Translation of W. Wunderlich’s “On a Developable Möbius Band”
Source abstract
Abstract
The following is a translation of Walter Wunderlich’s article “Uber ein abwickelbares Mobiusband”, which appeared in the Monatshefte fur Mathematik66 (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 fur Technische Mechanik, II (Stockholm, 1930), pp. 444–451, Sveriges Litografiska Tryckerier, Stockholm, 1931) on developable Mobius bands and improves Sadowsky’s upper bound of the dimensionally-reduced variational description for determining the configuration of a Mobius 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 Mobius 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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Research context
OpenAlex topics: Structural Analysis and Optimization · Engineering and Materials Science Studies · Structural Analysis of Composite Materials
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References (8)
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1. Möbius, F.A.: Über die Bestimmung des Inhaltes eines Polyeders (On the determination of the volume of a polyhedron). Ber. Verh. Sächs. Ges. Wiss. 17, 31–68 (1865); Gesammelte Werke, Band II (Collected Works, vol. II), p. 484. Hirzel, Leipzig (1886)
2. Maschke, H.: Note on the unilateral surface of Moebius. Trans. Am. Math. Soc. 1, 39 (1900)10.1090/S0002-9947-1900-1500522-7 ↗
3. Scheffers, G.: Einführung in die Theorie der Flächen (2. Aufl.) (Introduction to the Theory of Surfaces, 2nd edn.), pp. 41–43. Veit, Leipzig (1913)
4. Weyl, H.: Die Idee der Riemannschen Fläche (The Concept of the Riemann Surface), p. 26. Teubner, Leipzig (1913)
5. 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)
6. 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)10.21136/CPMF.1936.109339 ↗
7. 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
8. 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