Fabienne Serriere has used math to inspire her knitting for several years, and right now she’s raising money on Kickstarter to do it on a larger scale. As of July 2, the project, KnitYak, was about 1/3 of the way to the goal and 1/3 of the way through the campaign, which closes on July 23. I’d recommend heading over to Kickstarter to give her some money right now because KnitYak needs to happen.
Serriere was looking for a way to make patterns that were “beautifully non-repeating,” somewhere between strict order and total chaos. After some experimentation, she landed on cellular automata.
A cellular automaton is a grid in which every cell can be in one of a finite number of states (for example, on/off or black/white). In the elementary cellular automata Serriere is using, the state of a cell can be either white or black, and the state of a cell depends on the three cells adjacent to it on the left. (The cell to the left and the cells directly above and below that cell.) There is a nice explanation of cellular automata at the mathematics-fiber arts blog Botanica Mathematica.
Serriere initially thought about using John Conway’s Game of Life, a famous two-dimensional cellular automaton, but it didn’t look particularly appealing when it wasn't moving. Elementary cellular automata, on the other hand, are fundamentally one-dimensional; they only move in one direction, so a two-dimensional object like a scarf can capture the evolution of the system more effectively.
Cellular automata are some of the simplest places chaos can arise. A chaotic system is deterministic, meaning the evolution of the system is set at the outset, but small changes in initial conditions (in this case, the starting row) lead to large changes in results. Not all cellular automata are chaotic in all widths, which is interesting in and of itself (link goes to pdf).
Elementary cellular automata are important in computer science as well. “Most people who study or love computer science have seen the output from an elementary cellular automaton and recognize my scarf,” Serriere wrote. Rule 110, the rule her scarf uses, is particularly important because it is Turing complete, meaning theoretically one could program it to perform computations. (See this paper by Matthew Cook for more details.)
If you back KnitYak on Kickstarter at the scarf level or higher, at the end of the campaign you will have the opportunity to choose the rule and starting seed for the black and white merino wool scarf you’ll get. Each scarf is “provably unique”: Serriere will be keeping track of the rule and starting seed for each piece, and she guarantees that no one will have the same scarf as you do. She will also ship the source code with the scarf in case you need it someday to replace or augment your cellular automata wardrobe with a coordinating design. Not all cellular automata generate attractive designs with all possible starting states, so Serriere is going to cull the designs that aren’t up to snuff. “I am making a code-curated collection of possible knits, where all of the knits are as interesting as possible,” she says.
While you're eagerly anticipating your scarf, you can check out the Wikipedia page on elementary cellular automata to find a rule you like. You can also check out Serriere's rule 110 code to see what happens with some different starting seeds.