*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.