Abstract
The original motivation to study this class of geometrical objects of Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the structure of the boundary of 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 [1]. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.
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Gielis, J., Caratelli, D., Tavkhelidze, I. (2020). The General Case of Cutting GML Bodies: The Geometrical Solution. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_31
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