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

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.

The Möbius Phenomenon in Generalized Möbius-Listing Bodies with Cross Sections of Odd and Even Polygons

“…In the notation GML , n m m relates to the symmetry of the cross section, and n to the number of twists relative to m. The choice of regular polygons and of straight knives can be generalized to any convex or concave m-symmetrical cross section. For the functions R(q ) and p(t, y) (path and cross section of the GML , n m respectively) Gielis transformations defined by (4) can be used [2][3][4][5][6][7][8]. They provide for a unifying description for a wide range of natural and abstract shapes, including regular polygons [9,10] ( ; , , , ,…”

“…In case the cylinder is a strip and it is given a half twist (180°) before joining, the classic Möbius band results, a GML n surfaces based on a strip, have been classified in full generality for any integer value of n (for all multiples of n or 180°), for all cutting lines or strips (containing the basic line or not) of and for any number of cuttings [1]. Results have been reported for cutting GML with particular symmetries [2][3][4][5][6][7] and the general case was solved in Gielis and Tavkhelidze [8].…”

“…Now, the d m knife is a line of infinite length (1), rotated m times or a divisor thereof. with a the rotation parameter, and d the translation parameter of this infinite line [8,11,12]. Indeed, with the analytical definition of the geometrical d m knife [11] all possible cuttings of regular polygons can be studied with the number of blades on the knife either m or a divisor of m. This d m knife with m blades cuts from vertex to vertex, vertex to side or side to side, with each blade cutting exactly two points on the boundary of the polygon.…”

“…The Annex in Gielis and Tavkhelidze [8] shows all possible cuts for m = 6,…,10. These methods are intimately linked to combinatorial problems, for example the Euler problems of drawing nondissecting diagonals in polygons.…”

“…The number m has to be even (m = 2, 4, 6, …). The Möbius phenomenon never occurs when the polygon has an odd number of vertices and sides [6][7][8].…”