Growth and Form

Volume 2, Issue 1, March 2021, Pages 1 - 10

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

Authors
J. Gielis1, *, I. Tavkhelidze2
1Department of Biosciences Engineering, University of Antwerp, Belgium
2Faculty of Exact and Natural Sciences, Tbilisi State University, University Street 13, Tbilisi 0186, Georgia
*Corresponding author. Email: johan.gielis@uantwerpen.be
Corresponding Author
J. Gielis
Received 1 December 2020, Accepted 6 December 2020, Available Online 5 January 2021.
DOI
https://doi.org/10.2991/gaf.k.201210.001How to use a DOI?
Keywords
Generalized Möbius-Listing bodies and surfaces, Möbius phenomenon, regular polygons, Gielis transformations
Abstract

In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, it is well known that the Möbius phenomenon, whereby the cutting process yields only one body, occurs only in even polygons with an even number of vertices and sides, and only in the specific when the knife cuts through the center of the polygon. This knife cuts from vertex to vertex, vertex to side or side to side, cutting exactly two points on the boundary of the polygon. This is called a chordal knife, in connection to the chord cutting a circle. If the knife is a radial knife, i.e. it cuts only one point of the boundary, the Möbius phenomenon can occur both in odd and even polygons, but only when the radial knife cuts the center of the polygon. One finding is the reduction of a problem in 3D (with internal geometry) to a planar problem and the concomitant reduction of the analytic representation with multiple parameters to a few only. The shape of the cross section and number of twisting in the 3D representation suffice and reduce the problem to cutting of regular polygons and cyclic permutations.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Growth and Form
Volume-Issue
2 - 1
Pages
1 - 10
Publication Date
2021/01
ISSN (Online)
2589-8426
DOI
https://doi.org/10.2991/gaf.k.201210.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - J. Gielis
AU  - I. Tavkhelidze
PY  - 2021
DA  - 2021/01
TI  - The Möbius Phenomenon in Generalized Möbius-Listing Bodies with Cross Sections of Odd and Even Polygons
JO  - Growth and Form
SP  - 1
EP  - 10
VL  - 2
IS  - 1
SN  - 2589-8426
UR  - https://doi.org/10.2991/gaf.k.201210.001
DO  - https://doi.org/10.2991/gaf.k.201210.001
ID  - Gielis2021
ER  -