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Mathematics 2019, 7(2), 165; https://doi.org/10.3390/math7020165

A 6-Letter ‘DNA’ for Baskets with Handles

Sculptor, Washington, DC 20002, USA
This paper is an extended version of our paper published in Mallos, J. Walking Around Trees: A 6-Letter ‘DNA’ for Baskets with Handles. In Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture; Torrence, E.T.B., Séquin, C., Fenyvesi, K., Eds.; Tessellations Publishing: Phoenix, AZ, USA, 2018; pp. 219–224.
Received: 12 December 2018 / Revised: 1 February 2019 / Accepted: 3 February 2019 / Published: 13 February 2019
(This article belongs to the Special Issue Topological Modeling)
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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. View Full-Text
Keywords: Mullin encoding; shuffled Dyck language; chord diagrams; basket making; crochet; basket map; A-trail; non-crossing Eulerian circuit; topological surface; polygonal schema Mullin encoding; shuffled Dyck language; chord diagrams; basket making; crochet; basket map; A-trail; non-crossing Eulerian circuit; topological surface; polygonal schema
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Mallos, J. A 6-Letter ‘DNA’ for Baskets with Handles. Mathematics 2019, 7, 165.

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