Skip to main content
SpringerLink
Log in

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies, these are non-degenerate maxima, minima, and saddle points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work, it was shown that these classifications are complete in that no class is empty. Here, we construct 1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes and show that transitions between them can be realized by codimension 1 saddle-node and saddle–saddle (heteroclinic) bifurcations in the gradient vector fields. Our results indicate that all combinatorially possible transitions can be realized in physical shape evolution processes, e.g., by abrasion of sedimentary particles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Archdeacon, D., Hutchinson, J., Nakamoto, A., Negami, S., Ota, K.: Chromatic numbers of quadrangulations on closed surfaces. J. Graph Theory 37, 100–114 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold, V.: Ordinary Differential Equations. MIT Press, Cambridge (1998)

    Google Scholar 

  • Bloore, F.J.: The shape of pebbles. Math. Geol. 9, 113–122 (1977)

    Article  MathSciNet  Google Scholar 

  • Brinkmann, G., Greenberg, S., Greenhill, C., McKay, B., Thomas, R., Wollan, P.: Generation of simple quadrangulations of the sphere. Discrete Math. 305, 22–54 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Cantarella, J.: Knot Probabilities in Random Diagrams. arXiv preprint. arXiv:1512.05749 (2015)

  • Domokos, G.: Monotonicity of spatial critical points evolving under curvature driven flows. J. Nonlinear Sci. 25, 247–275 (2015)

  • Domokos, G., Lángi, Z.: The robustness of equilibria on convex solids. Mathematika 60, 237–256 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Domokos, G., Lángi, Z., Szabó, T.: A topological classification of convex solids. Geom. Dedic. 182, 95–116 (2016)

    Article  MATH  Google Scholar 

  • Domokos, G., Szabó, T., Várkonyi, P., Sipos, A.: Pebbles, shapes and equilibria. Math. Geosci. 42, 29–47 (2010)

    Article  MATH  Google Scholar 

  • Domokos, G., Jerolmack, D., Sipos, A.A., Török, A.: How river rocks round: resolving the shape-size paradox. PloS One 9, e88657 (2014)

    Article  Google Scholar 

  • Dong, S., Bremer, P., Garland, M., Pasucci, V., Hart, J.: Spectral surface quadrangulation. ACM Trans. Graph. 25, 1057–1066 (2006)

    Article  Google Scholar 

  • Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse–Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geometry 30, 87–107 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Firey, W.: The shape of worn stones. Mathematika 21, 1–11 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Gross, J., Yellen, J.: Graph Theory and Its Applications. CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  • Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 1st edn. Springer, New York (1983); 7th corrected printing (2002)

  • Hilbert, D., Cohn-Vossen, S. (eds.): Geometry and the Imagination. AMS Chelsea Publishing, Providence (1952)

    MATH  Google Scholar 

  • Holmes, P., Rand, D.: Bifurcations of the forced van der Pol oscillator. Q. Appl. Math. 35, 495–509 (1978)

    MathSciNet  MATH  Google Scholar 

  • Illenberger, J.: Pebble shape (and size!). J. Sediment. Res. 61, 756–767 (1991)

    Google Scholar 

  • Jerolmack, D.: Pebbles on Mars. Science 340, 1055–1056 (2013)

    Article  Google Scholar 

  • Kápolnai, R., Domokos, G., Szabó, T.: Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes. Period. Polytech. Electr. Eng. 56(1), 11–20 (2012)

    Article  Google Scholar 

  • Matsumoto, N., Nakamoto, A.: The number of diagonal transformations in quadrangulations on the sphere. In: Akiyama, J., Kano, M., Sakai, T. (eds.) Computational Geometry and Graphs, volume 8296 of Lecture Notes in Computer Science, vol. 8296, pp. 110–119. Springer, Heidelberg (2013). Revised selected papers from the Thailand-Japan Joint Conference (TJJCCGG2012) held at Srinakharinwirot University, Bangkok, December 6–8, 2012

  • Miller, K., Szabó, T., Jerolmack, D., Domokos, G.: Quantifying the significance of abrasion and selective transport for downstream fluvial grain size evolution. J. Geophyis. Res./Earth Surf. 119, 2412–2429 (2014)

    Article  Google Scholar 

  • Milnor, J. (ed.): Morse Theory. Princeton University Press, Princeton (1963)

    MATH  Google Scholar 

  • Nakamoto, A.: Diagonal transformations in quadrangulations of surfaces. J. Graph Theory 21(3), 289–299 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Nakamoto, A.: Generating quadrangulations of surfaces with minimum degree at least 3. J. Graph Theory 30, 223–234 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Negami, S., Nakamoto, A.: Diagonal transformations of graphs on closed surfaces. Sci. Rep. Yokohama Nat. Univ., Sec. I 40, 71–97 (1993)

    MathSciNet  Google Scholar 

  • Rayleigh, L.: The ultimate shape of pebbles, natural and artificial. Proc. R. Soc. Lond. A 181, 107–118 (1942)

    Article  Google Scholar 

  • Rayleigh, L.: Pebbles, natural and artificial. Their shape under various conditions of abrasion. Proc. R. Soc. Lond. A 182, 321–334 (1944)

    Article  Google Scholar 

  • Rayleigh, L.: Pebbles of regular shape and their production in experiment. Nature 154, 161–171 (1944)

    Article  Google Scholar 

  • Sotomayor, J.: Generic one-parameter families of vector fields on two-dimensional manifolds. Bull. Am. Math. Soc. 74, 722–726 (1968)

    Article  MathSciNet  Google Scholar 

  • Várkonyi, P., Domokos, G.: Static equilibria of rigid bodies: dice, pebbles and the Poincaré–Hopf theorem. J. Nonlinear Sci. 16, 255–281 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Varkonyi, P., Domokos, G.: Static equilibria of rigid bodies: dice, pebbles, and the poincare-hopf theorem. J. Nonlinear Sci. 16(3), 255–281 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Williams, R., Grotzinger, J., Dietrich, W., Gupta, S., Sumner, D.: Martian fluvial conglomerates at Gale crater. Science 340, 1068–1072 (2013)

    Article  Google Scholar 

  • Zingg, T.: Beitrag zur schotteranalyse. Schweiz Mineral Petrogr Mitt. 15, 39–140 (1935)

    Google Scholar 

  • Zomorodian, A. (ed.): Topology for Computing. Cambridge University Press, Cambridge (2005)

Download references

Acknowledgments

This work was supported by OTKA Grant T119245 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Comments from an anonymous referee and Tímea Szabó are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Mechanics, Materials and Structures, Budapest University of Technology, Műegyetem rakpart 1-3., Budapest, 1111, Hungary

    Gábor Domokos

  2. Program in Applied and Computational Mathematics and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

    Philip Holmes

  3. Department of Geometry, Budapest University of Technology, Egry József u. 1., Budapest, 1111, Hungary

    Zsolt Lángi

Authors
  1. Gábor Domokos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philip Holmes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zsolt Lángi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zsolt Lángi.

Additional information

Communicated by Alain Goriely.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Domokos, G., Holmes, P. & Lángi, Z. A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields. J Nonlinear Sci 26, 1789–1815 (2016). https://doi.org/10.1007/s00332-016-9319-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-016-9319-4

Keywords

Mathematics Subject Classification

Search

Navigation