Walking Around Trees: A 6-Letter ‘DNA’ for Baskets with Handles
James Mallos

Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture
Pages 219–224
Regular Papers

Abstract

Fabric surfaces made by techniques like crochet and net making are typically worked in a linear order that meanders, without crossing itself, to ultimately visit and build the entire surface. For a closed basket whose surface is a topological sphere, it is known that the construction can be described by a codeword on a 4-letter alphabet via Mullin’s encoding of plane graphs. Mullin’s code exemplifies the formal language known as the Shuffled Dyck Language with 2 Types of Parenthesis (SDL2.) Besides its 4-letter alphabet, SDL2 has some other similarities to DNA: any word can be ‘evolved’ via sequence of local mutations (rewriting rules); and ‘gene-splicing’ two SDL2 words by insertion or a concatenation, produces another SDL2 word. But SDL2 comes up short when we attempt to make a basket with handles. I show that extending the language with a third type of parenthesis works for orientable surfaces with handles provided an appropriate choice of cutset is made. (The cutset is the connected network of fronts—termed cuts in topology—where normal fabric construction encounters previously worked fabric.)

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