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Folds! (English) Zbl 0493.10001

In this series of three papers (see Zbl 0493.10002, Zbl 0493.10003) the authors survey a varied collection of topics which are all related to the so-called paper-folding sequences. Such sequences arise from repeatedly folding a sheet of paper, unfolding it again and considering the sequence of “upward” and “downward” bends. They have a number of highly interesting properties. For example, plane-filling curves can be constructed from them. They can also be used to construct sequences of integers \(u(h)\) satisfying \[ \sup_{0\leq\theta\leq 2\pi}|\sum_0^{n-1} (1)^{u(h)}e^{ih\theta}|\leq (2+\sqrt 2)\sqrt n, \] the lower bound \(\sqrt n\) being trivial. Some alternative ways of generating related sequences are generation by automatons and by symmetry operations. For example, \(\sum g_hX^h\) is algebraic over \(\mathbb F_p[X]\) if and only if the sequence \(g_h\) can be generated by a so-called \(p\)-automaton. Moreover, \(\sum g_hp^{-h}\) is a transcendental number in that case. Furthermore, the continued fraction of the Fredholm series \(g^{-2^h}\) can be given by a sequence generated by symmetry operations.
By generalization of the paperfolding idea, one can construct bizarre, plane-filling curves, which, drawn on a piece of paper yield intricate patterns that arouse ones fantasy. In all, the paper contains much information, is written in an entertaining form and worth wile reading.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11B83 Special sequences and polynomials
11B85 Automata sequences
05B30 Other designs, configurations
68Q45 Formal languages and automata
11J81 Transcendence (general theory)
11A55 Continued fractions
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References:

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