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December 2021 The Möbius phenomenon in Generalized Möbius-Listing surfaces and bodies, and Arnold's Cat phenomenon
Johan Gielis, Paolo Emilio Ricci, Ilia Tavkhelidze
Adv. Studies: Euro-Tbilisi Math. J. 14(4): 17-35 (December 2021). DOI: 10.3251/asetmj/1932200812

Abstract

Möbius bands have been studied extensively, mainly in topology. Generalized Möbius-Listing surfaces and bodies providing a full geometrical generalization, is a quite new field, motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems. However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably. Our research is motivated by this reduction of complexity. In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, the conditions under which a single body results, displaying the Möbius phenomenon of a one-sided body, have been determined for even and odd polygons. These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin. The Möbius phenomenon is important, since the process of cutting (or separation of zones in a GML body in general) then results in a single body, not in different, intertwined domains. In all previous works it was assumed that the cross section of the GML bodies is constant, but the main result of this paper is that it is sufficient that only one cross section on the whole GML structure meets the conditions for the Möbius phenomenon to occur. Several examples are given to illustrate this.

Citation

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Johan Gielis. Paolo Emilio Ricci. Ilia Tavkhelidze. "The Möbius phenomenon in Generalized Möbius-Listing surfaces and bodies, and Arnold's Cat phenomenon." Adv. Studies: Euro-Tbilisi Math. J. 14 (4) 17 - 35, December 2021. https://doi.org/10.3251/asetmj/1932200812

Information

Received: 29 September 2020; Accepted: 20 March 2021; Published: December 2021
First available in Project Euclid: 16 December 2021

Digital Object Identifier: 10.3251/asetmj/1932200812

Subjects:
Primary: 51B10
Secondary: 53A05 , 57M25

Keywords: Generalized Möbius-Listing bodies and surfaces , Gielis transformations , Möbius phenomenon , Poincaré's recurrence theorem , regular polygons

Rights: Copyright © 2021 Tbilisi Centre for Mathematical Sciences

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Vol.14 • No. 4 • December 2021
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