InterJournal Complex Systems, 785 Status: Accepted |
Manuscript Number: [785] Submission Date: 2004 |
Polyhedral Pattern Formation |
Subject(s): CX.1
Category:
Abstract:
The problem of stable configurations of N electrons on a sphere minimizing the potential energy of the system is related to the mathematical problem of extremal configurations in distance geometry and to the problem of densest lattice packing of congruent closed spheres. Arrangement of points on a sphere in three-space leading to equilibrium or periodic solutions has been of interest since 1904 when J. J. Thomson tried to obtain stable equilibrium patterns of electrons moving on a sphere and subject to electrostatic force inversely proportional to square of the distance between them. The geometric problem of determining largest diameter of n equal non-overlapping circles on a sphere (Tammes problem) is related to the dynamic problem of periodic orbits on a sphere. Coulomb energy for three dimensional configurations with charges partitioned among vertices of regular polygons parallel to the equatorial plane shows optimal trends when polygonal faces are triangular, a result which is identical to the solution of densest packing of equal circles on a sphere. Observation of stable vortex patterns in rotating superfluid He has led to the numerical study of stationary vortex patterns in two dimensions using point vortex theory. Utilizing this point vortex theory, we will show that the problem of such vortex patterns on a sphere is related to the problem of extremal configurations and obtain polyhedral patterns and periodic vortex orbits on a sphere using numerical methods.
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