Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

iSci

the informatics of scientific computing

Sheng's Numerical Results

Sheng used Equation 2.8 to model a quasiperiodic minimal surface with icosahedral BCC symmetry. In other words, the order parameter changes sign when translated by a body centered vector ( ½ ½ ½ ½ ½ ½ ) . In the 6D BCC lattice, there are 64 such translations, 40 of which represent two groups of 20 three fold axes ( { $\bar{{\frac{1}{2}}}$ $\bar{{\frac{1}{2}}}$ $\bar{{\frac{1}{2}}}$ ½ ½ ½ } and { ½ ½ ½ ½ ½ ½ } ) and 24 representing two groups of 12 five-fold directions ( { ½ ½ $\bar{{\frac{1}{2}}}$ ½ ½ $\bar{{\frac{1}{2}}}$ } and { ½ ½ $\bar{{\frac{1}{2}}}$ ½ ½ ½ } ), so symmetry dictates that the zero surface will lie on the bisectors of these points along both three- and five-fold symmetry directions.

Sheng's numerical investigations revealed that the surface at the smallest calculated value of the anisotropy parameter t was dominated by two stars of Fourier vectors. One star is composed of 12 five-fold ({100000} ) wavevectors that have equal length when projected into para or perp space. The other star consists of 20 three-fold wavevectors ( {111000} ) whose projections are a factor of τ ³ larger in perp space than physical space. These wavevectors are sufficient to determine the topology of the Ψ( x ) = 0 isosurface and also approximate its location reasonably well. Numerical studies indicated the Fourier amplitudes for the two stars of wavevectors were roughly equal and opposite, so the topology of the 5D hypersurface is well-approximated by

Ψ( x ) ≈ $\displaystyle \tilde{{\Psi}}$ ( x ) ≡ $\displaystyle \sum_{i}^{}$ cos 2 π {100000} _i ^. x - $\displaystyle \sum_{i}^{}$ cos 2 π {111000} _i ^. x = 0

( 15 )

where { x } _i is used to denote the i th member of the orbit of x under the icosahedral group Y _h . Sheng used this basic form of the interface to study the first four rational approximants[ 31 ] to the surface in order to observe convergence of the values of the surfaces area and genus per unit volume in the limit of the full quasiperiodic surface.

Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu

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