Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
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Sheng's Critical Surface
The critical surface is the 2D submanifold of the Ψ( x ) = 0 surface defined by ∇ ∥ Ψ( x ) vanishing. At points on the critical surface, the normal vector to the hypersurface lies in perp space, so the critical surface is the analogue of the critical point for the stretched D surface (Section 2.2) where the surface normal is parallel to the z axis. As with the stretched D surface, the critical surface defines a locus of points where the isosurface exhibits singular behavior when cut by the physical plane.
Sheng triangulated the critical surface of
(
x
)
as an approximation to the critical surface of the
t
→ 0
limit of
Ψ(
x
)
. Sheng found that
the projection of the critical surface to perp space formed a
multi-sheeted, onion-like structure that decorated each BCC lattice point.
Surrounding each lattice point was a nearly spherical surface and slightly
further away was a self-intersecting surface with three sheets winding around
singularities along the five-fold directions.
Sheng conjectured that the outer surface
of the critical surface's projection to perp space was polyhedral,
possibly a
triacontahedron, in the limit
t
→ 0
. Although the basic hypothesis
that the lowest-order Fourier modes define the topology of the 5D
Ψ(
x
) = 0
surface is
tenable and well-supported, the exact nature (polyhedral, curved, fractal?)
and shape of the 2D critical surface is highly sensitive to the amplitudes
of each Fourier mode. Consequently, examination of the critical surface's
structure in more detail provides the most insightful views into the
existence and structure of the quasiperiodic minimal surface.
The minimal surface is defined by a totally irrational cut through 6D, and this cut is parameterized by its x ⊥ coordinate. By varying the x ⊥ parameter, the 3-plane defining physical space will cross through points on the critical surface, and there the minimal surface will exhibit singular behavior. Accompanying this journey through a critical point is a local change in the topology of the surface. Sheng classified the basic topology jump as the appearance of a bubble in physical space (corresponding to crossing the inner, spherical part of the critical surface) followed by the bubble's joining to the rest of the surface by three catenoidal necks (each corrresponding to a sheet of the outer, multi-layered part of the critical surface). Sheng speculated that the separate sheets of the critical surface would merge together in the limit t → 0 , and then the formation of the bubble and its necks would be a single, instantaneous process.
Next: Methodology Up: Constructing Quasiperiodic Minimal Surfaces Previous: Sheng's Numerical Results