
- Wolfram sorted the rules into four kinds of behaviour, those that die to a uniform row, those that settle to repeating stripes, those that look random and the rare ones that make lasting structures that interact, as in rule 110.
- Stephen Wolfram's 2002 book A New Kind of Science is filled with pictures like this one, which he argued show that very simple rules can account for the complexity of nature.
- Stephen Wolfram numbered the 256 elementary rules in the early 1980s; he used rule 30 as the random-number generator in his Mathematica software, and the pattern on the shell of the cone snail Conus textile looks very much like it.
- Rule 110 is complex enough to be a universal computer: Matthew Cook proved in the 1990s that it can carry out any computation, so that something as small as one row of cells and an eight-line table can do what any computer can.
The mathematics
- There are eight possible patterns of three cells, and each can give white or black, so 2 to the power 8, or 256, rules; rule 30 is 00011110 in binary, one digit for each pattern read from 111 down to 000.
- A ring of n cells has only 2 to the power n possible rows, so the picture must repeat eventually; for rule 30 on a ring as wide as this one the repeat is so long that it never matters on screen.
- Each row is the previous one passed through a fixed table, so the picture is deterministic: the same first row always gives the same picture, even for rules that look random.
In art
- Triangles within triangles like those of rule 90 appear in the floor mosaics of Roman churches from the twelfth and thirteenth centuries, made by Cosmati craftsmen with no theory behind them.
- The rows can be woven, knitted or printed: each row is a line of black and white stitches, so elementary rules make patterns for textiles with nothing but a table of eight answers.
More ways to use it
Related techniques
Reaction-diffusion · Cellular automata · Sandpile · Grain growth · Slime mould · Flocking boids · Attractor cloud · Synchrony