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Translation of W. Wunderlich’s “On a Developable Möbius Band”

An Erratum to this article was published on 21 May 2015

Abstract

The following is a translation of Walter Wunderlich’s article “Über ein abwickelbares Möbiusband”, which appeared in the Monatshefte für Mathematik 66 (1962), 276–289 and was dedicated to Prof. Dr. Paul Funk on the occasion of his 75th birthday. Wunderlich summarizes Sadowsky’s work (Sitzber. Preuss. Akad. Wiss. 22:412–415, 1930; Verhandlungen des 3. Internationalen Kongresses für Technische Mechanik, II (Stockholm, 1930), pp. 444–451, Sveriges Litografiska Tryckerier, Stockholm, 1931) on developable Möbius bands and improves Sadowsky’s upper bound of the dimensionally-reduced variational description for determining the configuration of a Möbius band whose width is small in comparison to its length. Attempting to reproduce the equilibrium depiction of a band of finite width, using a rational-algebraic developable, Wunderlich then extends Sadowsky’s results by presenting perhaps the first successful model of a closed, analytic, developable Möbius band with associated thinness bounds. This translation makes Wunderlich’s work accessible to the broader research community at a time of growing interest in and relevance of thin-walled structural elements.

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Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. The proportionality factor has the value Eh 3/24, where E is the modulus of elasticity and h the thickness.

  2. In the full interval 0≤sL, |V(s)|>b must be maintained to keep the singularities of depicted points of regression outside of the rectangular region.

  3. According to (22), (24), and (26), the loci of U,V,W are—also in the central elevation—rational curves of respective order 10, 11, and 21.

  4. The highlighted generators of the developable on the cardboard model were determined as the shadow boundaries in the cone of light of a projection device.

References

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Acknowledgements

I thank Eliot Fried for his suggestion to translate this important work as well as his considerable editing and technical help with the manuscript. Michael Ban and Denis Hinz also provided valuable linguistic clarifications.

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Correspondence to Russell E. Todres.

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Translator’s Notes

i. Wunderlich uses the term torse, which has herein been replaced by the more common and synonymous developable.

ii. Edge of regression (cuspidal line) is used for the German Gratlinie.

iii. The use of ϰ 0 in (5) in the original instead of \(\bar{\varkappa}_{0}\) appears to be an error.

iv. For clarity, periods are used here instead of the commas Wunderlich used to denote decimals.

v. Synonymous with ideal point.

vi. The order of Figs. 5 and 6 is herein reversed from the original for clarity.

vii. The use of \(\ddot{X}_{i}\) in the original instead of \(\ddot{X}\) appears to be an error.

viii. Imaginary circle was chosen for the German nullteiliger Kreis.

Superscripted Arabic numbers refer to footnotes in the original, while superscripted lowercase Roman numerals are used for translator’s notes appearing at the end. References are those which appear in the original paper, available at: https://eudml.org/doc/177173.

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Todres, R.E. Translation of W. Wunderlich’s “On a Developable Möbius Band”. J Elast 119, 23–34 (2015). https://doi.org/10.1007/s10659-014-9489-y

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Keywords

  • Möbius bands
  • Differential geometry
  • Developable surfaces

Mathematics Subject Classification

  • 53A04
  • 74G10
  • 74G55
  • 74K10
  • 74K20
  • 01A75