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2.6 Quasicrystalline Structure : The Quasiperiodic Penrose Tiling Pattern

The regular arrangement of plant parts resemble the newly identified (since 1984) quasicrystalline order in condensed matter physics (Nelson, 1986;Steinhardt,1997References) . Traditional (last 100 years) crystallography has defined a crystalline structure as an arrangement of atoms that is periodic in three dimensions. Crystals have lattice structure with identical arrangement of atoms ( Von Baeyer, 1990; Lord, 1991References) with space filling cubes or hexagonal prisms. Five fold symmetry was prohibited in classical crystallography. In 1984, an alloy of aluminum and magnesium was discovered which exhibited the symmetry of an icosahedron with five-fold axis. At the same time Paul Steinhardt of the University of Pennsylvania and his student Dov Levine (Von Baeyer, 1990References) had quite independently identified similar geometrical structure, now called quasicrystals(Levine and Steinhardt,1984; Mintmire,1996References) These developments were based on the important work on the mathematics of tilings done by Roger Penrose and others beginning in the 1970s. Penrose(1974,1979References) discovered a nonperiodic tiling of the plane, using two types of tiles, which is a quasiperiodic crystal with pentagonal symmetry (DiVincenzo, 1989References)). It is generally accepted that a quasicrystal can be understood as a systematic (but not periodic) filling of space by unit cells of more than one kind. Such extended structures in space can be orderly and systematic without being periodic. Penrose tiling pattern (Figure 6Fivefold and Spiral Symmetry Associated with Fibonacci Sequence) are two dimensional quasicrystals.

The geometric pattern is selfsimilar and exhibits long-range correlations and is quasiperiodic. It is shown in Section 4 that turbulent fluid flows can be resolved into the quasiperiodic Penrose tiling pattern with fractal selfsimilar geometry to spatial pattern and long-range temporal correlations for temporal fluctuations. Self-organized criticality is exhibited as the Penrose tiling pattern for spatial geometry which then incorporates temporal correlations for dynamical processes.

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