Duke Mathematical Journal

Rigidity and polynomial invariants of convex polytopes

Maksym Fedorchuk and Igor Pak

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of the Robbins conjecture on the degree of generalized Heron polynomials.

Article information

Duke Math. J. Volume 129, Number 2 (2005), 371-404.

First available in Project Euclid: 27 September 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52B10: Three-dimensional polytopes
Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 51M25: Length, area and volume [See also 26B15] 52C25: Rigidity and flexibility of structures [See also 70B15]


Fedorchuk, Maksym; Pak, Igor. Rigidity and polynomial invariants of convex polytopes. Duke Math. J. 129 (2005), no. 2, 371--404. doi:10.1215/S0012-7094-05-12926-X. https://projecteuclid.org/euclid.dmj/1127831442

Export citation


  • A. D. Aleksandrov, Convex Polyhedra, Springer Monogr. Math., Springer, Berlin, 2005. MR 2127379
  • —, Selected Works, Part II: Intrinsic Geometry of Convex Surfaces, Classics Soviet Math., CRC, Boca Raton, Fla., 2005.
  • L. Asimow and B. Roth, The rigidity of graphs, Trans. Amer. Math. Soc. 245 (1978), 279--289. MR 0511410
  • —, The rigidity of graphs, II, J. Math. Anal. Appl. 68 (1979), 171--190. MR 0531431
  • A. V. Astrelin and I. Kh. Sabitov, A canonical polynomial for the volume of a polyhedron, Russian Math. Surveys 54 (1999), 430--431. MR 1711247
  • M. Berger, Géométrie, Vol. 3: Convexes et polytopes, polyèdres réguliers, aires et volumes, CEDIC, Paris, 1977. MR 0536872
  • B. BollobáS, Modern Graph Theory, Grad. Texts in Math. 184, Springer, New York, 1998. MR 1633290
  • Yu. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvature bounded below, Russian Math. Surveys 47 (1992), no. 2, 1--58. MR 1185284
  • P. BüRgisser, M. Clausen, and M. Shokrollahi, Algebraic Complexity Theory, Grundlehren Math. Wiss. 315, Springer, Berlin, 1997. MR 1440179
  • R. Connelly, ``Conjectures and open questions in rigidity'' in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, 407--414. MR 0562634
  • —, The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces, Adv. in Math. 37 (1980), 272--299. MR 0591730
  • —, ``Flexing surfaces'' in The Mathematical Gardner, PWS, Boston, 1981, 79--89.
  • —, ``Rigidity'' in Handbook of Convex Geometry, Vols. A, B, North-Holland, Amsterdam, 1993, 223--271. MR 1242981
  • —, Comments on generalized Heron polynomials and Robbins' conjectures, preprint, 2004, http://www.math.cornell.edu/,\~ connelly
  • R. Connelly, I. Sabitov, and A. Walz, The bellows conjecture, Beiträge Algebra Geom. 38 (1997), 1--10. MR 1447981
  • H. Crapo and W. Whiteley, Statics of frameworks and motions of panel structures, a projective geometric introduction, Structural Topology 1982, 43--82. MR 0666680
  • N. P. Dolbilin, Pearls of Polyhedra Theorey (in Russian), MCCME, Moscow, 2000.
  • H. Edelsbrunner, in Open Problems Presented at SCG '98, http://www.cs.duke.edu/,\~ pankaj/scg98-openprobs
  • H. Gluck, ``Almost all simply connected, closed surfaces are rigid'' in Geometric Topology (Park City, Utah, 1974), Lecture Notes in Math., Springer, Berlin, 1975, 225--239. MR 0400239
  • J. Graver, B. Servatius, and H. Servatius, Combinatorial Rigidity, Grad. Studies in Math. 2, Amer. Math. Soc., Providence, 1993. MR 1251062
  • P. Gritzmann and V. Klee, ``On the complexity of some basic problems in computational convexity, II: Volume and mixed volumes'' in Polytopes: Abstract, Convex and Computational (Scarborough, Ontario, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 440, Kluwer, Dordrecht, 1994, 373--466. MR 1322071
  • J. Hadamard, Leçons de géométrie, II: Géométrie dans l'espace, Jacques Gabay, Sceaux, France, 1988. MR 1189556
  • R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977. MR 0463157
  • D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, New York, 1952. MR 0046650
  • M. Kapovich and J. J. Millson, Universality theorems for configuration spaces of planar linkages, Topology 41 (2002), 1051--1107. MR 1923214
  • A. B. Kempe, How to draw a straight line, Nature 16 (1877), 65--67., 86--89.
  • H. C. King, ``Planar linkages and algebraic sets'' in Proceedings of the 6th Gökova Geometry-Topology Conference, Turkish J. Math. 23, Sci. Tech. Research Council Turkey, Ankara, 1999, 33--56. MR 1701638
  • F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Birkhäuser, Basel, 1993; reprint of 1884 ed. MR 1315530
  • M. Kovalev, Geometric theory of linkages, Russian Acad. Sci. Izv. Math. 44 (1995), 43--68. MR 1271514
  • P. Landers, Dying Mathematician Spends Last Days on Area of Polygon, Wall Street Journal, July 29, 2003, 1.
  • S. Lawrencenko, S. Negami, and I. Kh. Sabitov, A simpler construction of volume polynomials for a polyhedron, Beiträge Algebra Geom. 43 (2002), 261--273. MR 1913784
  • F. M. Maley, D. P. Robbins, and J. Roskies, On the areas of cyclic and semicyclic polygons, Adv. in Appl. Math. 34 (2005), 669--689. MR 2128992
  • J. Martin, Geometry of graph varieties, Trans. Amer. Math. Soc. 355 (2003), 4151--4169. MR 1990580
  • —, On the topology of multigraph picture spaces, Adv. Math. 191 (2005), 312--338. MR 2103216
  • E. Miller, ``Icosahedra constructed from congruent triangles'' in The Branko Grünbaum Birthday Issue, Discrete Comput. Geom. 24, Springer, New York 2000, 437--451. MR 1758062
  • N. MnëV, ``The universality theorems on the classification problem of configuration varieties and convex polytopes varieties'' in Topology and Geometry: Rohlin Seminar, Lecture Notes in Math. 1346, Springer, Berlin, 1988, 527--543. MR 0970093
  • I. Pak, The area of cyclic polygons: Recent progress on Robbins' conjectures, Adv. in Appl. Math. 34 (2005), 690--696. MR 2128993
  • D. P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (1994), 223--236. MR 1283889
  • —, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (1995), 523--530. MR 1336638
  • L. RodríGuez and H. Rosenberg, Rigidity of certain polyhedra in $\mathbbR^3$, Comment. Math. Helv. 75 (2000), 478--503. MR 1793799
  • B. Roth, Rigid and flexible frameworks, Amer. Math. Monthly 88 (1981), 6--21. MR 0619413
  • B. Roth and W. Whiteley, Tensegrity frameworks, Trans. Amer. Math. Soc. 265 (1981), 419--446. MR 0610958
  • I. Kh. Sabitov, The generalized Heron-Tartaglia formula and some of its consequences, Sb. Math. 189 (1998), 1533--1561. MR 1691297
  • —, The volume as a metric invariant of polyhedra, Discrete Comput. Geom. 20 (1998), 405--425. MR 1651896
  • —, Calculation of polyhedra (in Russian), Dokl. Acad. Nauk 377 (2001), 161--164. MR 1833711
  • —, Algorithmic solution of the problem of isometric realization of two-dimensional polyhedral metrics, Izv. Math. 66 (2002), 377--391. MR 1918847
  • —, Volumes of Polyhedra, MCCME, Moscow, 2002.
  • J.-M. Schlenker, La conjecture des sourfflets (d'après I. Sabitov), Astérisque 294 (2004), 77--95. MR 2111640
  • V. V. Varfolomeev, Inscribed polygons and Heron polynomials, Sb. Math. 194 (2003), 311--331. MR 1992154
  • —, Galois groups of Heron-Sabitov polynomials for pentagons inscribed in a circle, Sb. Math. 195 (2004), 3--16. MR 2068949
  • W. Whiteley, ``Matroids and rigid structures'' in Matroid Applications in Encyclopedia Math. Appl. 40, Cambridge Univ. Press, Cambridge, 1992, 1--53. MR 1165538
  • G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, New York, 1995. MR 1311028