Biomedical Engineering Reference
In-Depth Information
Table 11.1
Divergence angles utilized for the irrational and
perturbed GA spirals
Divergence
Divergence
ξ
angle (
◦
)
ξ
angle (
◦
)
Label
Labeling
+
√
8)/2
+
√
5)/2
τ
1.1
(2
210.883118
GA
(1
137.507764
+
√
29)/2
μ
1.2
(5
290.670335
1.3
π
3.14159...
309.026645
α
1
137.300000
α
2
137.369255
β
1
137.523137
α
3
137.403882
β
2
137.553882
α
4
137.473137
β
3
137.569255
(1
+
√
5)/2
GA
137.507764
β
4
137.600000
differentdegreesofaperiodicstructuralorder.Previousstudieshave
focused on the three most investigated types of aperiodic spirals,
including the GA spiral and two other Vogel spirals obtained by
the following choice of divergence angles: 137.3
◦
(i.e.,
α
1
spiral)
and 137.6
◦
(i.e.,
β
4
spirals are
called “nearly golden spirals” because their divergence angles are
numericallyveryclosetotheGAvalue,buttheirfamiliesofdiverging
arms, known as parastichies, are considerably fewer. In a recent
study, Trevino et al. [37] extended the analysis of aperiodic Vogel
spiralstostructuresgeneratedwithdivergenceanglesthatareequi-
spaced between the
α
1
-spiral and the GA spiral, and between the
golden angle and
β
4
, as summarized in Table11.1.
Thesespiralstructurescanbeconsideredasone-parameter(i.e.,
thedivergenceangle
α
) structural perturbation of the GA spiral and
possess fascinating geometrical features, which are responsible for
uniquemode localizationproperties and optical spectra [37].
ItiswellknownthatintheGAspiral,therearemanyparastichies
in both clockwise (CW) and counter-clockwise (CCW) directions
[56] and that the numbers of parastichies are consecutive numbers
in the Fibonacci series, the ratio of which approximates the
golden ratio. However, as the divergence angle is varied either
above (supra-GA or
β
-series) or below the golden angle (sub-GA
or
β
4
spiral) [34, 43, 54]. The
α
1
and
-series), the center region of the spiral where both sets of
parastichies (CW and CCW) exist shrinks to a point. The outer
regions are left with parastichies that rotate only CW for divergence
α
