Centroidal Voronoi Tessellations: Grids from Nature (home)
What is the best way to tessellate the sphere? While "best" will certainly be application dependent, we can ask if there is a type or class of grids that would both tessellate the sphere and be applicable to a broad range of climate system models.
The properties we would look for are the flexibility to both uniformly discretize the sphere and to produce robust discretization of variable resolution. The grid class should also be as isotropic as possible to reflect the isotropy of the sphere's surface. Being able to understand the grid properties as solutions to certain applied math problems would bring rigor to what is sometimes an ad hoc process. And finally, the grids should be easy to create.
Spherical Centroidal Voronoi Tessellations (SCVT) seem to fit those requirements nicely. In the limit of uniformity, SCVTs produce the most uniform and isotropic tessellation of the sphere that we know of. The Gersho conjecture states that variable resolution CVT grids will asymptotically approach locally-uniform grids as the number of generating points is increased. This conjecture was proven in 2D by Newan in "The Hexagon Theorem." CVTs are also the result of certain variational problems related to optimal sampling and optimal quadrature rules.
Certain species of animals also find CVTs to be a solution to their territorial conflict. The mouthbreeder fish (Tilapia mossambica) creates CVTs during its breeding process.
