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Infinitesimally rigid polyhedra. II. Modified spherical frameworks


Author: Walter Whiteley
Journal: Trans. Amer. Math. Soc. 306 (1988), 115-139
MSC: Primary 52A25; Secondary 51K99, 70C99, 73K99
DOI: https://doi.org/10.1090/S0002-9947-1988-0927685-4
MathSciNet review: 927685
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Abstract | References | Similar Articles | Additional Information

Abstract: In the first paper, Alexandrov's Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz's Theorem, and a related coordinatization of the proper realizations of a $ 3$-connected spherical polyhedron. Sample results show that: (a) a single $ k$-gonal block and a $ k$-gonal hole yield static rigidity if and only if the block and hole are $ k$-connected in a vertex sense; and (b) a $ 4$-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.


References [Enhancements On Off] (What's this?)

  • [1] A. D. Alexandrov, Konvex polyeder, German translation, Akademie-Verlag, Berlin 1958.
  • [2] L. Asimov and B. Roth, The rigidity of graphs, Trans. Amer. Math. Soc. 245 (1978), 279-289. MR 511410 (80i:57004a)
  • [3] D. Barnette, A proof of the lower bound conjecture, Pacific J. Math. 46 (1973), 349-354. MR 0328773 (48:7115)
  • [4] D. Barnette and B. Grünbaum, On Steinitz's theorem concerning $ 3$-polytopes and some properties of planar graphs, The Many Facets of Graph Theory, Lecture Notes in Math., vol. 110, Springer-Verlag, Berlin and New York, 1969, pp. 27-40.
  • [5] R. Connelly, Rigidity of polyhedral surfaces, Math. Mag. 52 (1979), 275-283. MR 551682 (80k:53089)
  • [6] -, The rigidity of certain cabled framewororks and the second order rigidity of arbitrarily triangulated convex surfaces, Adv. in Math. 37 (1980), 272-298. MR 591730 (82a:53059)
  • [7] H. Crapo and W. Whiteley, Statics of frameworks and motions of panel structures: a projective geometric introduction, Structural Topology 6 (1982), 43-82. MR 666680 (84b:51029)
  • [8] G. A. Dirac, Extensions of Menger's theorem, J. London Math. Soc. 38 (1963), 146-163. MR 0151958 (27:1939)
  • [9] H. Gluck, Almost all simply connected surfaces are rigid, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York, 1975, pp. 225-239. MR 0400239 (53:4074)
  • [10] J. Graver, A combinatorial approach to infinitesimal rigidity, preprint, Syracuse Univ., Syracuse, New York, 1984.
  • [11] B. Grünbaum, Convex polytopes, Wiley, New York, 1968.
  • [12] -, Lectures in lost mathematics, mimeograph notes, Univ. of Washington, Seattle, Washington, 1976.
  • [13] G. Kalai, Rigidity and the lower bound theorem. I, Invent. Math. 88 (1987), 125-151. MR 877009 (88b:52014)
  • [14] N. Kuiper, Spheres polyedriques flexible dans $ {E^3}$, d'apres Robert Connelly, Séminaire Bourbaki 1977/78, Lecture Notes in Math., vol. 710, Springer-Verlag, Berlin and New York, 1979, pp. 147-168. MR 554219 (82a:53060)
  • [15] G. Laman, On graphs and the rigidity of plane skeletal structures, J. Engineering Math. 4 (1970), 331-340. MR 0269535 (42:4430)
  • [16] L. A. Lyusternik, Convex figures and polyhedra (Russian), Moscow, 1956; English transl., Dover, New York, 1963. MR 0161219 (28:4427)
  • [17] B. Roth and W. Whiteley, Tensegrity frameworks, Trans. Amer. Math. Soc. 265 (1981), 419-446. MR 610958 (82m:51018)
  • [18] G. T. Salee, Incidence graphs of convex polytopes, J. Combin. Theory 2 (1967), 466-506. MR 0216364 (35:7198)
  • [19] E. Steinitz and H. Rademacher, Vorlesungen über die Theorie der Polyeder, Springer-Verlag, Berlin and New York, 1934.
  • [20] T. Tarnoi, Simultaneous static and kinematic indeterminacy of space trusses with cyclic symmetry, Internat. J. Solids Struct. 16 (1980), 345-356.
  • [21] T-S. Tay and W. Whiteley, Recent advances in the generic rigidity of frameworks, Structural Topology 9 31-38. MR 759309 (86i:51020)
  • [22] -, Generating isostatic frameworks, Structural Topology 11 (1985), 21-69. MR 804977 (87e:05139)
  • [23] N. White and W. Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Algebraic Discrete Method 4 (1983), 481-511. MR 721619 (85f:52024)
  • [24] W. Whiteley, Introduction to structural geometry. I, II, notes, Champlain Regional College, Quebec, Canada, 1976.
  • [25] -, Infinitesimally rigid polyhedra. I: Statics of frameworks, Trans. Amer. Math. Soc. 285 (1984), 431-465. MR 752486 (86c:52010)
  • [26] -, Realizability of polyhedra, Structural Topology 1 (1979), 46-58. MR 621628 (82j:52016)
  • [27] -, Rigidity and polarity. I: Statics of sheetworks, Geom. Dedicata 22 (1987), 329-362. MR 887581 (88h:51020)
  • [28] -, Determination of spherical polyhedra (to appear).
  • [29] -, Vertex splitting in isostatic frameworks, Structural Topology (to appear). MR 1102001 (92c:52037)
  • [30] H. Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339-362. MR 1501641
  • [31] -, $ 2$-isomorphic graphs, Amer. J. Math. 55 (1933), 245-254. MR 1506961

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0927685-4
Keywords: Infinitesimal rigidity, static rigidity, generic rigidity, polyhedral framework, $ 4$-connected graph, Steinitz's Theorem
Article copyright: © Copyright 1988 American Mathematical Society

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