Biomedical Engineering Reference
In-Depth Information
Figure 11.1
(a) Penrose array, generation 12; (b) Danzer array, generation
4; (c) Pinwheel array, generation 5; (d) Delaunay-triangulated Pinwheel
centroid (DTPC) array, generation 5; (d) Penrose reciprocal space; (e)
Danzer reciprocal space; (f) Pinwheel reciprocal space (h) DTPC reciprocal
space.
such aperiodic tiling of the plane in the 1970s. He discovered that
only two planar shapes, or prototiles known as the kite and dart
tiles, are su
cient to aperiodically cover the entire plane without
leaving any gaps. Moreover, the Fourier spectrum of the Penrose
tiling, shown in Fig. 11.1e, features
δ
Bragg peaks arranged in a
pattern with 10-fold rotational symmetry. In Fig. 11.1b, we show
a deterministic point pattern obtained using the Danzer inflation
rule [39], which results in a 14-fold rotational symmetry. The
corresponding reciprocal space is shown in Fig. 11.1f.
Deterministicpointpatternswithincreasingdegreeofrotational
symmetry up to an infinite order (i.e., continuous circular sym-
metry) have also been demonstrated [40] using a simple iterative
procedure that decomposes atriangle into congruent copies.
The resulting structure, called the Pinwheel tiling, has triangular
elements (i.e., tiles) that appear in infinitely many orientations
and, in the limit of infinite-size, the diffraction pattern displays
continuous (“infinity-fold”) rotational symmetry. Radin has shown
that there are no discrete components in the Pinwheel diffraction
spectrum [40]. However, it is currently unknown whether the
spectrum is continuous or singular continuous. A point pattern
obtained from the Pinwheel tiling and the corresponding Fourier
