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The t → 0 Voronoi Construction

Since Equation  2.14 gives an isosurface that well approximates the minimal surface and is presumed to have the same topology as the limit structure, a natural way to obtain a cell complex for the limit surface is to attempt the Voronoi construction outlined above. A quick stab at the Voronoi cells of the extrema of $ \tilde{{\Psi}}$ ( x ) reveals that this cell complex does not capture the topology of the surface, so a closer examination is merited. The extrema of $ \tilde{{\Psi}}$ ( x ) closest to the origin are { ¼ , ¼ , 0, $ \bar{{\frac{1}{4}}}$ , $ \bar{{\frac{1}{4}}}$ , ¼ } and { ¼ , ¼ , ¼ , 0, 0, 0} along five- and three-fold directions, respectively. The extrema here have $ \tilde{{\Psi}}$ = 6 , which differs in sign from $ \tilde{{\Psi}}$ (000000) = - 4 .

The inflation symmetry of the icosahedral quasilattice provides a means to change the relative scales of parallel and perp space, and consequently, the topology of the cell complex in 6D formed by a Voronoi construction depends on the scale used. If the real space coordinates are deflated by τ according to

x = ( x || , x ⊥ ) → ( $\displaystyle {\frac{{1}}{{\tau}}}$ x || , - τ x ⊥ ) = ( $\displaystyle \tilde{{x}}_{\parallel}^{}$ , $\displaystyle \tilde{{x}}_{\perp}^{}$ ) ( 25 )

the order parameter field is invariant if the Fourier modes are inflated by τ

k = ( k || , k ⊥ ) → ( τ k || , - $\displaystyle {\frac{{1}}{{\tau}}}$ k ⊥ ) = ( $\displaystyle \tilde{{k}}_{\parallel}^{}$ , $\displaystyle \tilde{{k}}_{\perp}^{}$ ) ( 26 )

The new dominant stars of wavevectors corresponding to Equation  2.14 are ½ {1 $ \bar{{1}}$ $ \bar{{1}}$ $ \bar{{1}}$ 11} along 5-fold directions and ½ {11111 $ \bar{{1}}$ } along 3-fold directions. The surface in physical(phason) space is invariant under this operation except for the rescaling by $ {\frac{{1}}{{\tau}}}$ (- τ ) . Originally, Ψ ( x ) changed sign whenever translated by any vector with all half-integral coordinates, but the order parameter field obtained by inflating the wavevectors changes sign along the deflation of such vectors. From Table  2.1 , the sign changes occur through translations by integral lattice points with odd sums and half integral lattice points with even sums. The function defined by replacing the wavevectors in Equation  2.14 with their τ -inflated counterparts is labelled $ \tilde{{\Psi}}_{\tau}^{}$ .
Figure 2.11: Comparison of the t = 0.075 raw data and the Voronoi construction.
\begin{figure} \centering \epsffile{vorcomp.eps} \singlespacing \normalspacing \end{figure}

The extrema of $ \tilde{{\Psi}}$ ( x ) can be deflated to give the extrema of $ \tilde{{\Psi}}_{\tau}^{}$ ( x ) , and in this process the closest lattice point to each extremum changes. The maxima along 5-fold directions are now closest to {100000} lattice points, so they may be translated by these lattice vectors to obtain the minima closest to the origin $ {\frac{{1}}{{8}}}$ {11 $ \bar{{3}}$ $ \bar{{1}}$ $ \bar{{1}}$ 1} . The situation is slightly different for the maxima along 3-fold directions. The maxima are of the form $ {\frac{{1}}{{8}}}$ { $ \bar{{3}}$ $ \bar{{3}}$ $ \bar{{3}}$ 111} which are equidistant from the origin and a 3-fold lattice point ½ { $ \bar{{1}}$ $ \bar{{1}}$ $ \bar{{1}}$ 111} . In order to associate 12 + 20 maxima or minima with the closest lattice point with the same sign of $ \tilde{{\Psi}}_{\tau}^{}$ , the 3-fold maxima can be translated to obtain the minima $ {\frac{{1}}{{8}}}$ { $ \bar{{1}}$ $ \bar{{1}}$ $ \bar{{1}}$ 333} . By associating these translated minima with the origin, the Voronoi construction can be repeated with the new inflation conventions, and the topology formed by this cell complex agrees with the topology of the $ \tilde{{\Psi}}_{\tau}^{}$ ( x ) = 0 surface. At the FCC grid points used in the n = 32 calculation, the sign of Ψ ( x ) agrees with the sign predicted by the Voronoi construction at 96% of the points. Figure  2.11 illustrates the agreement between the real-space data and the Voronoi construction.

Once the appropriate inflation scale is identified, the Voronoi cells and Deloné triangulation can be calculated. These calculations were carried out using the free software qhull[ 32 ]. About each 6D BCC lattice point, there are 32 associated extrema of $ \tilde{{\Psi}}$ ( x ) (but only 2 symmetry-inequivalent extrema), and each extremum's Voronoi cell contains the central lattice point as part of its boundary. Surrounding this lattice point is a motif consisting of the bounding faces of the Voronoi cells. Interestingly, the Voronoi cells of these extrema along five- and three- fold directions have the same volume, which by symmetry is 1/64. Moreover, these two symmetry-inequivalent Voronoi cells are in fact related by a symmetry operation that interchanges parallel and perp space. A single extremum's Voronoi cell, as calculated by qhull, contains 202 5-faces, 142 of which separate extrema of different sign and belong to the cell complex describing the topology of the isosurface. As a rule, the Deloné edges dual to these faces are not along high symmetry directions, but there are exceptions. Extrema associated with the same lattice point share 5-faces that are not part of the discrete approximation to the Ψ = 0 surface, and a subset of these faces have normal vectors (a.k.a. Deloné edge directions) along 2-fold directions. For the 5-faces associated with Voronoi faces belonging to the cell complex, there are 2 5-faces with normal vectors along 5-fold directions. Their normal vectors are related by inversion, and these two faces turn out to have particular importance. A Voronoi cell has 5 other 5-faces in the cell complex with Deloné edges along two-fold directions; these faces will be important, too. The projection of the Voronoi cell of an extremum into parallel and perp spaces is shown in Figure  2.6.3 .

Figure 2.12: Projection of the Voronoi cell of a 5-fold extremum into para (left) and perp spaces
\begin{figure} \begin{center} \leavevmode \epsffile{vcellnoedge.eps} \singlespacing \normalspacing \end{center} \end{figure}

It is worth an aside to note that the Voronoi cells computed by qhull are computed using floating point arithmetic. This is much faster than using arbitrary precision arithmetic, but it has its price. Voronoi 5-faces are not planar to arbitrary precision and consequently bound a 6-dimensional volume. This imprecision makes it difficult to determine whether or not the face's normal vector is along a high-symmetry direction, so 5-face normal vectors are best-computed using the Deloné triangulation rather than the 5-faces of the Voronoi cells obtained from qhull.

next up previous
Next: The Critical Surface Obtained Up: Discrete Structure of the Previous: The Voronoi Construction at
Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu