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The Dehn invariants of the Bricard octahedra

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Abstract

We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.

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References

  1. Alexander R.: Lipschitzian mappings and total mean curvature of polyhedral surfaces. I. Trans. Amer. Math. Soc 288, 661–678 (1985)

    Article  MATH  Google Scholar 

  2. Berger M.: Geometry. I, II. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  3. Boltyanskii V.G.: Hilbert’s third problem. Winston, Washington D.C. (1978)

    MATH  Google Scholar 

  4. Bricard, R.: Mémoire sur la théorie de l’octaèdre articulé. J. de Math. (5) 3, 113–148 (1897)

    Google Scholar 

  5. Connelly R.: A counterexample to the rigidity conjecture for polyhedra. Publ. Math. IHES 47, 333–338 (1977)

    MathSciNet  MATH  Google Scholar 

  6. Connelly R.: The rigidity of polyhedral surfaces. Math. Mag 52, 275–283 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Connelly R., Sabitov I., Walz A.: The bellows conjecture. Beitr. Algebra Geom 38, 1–10 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Dehn, M.: Über raumgleiche Polyeder. Gött. Nachr. 345–354 (1900)

  9. Gluck H.: Almost all simply connected closed surfaces are rigid. In: Glaser, L.C., Rushing, T.B. (eds) Lect. Notes Math 438., pp. 225–239. Springer, Berlin (1975)

    Google Scholar 

  10. Jessen B.: The algebra of polyhedra and the Dehn—Sydler theorem. Math. Scand 22, 241–256 (1968)

    MathSciNet  MATH  Google Scholar 

  11. Kuiper, N.H.: Sphères polyédriques flexibles dans E 3, d’après Robert Connelly. In: Séminaire Bourbaki, vol. 1977/78, Exposés 507–524, Lect. Notes Math. 710, pp. 147–168. Springer, Berlin (1979)

  12. Lebesgue H.: Octaèdres articulés de Bricard. Enseign. Math. II. Sér 13, 175–185 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Maksimov I.G.: Nonflexible polyhedra with a small number of vertices. J. Math. Sci., New York 149, 956–970 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sabitov, I.Kh.: Local theory of bendings of surfaces. In: Geometry III, Theory of surfaces, Encycl. Math. Sci. 48, pp. 179–250. Springer, Berlin (1992)

  15. Sabitov I.Kh.: The volume of a polyhedron as a function of its metric (in Russian). Fundam. Prikl. Mat 2, 1235–1246 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Sabitov I.Kh.: The volume as a metric invariant of polyhedra. Discrete Comput. Geom 20, 405–425 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sabitov, I.Kh.: The volumes of polyhedra (in Russian). Moscow Center for Continuous Mathematical Education, Moscow (2002)

  18. Schlenker, J.-M.: La conjecture des soufflets (d’après I. Sabitov). In: Seminaire Bourbaki, vol. 2002/03, Exposes 909–923. Société Mathématique de France, Paris. Astérisque 294, 77–95, Exp. No. 912 (2004)

  19. Smart W.M.: Text-book on spherical astronomy, 6th edn. Cambridge University Press, Cambridge (1960)

    Google Scholar 

  20. Stachel H. et al.: Flexible octahedra in the hyperbolic space. In: Prékopa, A (eds) Non-Euclidean geometries, János Bolyai memorial volume., pp. 209–225. Springer, New York (2006)

    Google Scholar 

  21. Sydler J.P.: Conditions necessaires et suffisantes pour l’équivalence des polyedres de l’espace euclidien à trois dimensions. Comment. Math. Helv 40, 43–80 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Victor Alexandrov.

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The author is supported in part by the Russian Foundation for Basic Research (Grant 10–01–91000–ANF), the Federal Program ‘Research and educational resourses of innovative Russia in 2009–2013’ (contract 02.740.11.0457), and the Russian State Program for Leading Scientific Schools (Grant NSh-6613.2010.1).

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Alexandrov, V. The Dehn invariants of the Bricard octahedra. J. Geom. 99, 1–13 (2010). https://doi.org/10.1007/s00022-011-0061-7

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  • DOI: https://doi.org/10.1007/s00022-011-0061-7

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