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Abstract
Fabric surfaces, made using techniques such as 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 ( S D L 2 ). Besides its 4-letter alphabet, S D L 2 has some other similarities to DNA: Any word can be ‘evolved’ via a sequence of local mutations (rewriting rules), and ‘gene-splicing’ two S D L 2 words, by an insertion or concatenation, produces another S D L 2 word. However, S D L 2 comes up short when we attempt to make a basket with handles. I show that extending the language to S D L 3 , by addition of a third type of parenthesis, succeeds for orientable surfaces with handles—provided an appropriate choice of cut graph is made.
Highlights
As a sculptor, I have been interested in character sequences that code for the construction of shapes [1,2,3]
SDL2 is the subset of Σ∗SDL2 consisting of all words, and only those words, that can generated from e via the following rewriting rules: Insert a matched pair of parentheses, ‘()’ or ‘[]’, anywhere
If the chords in the schema can be drawn without crossings (Figure 10a), the polygonal opening closes up to a cut graph which is a tree, which shows that the surface is a topological sphere and can be coded in SDL2
Summary
I have been interested in character sequences that code for the construction of shapes [1,2,3]. Some fun can be had making shapes this way. I have done fair projects where participants learn to make a basket from a word (Figure 1a), or ‘evolve’ a new character sequence (via rewriting rules) and discover the shape that it describes by snapping pieces together (Figure 1b) or twisting balloons (Figure 1c). At some length, the coding and making of baskets without handles, and show a limited way to extend these techniques to baskets with handles. This paper extends the work presented in [3]
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