Farey boat: continued fractions and triangulations, modular group and polygon dissections. (English) Zbl 1437.11097
In this article the authors introduce an interesting combinatorial interpretation of several known results on continued fractions. In particular they consider two theorems relating continued fractions and triangulations studied by J. H. Conway and H. S. M. Coxeter [Math. Gaz. 57, 87–94 (1973; Zbl 0285.05028)], and C. Series [J. Lond. Math. Soc. (2) 31, 69–80 (1985; Zbl 0545.30001)]. The authors have obtained more general polygon dissections when extending these theorems for elements of the modular group \(\mathrm{PSL}(2,\mathbb Z)\). These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of \(\mathrm{PSL}(2,\mathbb Z)\).
Reviewer: Oleg Karpenkov (Liverpool)
MSC:
| 11J70 | Continued fractions and generalizations |
| 11A55 | Continued fractions |
| 11-03 | History of number theory |
Keywords:
continued fractions; Farey graph; polygon dissections; Ptolemy rule; Pfaffians; modular groupReferences:
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