Infinitesimally rigid polyhedra. I. Statics of frameworks
Author:
Walter Whiteley
Journal:
Trans. Amer. Math. Soc. 285 (1984), 431465
MSC:
Primary 52A25; Secondary 51K99, 70C99, 73K99
MathSciNet review:
752486
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Abstract: From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in space, detailed analogues of Alexandrov's theorem are presented for convex polytopes built as bar and joint frameworks in space.
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 A. D. Alexandrov, Konvex polyeder, German transl., AkademieVerlag, Berlin, 1958.
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 , Rigidity of graphs. II, J. Math. Anal. Appl. 68 (1979), 171190. MR 531431 (80i:57004b)
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 , Introduction, Structural Topology 1 (1979), 812.
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 R. Bricard, Memoire sur la theorie de l'octaedre articulé, J. Math (Liouville) (5) 3 (1897), 113148.
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 E. D. Bolker and B. Roth, When is a bipartite graph a rigid framework? Pacifie J. Math. 90 (1981), 2744. MR 599317 (82c:57003)
 [8]
 A. Cauchy, Deuxieme memoire sur les polygons et les polyedres, J. Ecole Polytechnique XVIe Cahier (1831), 8798.
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 , The rigidity of certain cabled frameworks and the second order rigidity of arbitrarily triangulated convex surfaces, Adv. in Math. 37 (1980), 272298. MR 591730 (82a:53059)
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 H. Crapo and W. Whiteley, Statics of frameworks and motions of panel structures: a projective geometric introduction, Structural Topology 6 (1982), 4282. MR 666680 (84b:51029)
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 H. S. M. Coxeter, Regular polytopes, Dover. New York, 1973. MR 0370327 (51:6554)
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 B. Grünbaum, Convex polytopes, Wiley, New York, 1968.
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 , Lectures in lost mathematics, University of Washington, Seattle, Washington, 1976, preprint.
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 H. Gluck, Almost all simply connected closed surfaces are rigid, Geometric Topology, Lectures Notes in Math., vol. 438, SpringerVerlag, Berlin and New York, 1975, pp. 225239. MR 0400239 (53:4074)
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 L. Henneberg, Die Graphische Statik der Starren Systeme, Liepzig, 1911; Johnson reprint, 1968.
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 N. H. Kuiper, Sphères polyedriques flexible dans , d' après Robert Connelly, Sem. Bourbaki 514 (1978), 51410151412. MR 554219 (82a:53060)
 [18]
 J. C. Maxwell, On reciprocal figures and diagrams of forces, Philos. Mag. Ser. (4) 27 (1864), 250261.
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 B. Roth, Rigid and flexible frameworks, Amer. Math. Monthly 88 (1981), 620. MR 619413 (83a:57027)
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 B. Roth and W. Whiteley, Rigidity of tensegrity frameorks, Trans. Amer. Math. Soc. 256 (1981), 419446. MR 610958 (82m:51018)
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 J. J. Stoker, Geometric problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1968), 119168. MR 0222765 (36:5815)
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 N. White and W. Whiteley, The algebraic geometry of stresses in frameworks, J. Algebraic Discrete Methods 4 (1983), 481511. MR 721619 (85f:52024)
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 W. Whiteley, Motions, stresses and projected polyhedra, Structural Topology 7 (1983), 1338. MR 721947 (85h:52010)
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 , Realizability of polyhedra, Structural Topology 1 (1979), 4658. MR 621628 (82j:52016)
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 , Introduction to structural geometry. I. Infinitesimal motions and infinitesimal rigidity, Champlain Reg. Coll., St. Lambert Quebec, preprint 1977.
 [26]
 , Introduction to structural geometry. II. Statics and stresses, Champlain Reg. Coll. St. Lambert, Quebec, preprint 1978.
 [27]
 , Infinitesimally rigid polyhedra. II. Articulated panels (to appear).
 [28]
 , The static and infinitesimal rigidity of sheet structures (to appear).
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DOI:
http://dx.doi.org/10.1090/S00029947198407524866
PII:
S 00029947(1984)07524866
Article copyright:
© Copyright 1984
American Mathematical Society
