Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Approximating the Quasiperiodic Surface
Nothing in Chapter 2 raised doubts about the existence of the quasiperiodic
minimal surface, but the methods used there provide no way to ``experimentally''
verify that an interface with the given topology may in fact lead to
a minimal surface. It is then worthwhile and imperative to quantify
the minimality that can be obtained for a surface with the given topology.
Although the topology of the surface seems to be reasonably well-known based
on the Voronoi construction and the function
(
x
)
, it is
useful to quantify the geometry of the surface in physical space.
Of course, the quasiperiodic surface has no unit cell in physical space so that the entire surface cannot be simulated in three dimensions, but the computational burden may be greatly eased by studying the surface in a fundamental domain of the icosahedral group containing 1/120 of physical space. Also, the surface may be cut by a series of ``pseudo-mirror planes'' normal to a two-fold direction. In this way, regions of space congruent up to a scale factor to the shape of the O 2 tile (or 1/120 of a triacontahedron) will contain a triangulated surface patch that can be relaxed. As the pseudo-mirror planes are scaled successively further away from the origin by powers of τ , the area and genus per unit volume of the minimal surface should converge to the bulk values for the surface.
A variety of methods exist for creating the initial triangulated surface,
and they all lead to surfaces with the same topology. First, the
approximate interface specified by
(
x
) = 0
may be used by taking a cut with
x
⊥
constant. This
method has been used mainly as a check on the topology formed by the
methods to follow and also as an aid in visualizing the structure of
the real space surface.
The Voronoi construction outlined in Section 2.6 may
be used as a way to come up with a very good guess for the sign of the order
parameter at a point in physical space, but care must be taken that bubbles
are not formed in physical space when the 6D cell complex is cut. For
this reason, it is good to compare a variety of methods such as the Voronoi
construction and
(
x
)
to verify that there is agreement about
the real-space topology of the surface.
The Voronoi construction can
be used to arrive at a triangulated surface most effectively by projecting
the extrema of
(
x
)
associated with particular lattice points
into physical space and re-implementing the Voronoi construction with this
subset of extrema by computing the 3D Voronoi cell complex and deleting
faces shared by extrema of the same sign. This method generates
triangulated surfaces with the correct topology provided the initial set
of extrema are well-chosen for the surface in that region of
space, and in fact, the tiling of Chapter 3 was
initially conceived as the scaffolding on
which the Voronoi construction could be carried out in physical space.
The tiling would provide the lattice points, and the Voronoi construction would
give the triangulated surface. Fortunately, every tile has the same
surface patch associated with it, so the Voronoi construction needs to
be computed in physical space only once for each tile. It is also
fortunate that local configurations of tiles decorated with their
surface patches are capable of reproducing the conjectured topology of
the quasiperiodic surface in physical space. Thus, the four surface patches
provide an extremely useful way to quickly build arbitrarily large regions
of the quasiperiodic surface if there is confidence that the
aggregate topology generated is the proper one.
The final method is the lazy man's way of making a triangulation by simply getting the correct topology and letting the Surface Evolver do the rest. If a surface patch's topology is known, the simplest (or easiest to generate) triangulation satisfying that topology can be used. For instance, the surface patch for the T 1 tile bisects four of the tile edges but is curved on the tile faces (see Section 4.5). The topology can be reproduced by a plane joining the four points on the tile edges, and this triangulation is more successful in terms of its numerical curvature properties than the triangulation obtained using the Voronoi construction. Applications of this quick method to other tiles sometimes works as well as or better than the more erudite methods, and sometimes it does not. Since all methods yield the equivalent topology, one may take the pick of the litter to use the best one. Not all triangulations are created equal once they are evolved, and it is found that no one method always obtains the best numerical results.
Chris Kimmer 2011-06-01