Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

The Critical Surface Obtained from the t → 0 Voronoi Construction

The 5-faces of this cell complex provide an approximation to the Ψ ( x ) = 0 hypersurface in 6D, but the interesting structure is still the surface in physical space. The intersection of the 5-faces with the three dimensional physical space is generically a two dimensional surface. Since the limiting process is expected to produce a surface lying entirely in physical space, the most accurate cell complex would mimic this notion; it should have 5-faces that are the direct product of 2-faces in physical space and 3-faces in perp space, as well as products of 3-faces in physical space and 2-faces in perp space corresponding to phason jumps. For this latter case, the normal vector of the 5-face will lie in perp space, and this 5-face will be associated with the critical surface.

Although the critical surface is two-dimensional, it is not incorrect to attempt to relate a 5-face to the critical surface, and this concept is clear in the example of the stretched D surface. The critical submanifold in one unit cell is the single point with a normal in perp space, but there is an entire two-face of the stretched and rescaled D surface that is transversal to the z -axis. In the limit of infinite stretch, this two face projects down to a single point on the z axis. Likewise, a 5-face with a normal vector ``almost lying in perp space'' would be expected to project to a plane in perp space in the limiting case. The notion of ``almost lying in perp space'' can be made quantitative by measuring the angle θ = cos ^-1 with n _⊥ the component of the 5-faces normal in perp space. Applying this measure to the Voronoi cell of a single 5-fold extremum yields seven faces with angles less than 21 ^o (The next largest angle is 35 ^o ). Five of these seven are related by a rotation about the five-fold direction fixed by the extremum, and these five have normal vectors along two-fold directions. The other two are symmetry-inequivalent and have normals along five-fold directions. The faces with two-fold normals are shared by 3-fold extrema from neighbouring lattice points while the ones with five-fold normals are shared with other 5-fold extrema (relative to their respective associated lattice points). Specific coordinates of the extrema related by the critical faces are given in Table  2.2 .

Table  2.2 also shows the distances in perp space of the projected facet centroids from the origin. The facet with the closest centroid in perp space is also the one with the normal vector lying most in perp space, so it is reasonable to expect that this facet is related to the critical surface. This facet's centroid agrees well with the location of the inner surface of the critical surface based on the ( x ) = 0 isosurface, and this 5-face is not connected to the other 5-faces with normal vectors mostly in perp space. This facet generates the inner surface of the critical surface under the action of the icosahedral group.

The other 5-faces with mostly-perp normal vectors are equally interesting. The faces with the two-fold normals decorate a five-fold direction, and adjacent faces share 4-faces along their boundaries. Each of these five 5-faces also shares a 4-face with the sole remaining face in Table  2.2 .

These three classes of faces with favorable normal vectors exhaust the possibilities for the approximate critical surface using the cell complex from the Voronoi construction. They were all found by examing the Voronoi cell of a 5-fold extremum, but examination of the other kind of 3-fold extremum reveals no new faces. The only ones found are the ones that connect five-fold to three-fold extrema associated with different lattice points.

The 5-faces outlined here are given as candidates for describing the structure of the critical surface, but they cannot form the critical surface by themselves. To illustrate, the symmetry orbit of the face that projects into perp space closest to the origin (as listed in Table  2.2 ) creates intersecting faces in perp space rather than an inner polyhedron. These intersecting faces enclose a volume centered at the origin in perp space that corresponds to Sheng's critical surface's inner bubble, but determining the exact form of the inner polyhedron of the critical submanifold involves considering all the 2-faces of the cell complex formed by the Voronoi construction and actually computing the critical submanifold of this complex. This is a well-defined topological problem and consequently a most promising area for future work. Such a tactic may be overkill, however, for this inner face has not merged with the outer faces conjectured to be relevant to the critical surface, and the failure to merge indicates that a critical surface found by a rigorous computation could not support a minimal surface in physical space. It appears that merely getting the correct topology, as the cell complex has done, is insufficient to predict what critical surface should be expected in the limit t → 0 .

Chris Kimmer 2011-06-01

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