Periodic Triangulations from Pattern Forming Equations Ruggero Gabbrielli - School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom Pattern forming partial differential equations and reaction-diffusion systems have been widely used to study and simulate biological processes and chemical reactions where two antagonist components lead to a multitude of different spatial textures. Here the focus has been directed to the generation of homogenously distributed points in T3, so that the periodic sets produced can be used to form a triangulation. The method has also been extended to higher dimensions. Here we show a recent result obtained with the Swift-Hohenberg equation. A new partition of space with lower cost than the Voronoi diagram of the BCC lattice has been found using methods adopted from pattern formation theory. The partition, called P42, contains two 14-hedra and twelve 13-hedra. ∂u = au − (∇2 + 1)2u + bu2 − u3 ∂t The function u(t) is computed inside a cube of given length with periodic boundary conditions and random initial values. Once the system has reached a configuration close to a stationary state, the coorindates of the maxima are stored into a file. Coefficients a and b have been tuned so that the system evolves to blobs instead of pipes or sheets. All the known three-dimensional lattices, a large number of known non-lattices and a few new non-lattices have been obtained using this technique. We report here one case that has been of intrest for its relation to the geometry of foams, cell aggregates, tetrahedrally close packed structures and - more formally - to an optimisation problem in geoemtric measure theory known as the Kelvin problem, which deals with the Voronoi diagrams of such sets. This problem asks for the partition of space into regions of equal volume having the least surface area. The currently conjectured solution is the Weaire-Phelan structure. P42 and its periodic unit Only 13 and 14-hedra 4 octahedra The Weaire-Phelan structure Voronoi centres The dual structure is a tiling by tetrahedra. There are three different kinds of tetrahedra. 12 type 1 Voronoi centres 24 type 2 6 type 3 24 type 4 The four octahedra can be split into four tetrahedra each, this way introducing two additional quadrilateral faces in the associated Voronoi diagram. Since the splitting operation can be done along any of three diagonals of the octahedron, the combinations are 81. If the periodic unit is enlarged to a multiple of itself, then the number of possible structures approaches infinity. It is also possible to extend the already known tetrahedrally close packed domain to include this infinite new family of structures. Another pattern forming model is given by the Brusselator equations. This is a two component reaction-diffusion system described by the following system of equations: 2u ∂u1 ∂ = a − (b + 1)u1 + u21u2 + D1 21 ∂t ∂x 2u ∂u2 ∂ = bu1 − u21u2 + D2 22 ∂t ∂x Type 1 Type 2 Type 3 Acknowledgments I would like to thank Michael Cross for the inspiring online demos on pattern formation. I wish to thank also Ken Brakke, Olaf DelgadoFriedrichs, David Lloyd and John Sullivan for the code provided and their invaluable help. 24 tetrahedra 16 tetrahedra 6 tetrahedra All the images have been generated with 3dt, which is part of the Gavrog Project.
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