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Wire bending. (English) Zbl 0663.10056
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

MSC:
 11B99 Sequences and sets 11A55 Continued fractions 68Q42 Grammars and rewriting systems 11A99 Elementary number theory
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References:
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