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.