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Möbius bands, unstretchable material sheets and developable surfaces.

Yi-Chao Chen1, Eliot Fried2

  • 1Department of Mechanical Engineering , University of Houston , Houston, TX 77204-4006, USA.

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|September 13, 2016
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Summary
This summary is machine-generated.

This study clarifies that pure bending into a Möbius band requires more than just developable surfaces. It corrects the misconception that rectifying developables accurately model this process, highlighting the importance of unstretchability.

Keywords:
bending elasticityinextensible space curvesisometric mappingsruled surfaces

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Area of Science:

  • Geometry
  • Material Science
  • Mathematical Physics

Background:

  • Möbius bands are formed by twisting and joining rectangular sheets, often modeled by isometric mappings to developable surfaces.
  • Previous attempts to find the equilibrium shape of Möbius bands focused on minimizing bending energy within developable surfaces.

Purpose of the Study:

  • To analyze the relationship between isometric mappings and developable surfaces in the context of Möbius band formation.
  • To correct the erroneous use of rectifying developables in modeling the pure bending of rectangular regions into Möbius bands or ribbons.

Main Methods:

  • Investigating the properties of mappings from a planar region to developable surfaces.
  • Analyzing the conditions under which a mapping to a rectifying developable surface is isometric.
  • Comparing the geometric constraints of developability with the physical constraint of unstretchability.

Main Results:

  • A surface from an isometric mapping is developable, but a mapping to a developable surface is not necessarily isometric.
  • The notion of a rectifying developable is not suitable for describing the pure bending of a rectangular region into a Möbius band.
  • Isometric mapping to a rectifying developable surface is only valid for cylindrical surfaces with the midline as the generator.

Conclusions:

  • The study highlights a critical distinction between physical unstretchability and geometric developability in modeling curved surfaces.
  • It proposes alternative modeling strategies that accurately account for the constraints of pure bending and material properties.
  • This work provides a more rigorous mathematical framework for understanding the formation and shape of Möbius bands and similar structures.