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      The general case of cutting of Generalized Möbius-Listing surfaces and bodies

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          Abstract

          The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.

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          Most cited references 23

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          A generic geometric transformation that unifies a wide range of natural and abstract shapes.

           Johan Gielis (2003)
          To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.
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            The shape of a Möbius strip.

            The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 degrees, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1,2. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This could give new insight into energy localization phenomena in unstretchable sheets, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures.
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              Semi-analytic geometry with R-functions

               Vadim Shapiro (2007)
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                Author and article information

                Journal
                fopen
                https://www.4open-sciences.org
                4open
                4open
                EDP Sciences
                2557-0250
                31 August 2020
                31 August 2020
                2020
                : 3
                : ( publisher-idID: fopen/2020/01 )
                Affiliations
                [1 ] University of Antwerp, Department of Bioscience Engineering, , 2020 Antwerpen, Belgium,
                [2 ] Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, , 0179 Tbilisi, Georgia,
                Author notes
                Article
                fopen200013
                10.1051/fopen/2020007
                © J. Gielis and I. Tavkhelidze, Published by EDP Sciences, 2020

                This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

                Page count
                Figures: 52, Tables: 14, Equations: 184, References: 38, Pages: 48
                Product
                Self URI (journal page): https://www.4open-sciences.org/
                Categories
                Research Article
                Mathematics - Applied Mathematics
                Custom metadata
                4open 2020, 3, 7
                2020
                2020
                2020
                yes

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