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Construction of Unknotted and Knotted Symmetric Developable Bands

Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
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Academic Editor: Raffaele Barretta
Symmetry 2021, 13(3), 431; https://doi.org/10.3390/sym13030431
Received: 17 February 2021 / Revised: 1 March 2021 / Accepted: 3 March 2021 / Published: 7 March 2021
We describe a method for constructing developable bands with N ≥ 3 half twists. Each band is formed by threading a flat rectangular strip through a scaffold made from identical circular cylinders and smoothly connecting its short ends. The N cylinders in a scaffold are arranged with N-fold rotational symmetry. The number of half twists in a band is equal to the number N of cylinders in its scaffold and each band inherits the symmetry of its scaffold. Each scaffold admits a family of bands of the same length but variable width up to a maximum value determined by the features of the scaffold. Apart from orientable and nonorientable unknots, our method allows for the construction of bands with the topology of torus knots. We detail the geometric properties of the construction, discuss certain fundamental restrictions that must be met to ensure constructability, and calculate the elastic bending energy of each band. The rotational symmetry underlying the construction is essential for obtaining the presented bands, as the general non-symmetric problem is even more complex and has not yet been investigated. The bands and their corresponding scaffolds can be used as structural elements in practical applications, one of which we describe and analyze. The construction serves as a basis for a general framework for building a large variety of scaffolds and the corresponding unstretchable bands. Together, these assemblies can be used in architectural, interior, and machine design. They also open new avenues for the layout of conveyor belts in factories, airports, and other settings. View Full-Text
Keywords: structural scaffolds; orientable and nonorientable surfaces; torus knots; interior design; pure bendings; elastic bending energy structural scaffolds; orientable and nonorientable surfaces; torus knots; interior design; pure bendings; elastic bending energy
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MDPI and ACS Style

Schönke, J.; Grunwald, M.; Fried, E. Construction of Unknotted and Knotted Symmetric Developable Bands. Symmetry 2021, 13, 431. https://doi.org/10.3390/sym13030431

AMA Style

Schönke J, Grunwald M, Fried E. Construction of Unknotted and Knotted Symmetric Developable Bands. Symmetry. 2021; 13(3):431. https://doi.org/10.3390/sym13030431

Chicago/Turabian Style

Schönke, Johannes, Michael Grunwald, and Eliot Fried. 2021. "Construction of Unknotted and Knotted Symmetric Developable Bands" Symmetry 13, no. 3: 431. https://doi.org/10.3390/sym13030431

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