The numbers lead a dance. Mathematics of the Sestina

• Alan R. Champneys

University of Bristol, UK
• Poul G. Hjorth

Technical University of Denmark, Kgs. Lyngby, Denmark
• Harry Man

Oxford Brookes University, UK

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Abstract

Sestinas are poems of 39 lines comprising six verses of six lines each, and a three line final verse or ‘envoi’. The structure of the sestina is built around word repetition rather than strict rhyme. Each verse uses the same set line ending words, but in a permuted order. The form of the permutation is highly specific, and is equivalent to iteration of the tent map. This paper considers for which number $N$ of verses, other than 6, can a sestina-like poem be formed. That is, which $N$ will the prescribed permutation lead to a poem of $N$ verses where no two verses have the same order of their end words. In so doing, a link is found between permutation groups, chaotic dynamics, and Cunningham numbers.