Folds!
type: Item
instance of: scholarly article
title: Folds!
zbMATH Open document ID: 0493.10001
DOI: 10.1007/BF03024244
author: Michel Mendès France
author: F. Michel Dekking
author: Alfred J. van der Poorten
published in: The Mathematical Intelligencer
publication date: 1982-01-01T00:00:00Z
review text: 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.
reviewed by: Frits Beukers
Mathematics Subject Classification ID: 11-01
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11B85
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 68Q45
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 3775607
zbMATH Keywords: paperfolding sequences
zbMATH Keywords: automata
zbMATH Keywords: systems of functional equations
zbMATH Keywords: dragon curves
zbMATH Keywords: dimension of plane curves
zbMATH Keywords: Rudin-Shapiro sequences
zbMATH Keywords: Fredholm series
MaRDI profile type: Publication
cites work: Suites algébriques, automates et substitutions
cites work: Uniform tag sequences
cites work: Q5859826
cites work: Uniform distribution modulo one: a geometrical viewpoint.
cites work: Q3241504
cites work: Arithmetic and analytic properties of paper folding sequences
cites work: Dimension des courbes planes, papiers plies et suites de Rudin-Shapiro
cites work: Some Theorems on Fourier Coefficients
full work available at URL: https://doi.org/10.1007/bf03024244
OpenAlex ID: W4242109952
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description: scientific article; zbMATH DE number 3775607