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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Notes
The proportionality factor has the value Eh 3/24, where E is the modulus of elasticity and h the thickness.
In the full interval 0≤s≤L, |V(s)|>b must be maintained to keep the singularities of depicted points of regression outside of the rectangular region.
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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Maschke, H.: Note on the unilateral surface of Moebius. Trans. Am. Math. Soc. 1, 39 (1900)
Scheffers, G.: Einführung in die Theorie der Flächen (2. Aufl.) (Introduction to the Theory of Surfaces, 2nd edn.), pp. 41–43. Veit, Leipzig (1913)
Weyl, H.: Die Idee der Riemannschen Fläche (The Concept of the Riemann Surface), p. 26. Teubner, Leipzig (1913)
Krames, J.: Die Regelfläche dritter Ordnung, deren Striktionslinie eine Ellipse ist (The ruled surface of order 3 whose line of striction is an ellipse). Sitzber. Akad. Wiss. Wien 127, 563–568 (1918) (Wunderlich comment: The strange and distinguishing feature determined therein is that the line of striction, which in general is of eighth order for cubic ruled surfaces, reduces to an ellipse in the presented special case by means of splitting of minimal generators)
Klíma, J.: O zborcené ploše, jejíž část je topologicky ekvivalentní s Möbiovým listem (On a skew surface, part of which is topologically equivalent to the Möbius band). Čas. Pěst. Mat. Fys. 65(4), 211–216 (1936)
Sadowsky, M.: Ein elementarer Beweis für die Existenz eines abwickelbaren Möbiusschen Bandes und Zurückfürhing des geometrischen Problems auf ein Variationsproblem (An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem). Sitzber. Preuss. Akad. Wiss. 22, 412–415 (1930). See English translation by Hinz and Fried in this issue
Sadowsky, M.: Theorie der elastisch biegsamen undehnbaren Bänder mit Anwendungen auf das Möbius’sche Band (Theory of elastically bendable inextensible bands with applications to the Möbius band). In: Oseen, C.W., Weibull, W. (Hrsg.) Verhandlungen des 3. Internationalen Kongresses für Technische Mechanik, Teil II: Elastizität, Plastizität, Festigkeit, Ballistik und rationelle Mechanik, Stockholm, 24–29 August 1930, S. 444–451 (Oseen, C.W., Weibull, W. (Eds.), Proceedings of the 3rd International Congress for Applied Mechanics, Part II: Elasticity, Plasticity, Strength, Ballistics, and Rational Mechanics, Stockholm, 24–29 August 1930, pp. 444–451. Sveriges Litografiska Tryckerier, Stockholm, 1931). See English translation by Hinz and Fried in this issue
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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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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DOI: https://doi.org/10.1007/s10659-014-9489-y