Biomedical Engineering Reference
In-Depth Information
light-matter coupling in photonic-plasmonic nanostructures [34,
36, 37]. Deterministic aperiodic media with increasing degree
of rotational symmetry in their reciprocal space will be briefly
reviewed in the next section providing the necessary background to
introduce the fascinating class of Vogel spiral structures.
11.2 Rotational Symmetry: From Tilings to Vogel Spirals
Oneofthedeepestresultsofclassicalcrystallographystatesthatthe
combinationoftranslationswithrotationsrestrictsthetotalnumber
of available rotational symmetries to the ones compatible with the
periodicity of the lattice [27]. This important result is known as the
crystallographic restriction. We say that a structure possesses an
n
-
fold rotational symmetry if it is left unchanged when rotated by an
angle 2
/
n
, and the integer
n
is called the order of the rotational
symmetry (or the order of its symmetry axis). It can be shown
that only rotational symmetries of order
n
π
=
2, 3, 4, 6 fulfill the
translational symmetry requirements of2D and3Dperiodic lattices
in Euclidean space [25-27], therefore excluding
n
=
>
6.
As a result, the pentagonal symmetry, very often encountered in
nature as in the pentamerism of viruses, micro-organisms such as
radiolarians, plants, and a number of marine animals (i.e., sea stars,
urchins, crinoids, and so on) has been traditionally excluded from
the mineral kingdom until the non-crystallographic symmetries
were discovered in quasicrystals. Moreover, it was recently shown
that aperiodic tilings displaying an arbitrary degree of rotational
symmetry can be deterministically constructed using a purely
algebraic approach [38]. In Fig. 11.1, we display four remarkable
aperiodic point patterns featuring increasing rotational symmetry,
along with their corresponding reciprocal space, obtained by
Fourier transformation. Wenotice that aperiodic structures possess
anon-periodicreciprocalspace.Asaresult,thediffractionkvectors
of aperiodic Fourier space lose their global meaning and should be
regarded merely as locally defined spatial frequency components.
In Fig. 11.1a, we show a point pattern obtained by positioning
particlesattheverticesofthecelebratedPenrosetiling,namedafter
the mathematician and physicist Roger Penrose who investigated
5and
n
