Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-30T18:56:00.074Z Has data issue: false hasContentIssue false
  • English
  • Français

Arithmetic and analytic properties of paper folding sequences

Published online by Cambridge University Press:  17 April 2009

M. Mendès France  and
A.J. van der Poorten
M. Mendès France
Affiliation:
UER de Mathématiques et d'Informatique, Université de Bordeaux I, 351 Cours de la Libération, 33405 Talence – Cédex, France;
A.J. van der Poorten
Affiliation:
School of Mathematics and Physics, Macquarie University, North Ryde, New South Wales 2113, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The mechanical procedure of paper folding generates an uncountable family of infinite sequences of fold patterns. We obtain the associated Fourier series and show that the sequences are almost periodic and hence deterministic. Further, we show that paper folding numbers defined by the sequences are all transcendental.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 1 , August 1981 , pp. 123 - 131
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Carlson, Fritz, “Über Potenzreihen mit ganzzahligen Koeffizienten”, Math. Z. 9 (1921), 1–13.Google Scholar
[2]Davis, Chandler, Knuth, Donald E., “Number representations and dragon curves – I”, J. Recreational Math. 3 (1970), 61–81; “Number representations and dragon curves – II”, J. Recreational Math. 3 (1970), 133–149.Google Scholar
[3]Dekking, F.M., “Constructies voor o-l-rijen met strikt ergodische afgesloten baan”, (Doctoraalscriptie, 1974).Google Scholar
[4]Kamae, Teturo, “Subsequences of normal sequences”, Israel J. Math. 16 (1973), 121–149.Google Scholar
[5]Loxton, J.H. and Poorten, A.J. van der, “Arithmetic properties of certain functions in several variables III”, Bull. Austral. Math. Soc. 16 (1977), 15–47.Google Scholar
[6]Loxton, J.H. and Poorten, A.J. van der, “Transcendence and algebraic independence by a method of Mahler”, Transcendence theory: advances and applications, 211–226 (Proc. Conf. Univ. Cambridge, Cambridge, 1976. Academic Press, London, 1977).Google Scholar
[7]Mahler, Kurt, “On the translation properties of a simple class of arithmetical functions”, J. Math. Phys. 6 (1972), 158–163.Google Scholar
[8]Mahler, Kurt, “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann. 101 (1929), 342–366.Google Scholar
[9]Mendès-France, M., “Principe de la symétrie perturbée”, Séminaire de Théorie des Nombres, Paris 1979–80, 77–98 (Séminaire DelangePisot-Poitou. Birkhäuser, Boston, Basel, Stuttgart, 1981).Google Scholar
[10]France, M. Mendès et Tenenbaum, G., “Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro”, Bull. Soc. Math. France (to appear).Google Scholar
[11]Rauzy, G., “Nombres normaux et processus déterministes”, Acta Arith. 29 (1976), 211–225.Google Scholar
[12]Wiener, Norbert, “Generalized harmonic analysis”, Acta Math. 55 (1930), 117–258.CrossRefGoogle Scholar