Advertisement

Arnold’s Weak Resonance Equation as the Model of Greek Ornamental Design

  • Faina BerezovskayaEmail author
  • Georgiy P. Karev
Chapter
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)

Abstract

We propose and study a mathematical model that reproduces qualitatively several ancient ornamental designs that one can see in archeological and historical museums of Crete and Athens. The spiraling wave pattern can be seen throughout the full spectrum Classical tradition. The designs contain several rings that circumscribe a fixed number of “flowers” (centers or spirals), specific to each design.

The model proposed is based on a complex differential equation of “weak resonance” (Arnold 1977). We analyze the role of the model parameters in giving rise to different peculiarities of the repeated designs, in particular, the “dynamical indeterminacy.” The model allows tracing design changes under parameter variation, as well as to construct some new ornamental designs. We discuss how observed ornamental design may reflect some philosophical ideas of ancient inhabitants of Greece.

References

  1. 1.
    V.I. Arnold, Loss of stability of self-oscillation close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11, 85–97 (1977)CrossRefGoogle Scholar
  2. 2.
    V. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, Heidelberg, Berlin, 1983)CrossRefGoogle Scholar
  3. 3.
    F.S. Beresovskaya, A.I. Khibnik, On separatrices bifurcations in the problem of auto-oscillations stability loss at resonance 1: 4. Prikl. Math. Mech. 44, 663–667 (1980)Google Scholar
  4. 4.
    F.S. Beresovskaya, A.I. Khibnik, On the problem of bifurcations of self-oscillations close to a 1:4 resonance. Selecta Mathematica formerly Sovietica 13(2), 197–215 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    R.I. Bogdanov, A versal deformation of a singular point of a vector field in the plane in the case of zero eigenvalues. Selecta Math. Sov. 1(4), 389–421 (1981). (Transl. from Russian, “Proceeding of Petrovskii Seminar” v.2 Moscow University, 37-65, 1976)zbMATHGoogle Scholar
  6. 6.
    E.I. Horozov, Versal deformations of equivariant vector fields for the cases of symmetries of order 2 and 3. Trudy Sem. I. G. Petrovskogo 5, 163–192 (1979)Google Scholar
  7. 7.
    A.I. Neistadt, Bifurcations of phase portraits of a system of differential equations, arising in the problem of auto-oscillations stability loss at 1: 4 resonance. Prikl. Math. Mech. 42, 830–840 (1978)Google Scholar
  8. 8.
    B. Krauskopf, The bifurcation set for the 1:4 resonance problem. Exp. Math. 3(2), 107–128 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1995)CrossRefGoogle Scholar
  10. 10.
    A. Andronov, E. Leontovich, I. Gordon, A. Maer, Theory of Bifurcations of Dynamical Systems on a Plane (NASA TT F-556, Israel Program for Scientific Translations, Jerusalem, 1973)Google Scholar
  11. 11.
    F. Berezovskaya, G. Karev, Arnold Weak Resonance Equation as the Model of Greek Ornamental Design. arXiv preprint arXiv:1811.09880 (2018)Google Scholar
  12. 12.
    K.R. Popper, The Open Society and its Enemies, The Spell of Plato, vol I (Routledge and Kegan Paul, London, 1966)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsHoward UniversityWashington, DCUSA
  2. 2.National Centre for Biotechnology Information, National Institutes of HealthBethesdaUSA

Personalised recommendations