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Pattern formation on a sphere.

P C Matthews1

  • 1School of Mathematical Sciences, University of Nottingham, University Park, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2003
PubMed
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Pattern formation on spheres, described by spherical harmonics, uniquely determined leading equations for even degrees. Icosahedral symmetry emerged as the preferred pattern for degrees 6, 10, and 12, confirmed by simulations.

Area of Science:

  • Mathematics
  • Physics
  • Complex Systems

Background:

  • Spherical harmonics describe patterns on sphere surfaces.
  • Interactions of spherical harmonics of degree l govern pattern formation.
  • For even l, leading-order equations are symmetry-determined, independent of physical context.

Purpose of the Study:

  • To investigate pattern formation on a sphere for even degrees of spherical harmonics.
  • To determine the existence and stability of solutions for even l up to l=12.
  • To identify preferred symmetries in pattern formation.

Main Methods:

  • Analysis of equations involving spherical harmonics interactions.
  • Application of variational or eigenvalue criteria for stability analysis.

Related Experiment Videos

  • Numerical simulations of a model pattern-forming equation.

    • Main Results:

    • Leading-order equations for even l are uniquely determined by symmetry.
    • Existence and stability results were established for even l up to l=12.
    • Icosahedral symmetry was identified as the preferred solution for l=6, l=10, and l=12.
    • Numerical simulations corroborated theoretical predictions near onset.

    Conclusions:

    • Symmetry plays a crucial role in determining pattern formation on spheres.
    • Icosahedral symmetry is a robust feature for specific degrees of spherical harmonics.
    • Model simulations align with theoretical findings, extending to more complex patterns.