Abstract

Symmetry: Culture and Science
Volume 34, Number 1, pages 111-112 (2023)
https://doi.org/10.26830/symmetry_2023_1_111

CUTTING OF GENERALIZED MÖBIUS-LISTING SURFACES AND BODIES

Johan Gielis

Email: johan.gielis@gmail.com

The Figurae Mathematicae, as designer Albert Kiefer called them, illustrate all possible ways of cutting Generalized Möbius-Listing GML surfaces and bodies with a twists. This class of surfaces and bodies (with Möbius band and cylinder as special cases) has been introduced by Ilia Tavkhelidze (Tbilisi, Georgia) in 1997.Cutting can be done from vertex-to-vertex VV, vertex-to-side VS, and side-to-side SS and can result in complex configurations. Fortunately, the general problem could be reduced to the study of planar shapes, in particular regular polygons. The complete solution is related to many other studies of planar geometry by Euler, Lamé, Segner, Catalan and Cayley. Direct relations unveiled with Catalan numbers, the entries A006561 and A000203 in the Online Encyclopedia of Integers (OEIS.org), partial Bell polynomials and so forth.

References:
Tavkhelidze, I., Cassisa, C., Gielis, J., & Ricci, P. E. (2013) About" Bulky" Links, generated by Generalized Möbius Listing's bodies GMLN3, Rendiconti Lincei, 24(1), 11-38. https://doi.org/10.4171/RLM/643

Tavkhelidze, I., & Gielis, J. (2018) The process of cutting bodies with d m knives, In Reports of the Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, 32.

Gielis, J., & Tavkhelidze, I. (2020) The general case of cutting of Generalized Möbius-Listing surfaces and bodies, 4Open, 3, 7. https://doi.org/10.1051/fopen/2020007