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,1997 References
) . 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, 1991 References )
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, 1990 References
) had quite independently identified similar geometrical structure, now
called quasicrystals (Levine and Steinhardt,1984; Mintmire,1996
References ). These developments were
based on the important work on the mathematics of tilings done by Roger
Penrose and others beginning in the 1970s.
Penrose (1974,1979
References ) discovered a nonperiodic
tiling of the plane, using two types of tiles, which is a quasiperiodic
crystal with pentagonal symmetry (DiVincenzo, 1989 References
). 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 6 Fivefold
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.