A formal language for describing knitted textiles using swatches

Knitted textiles are a type of 2D programmable material. Their emergent elastic behavior can be 'coded' in by changing the entanglement of yarn at the stitch level. Therefore, the natural formulation to describe and study complex knitted textiles is a language with a grammar. We propose such a framework, where the alphabet is composed of irreducible swatches -- knitable textile knots and links -- that can be concatenated leading to higher order or compound swatches. In this talk, first we define what are irreducible and reducible/compound swatches -- analogous to prime factorization of natural numbers. Then a set of operations acting on two or more swatches is described to define the syntax and grammatical structures of the proposed language of knitted textiles. Using this approach, we then quantify an upper bound on the number of words of fixed length in the vocabulary of the language corresponding to a knitting protocol capable of generating a finite alphabet. This work is supported by the NSF grant, NSF DMR1847172.