Mendès France, Michel; Shallit, J. O. Wire bending. (English) Zbl 0663.10056 J. Comb. Theory, Ser. A 50, No. 1, 1-23 (1989). The 2-dimensional dragon curves obtained from paper-folding sequences have been studied in several papers [see Davis and Knuth, J. Recreat. Math. 3, 61-81 and 133-149 (1970); F. M. Dekking, M. Mendès France and A. J. van der Poorten, Math. Intell. 4, 130- 138, 173-195 (1985; Zbl 0493.10001, Zbl 0493.10002); M. Mendès France and A. J. van der Poorten, Bull. Austr. Math. Soc. 24, 123- 131 (1981; Zbl 0451.10018)]. In the first of these papers, Davis and Knuth ask whether there are 3-D “dragon curves” which have both aesthetic and interesting properties. The paper under review gives an answer to this question by studying the sequences obtained from bending wires. The authors study these sequences from the point of view of binary expansion of integers, of finite automata and of continued fractions. One very surprising result is that the curves traced out in 3-D are very often bounded curves, which is in sharp contrast to the 2-D case where the curves are not self-intersecting. Let us finally say that the related “handkerchief-folding” sequences, and the “p-paperfolding” sequences have been recently studied, respectively, by O. Salon [Sémin. Théor. Nombres, Univ. Bordeaux I 1986/1987, Exp. No.4 (1987; Zbl 0653.10049)] and by D. Razafy Andriamampianina [Ann. Fac. Sci. Toulouse (to appear); see the following preview Zbl 0663.10057)], and that the paper-folding regular polygons [see for instance J. Froemke and J. W. Grossman, Am. Math. Mon. 95, No.4, 289-307 (1988; Zbl 0651.10002)] seem to have no relationship with the above sequences. Reviewer: J.P.Allouche Cited in 1 ReviewCited in 15 Documents MSC: 11B99 Sequences and sets 11A55 Continued fractions 68Q42 Grammars and rewriting systems 11A99 Elementary number theory Keywords:wire-bending; paper-folding sequences; dragon curves; binary expansion; finite automata; continued fractions Citations:Zbl 0493.10001; Zbl 0493.10002; Zbl 0451.10018; Zbl 0653.10049; Zbl 0663.10057; Zbl 0651.10002 PDFBibTeX XMLCite \textit{M. Mendès France} and \textit{J. O. Shallit}, J. Comb. Theory, Ser. A 50, No. 1, 1--23 (1989; Zbl 0663.10056) Full Text: DOI References: [1] Allouche, J.-P, Automates finis et théorie des nombres, Exposition Math., 5, 239-266 (1987) · Zbl 0641.10041 [2] Blanchard, A.; Mendès France, M., Symétrie et transcendance, Bull. Sci. Math. (2), 106, 325-335 (1982) · Zbl 0492.10027 [3] Christol, G.; Kamae, T.; Mendès France, M.; Rauzy, G., Suites algébriques, automates, et substitutions, Bull. Soc. Math. France, 108, 401-419 (1980) · Zbl 0472.10035 [4] Davis, C.; Knuth, D. E., Number representations and dragon curves I-II, J. Recreational Math., 3, 133-149 (1970) · Zbl 1473.11066 [5] Dekking, M.; Mendès France, M.; van der Poorten, A., FOLDS!, Math. Intell., 4, 173-195 (1982) · Zbl 0493.10002 [6] Gardner, M., Mathematical games, Sci. Amer., 217, 115 (July 1967) [7] Kmošek, M., Rozwiniȩcie niektórych liczb niewymiernych na ułamki łańcuchowe, (Master’s thesis (1979), Uniwersytet Warszawski: Uniwersytet Warszawski Warsaw) [8] Mendès France, M., Principe de la symétrie perturbée, (Séminaire de Théorie des Nombres. Séminaire de Théorie des Nombres, Paris, 1979-1980 (1981), Birkhäuser: Birkhäuser Basel), 77-98 · Zbl 0451.10019 [9] Mendès France, M.; van der Poorten, A., Arithmetic and analytic properties of paperfolding sequences, Bull. Austral. Math. Soc., 24, 123-131 (1981) · Zbl 0451.10018 [10] Mendès France, M.; Tenenbaum, G., Dimensions des courbes planes, papiers pliés et suites de Rudin-Shapiro, Bull. Soc. Math. France, 109, 207-215 (1981) · Zbl 0468.10033 [11] Pólya, G., Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Annal., 84, 149-160 (1921) · JFM 48.0603.01 [12] Shallit, J. O., Simple continued fractions for some irrational numbers, J. Number Theory, 11, 209-217 (1979) · Zbl 0404.10003 [13] Shallit, J. O., Simple continued fractions for some irrational numbers II, J. Number Theory, 14, 228-231 (1982) · Zbl 0481.10005 [14] Shallit, J. O., Explicit descriptions of some continued fractions, Fibonacci Quart., 20, 77-81 (1982) · Zbl 0472.10012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.