fractal and fractional
Article
The Unexpected Fractal Signatures in
Fibonacci Chains
Fang Fang
*, Raymond Aschheim and Klee Irwin
Quantum Gravity Research, Los Angeles, CA 90290, USA; raymond@quantumgravityresearch.org (R.A.);
Klee@quantumgravityresearch.org (K.I.)
* Correspondence: Fang@quantumgravityresearch.org
Received: 2 August 2019; Accepted: 29 October 2019; Published: 6 November 2019






Abstract: In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set
is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = φ.
The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the
head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on
the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is
very sensitive to changes in the Fibonacci order but not to the L/S ratio.
Keywords: Fibonacci chain; fractal signature; Fourier space
1. Introduction
Quasicrystals possess exotic and sometimes anomalous properties that have interested the
scientific community since their discovery by Shechtman in 1982 [1]. Of particular interest in this
manuscript is the self similar property of quasicrystals that links them to fractal. Historically, research
on the fractal aspect of quasicrystalline properties has revolved around spectral and wave function
analysis [2–5]. Mathematical investigation [6–9] of the geometric structure of quasicrystals is less
represented in the literature than experimental work.
In this paper, a new framework for analyzing the fractal nature of quasicrystals is introduced.
Specifically, the fractal properties of a one-dimensional Fibonacci chain and its variations are studied in
the complex Fourier space. The results may also be found in two and three dimensional quasicrystals
that can be constructed using a network of one dimensional Fibonacci chains [10].
2. The Fractal Signature of the Fibonacci Chain in Fourier Space
The Fibonacci chain is a quasiperiodic sequence of short and long segments where the ratio
between the long and the short segment is the golden ratio [11]. It is an important 1D quasicrystal that
uniquely removes the arbitrary closeness in the non-quasicrystalline grid space of a quasicrystal and as
a result the non-quasicrystalline grid space is converted into a quasicrystal as well [10]. The Fourier
representation of a Fibonacci chain is given as follows:
zs =
1√
n
n
∑
r=1
ure
2πi(r−1)(s−1)
n
, s = 1, 2, ...n,
(1)
where
ur = (φ− 1)(b(r + 1)φc − brφc − 2) + φ = (φ− 1)(b(r + 1)φc − brφc) + (2− φ), r = 1, 2, ...n
(2)
is a Fibonacci chain of length n with units of length φ and 1, where x → bxc is the floor function.
Here φ =
√
5+1
2
is the golden ratio.
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Note that the Fibonacci chain can also be generated using substitution rules {L→ LS, S→ L} [11]
where m− 1 iterations of the substitution method produces a Fibonacci chain of length (its number of
tiles) equal to the mth Fibonacci number Fm: |S| = 1 = F1, |L| = 1 = F2, |LS| = 2 = F3, |LSL| = 3 =
F4, |LSLLS| = 5 = F5, ....Tiles L and S (for Long and Short) are of respective length φ and 1 and ur
given by Equation (2) is the coordinate of the right point of the rth tile.
Consecutive iterations of the Fibonacci chain are plotted in the Fourier space in Figure 1. Figure 2
shows that the main cardioid (heart shape) in the Mandelbrot set [12] appeared at the center of each
plot. It is known that the denominators of the periods of the circular bulbs at sequential scales in the
Mandelbrot set follow the Fibonacci number sequence [13].
Figure 1. The rescaled Fourier space representation of a Fibonacci chain is shown here using the
substitution method with different iterations, starting with 20. The horizontal axis represents the real
part and the vertical axis represents the imaginary part of the Fourier coefficients.
Figure 2. (a) The Mandelbrot set, (b) overlays of the Mandelbrot set and the Fourier space with matching
cardioid, and (c) the fractal structure in the Fourier space of a Fibonacci chain with 25 iterations.
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The plots reveal that the size of the cardioid decreases with successive iterations and the
scaling factor for the size change is approximately 1/
√
φ and precisely
√
Fm−1
Fm , so we can rescale
the Equation (1) by omitting the normalisation factor
√
1
Fm .
zs,m =
Fm
∑
r=1
ure
2πi(r−1)(s−1)
Fm
, s = 1, 2, ...Fm,
(3)
Section 2.3 will show that each momentum (point in the complex plane of the Fourier transform)
zs,m is very close to a point zp,m+4 in the point set obtained with four more iterations. Conveniently
rearranging the terms from Equation (3) gives:
z1+Fk ,m ≈ (−φ)
2k−1−m
(4)
Due to the minus sign before φ, the orientation of the cardioid undergoes mirror flips with every
odd and even numbered iteration. Upon magnification, the fractal nature becomes apparent near the
real axis, as shown in Figure 3.
Figure 3. Appearance of the fractal pattern at the 34th iteration of the Fibonacci chain: (a) general view
showing the cardioid; (b) detail of the central part; (c) zoom near the real axis, some lines are getting
very close and perpendicular to the real axis, they are called ‘trunks’ in Figure 6.
2.1. Fractal Dimension
The fractal dimension of the complex Fourier representation of the Fibonacci chain is calculated
as the pointwise dimension proposed by Grassberger and Procaccia [14]. The system we are studying
here is a chaotic system that settles down to a strange attractor in phase space. Fixing a point x on the
attractor A, the number of points, Nx (e), on A inside a ball of radius e about x, typically grows as a
power law: Nx (e) ∝ ed, where d is called the pointwise dimension. It is calculated to be approximately
1.7 for the complex Fourier representation of the Fibonacci chain (Figure 4).
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Figure 4. Pointwise dimension of the of the Fibonacci chain in Fourier space.
2.2. Universality Near the Real Line
The well-known Mandelbrot set is a collection of complex numbers c satisfying the equation
zn+1 = z2n − c where zn+1 < 2. With each fixed point c there is a Julia set. The universality emerging
near the real line is related to the non-fractal limit of the Julia set, at the specific point c = −2,
which corresponds tp a fixed point for z0 = −2. There is no more Julia set at this point, like at the
trivial fixed point c = 0. But there is a very chaoticly looking fractal concentrated around the real line
when c = −2 + e. When sliding c along the real line from c = −1 to c = −2, a set of attractors and
anti-attractors are also sliding and converging to a limit (Figure 5).
The sequence zs seems to be sparse and not organized locally, as the neighbours are not local
but as in every quasicrystal there is a non local order. Near the real line are seeded trees of a non
planar grid which self-organize in modular geodesics of lengths of the Fibonacci numbers, which are
measured as the difference of indices.
Cvm = z1 + vFm ;
Cv,+h
m = z1 + vFm + hFm+2 ;
Cv,−h
m = z1 + vFm + hFm−2
(5)
Cvm is a seed for any m and v=1 and it equals to the complex number zs,m defined by Equation (3)
where s = 6, 8, 14, 22, 56, 90... close to the real line because of Equation (4). For C1m(k) = z1+Fk ,m, a
vertical trunk emerges from this seed perpendicular to the real line and realizes a modular geodesic
of zs,m where the index s grows by a multiple of Fm. Therefore from the seed z56 grows the trunk
z111, z166, z221, z276 and from the seed z35 grows the trunk z69, z103, z137, z171 (Figure 6).
2.3. Self Similarity
Equation (4) enables a map from C to itself, mapping the iteration m to a subset of the iteration
m + 4. It maps seeds of different periodicity and also their corresponding trees. This proves that
the iteration m + 4 contains a scaled version of the iteration m and that the sequence of spectrum is
self-similar and becomes fractal at the infinite limit m→ ∞.
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Figure 5. The Julia sets on the real line, while sliding from the Golden julia set (on the top) where c =−1
to the limit of non fractality, c = −2 on the bottom, shows a sliding of the attractors and anti-attractors.
For example the left end slides from −φ to −2.
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Figure 6. The Fourier space representation of a Fibonacci chain is shown here with the geodesics of
Fibonacci steps: 5 in red, 8 in orange, 13 in yellow, 21 in green, 34 in cyan, 55 in blue, 89 in magenta and
144 in purple. The “seeds” are the point closest to the horizontal axis: 56 on the left, 22 spreading a
green trunk, 14 with a yellow trunk and 35 with a cyan trunk. The other trees are omitted.
3. The Variations of the Fibonacci Chain in Fourier Space
It is tempting to conjecture that this kind of Fractal signature will appear in any finite portion
of the Fibonacci chain or a Fibonacci chain that is nearly perfect (a Fibonacci chain that is generated
strictly from the substitution rules mentioned earlier). A series of tests has been conducted to verify
this conjecture. The results are surprising. Figure 7 shows the results of the Fourier space of the
Fibonacci chain of 25th iteration with various, mostly minor modifications, such as, (Figure 7a–f) small
truncation from the head or tail of the chain, (Figure 7l–p) scrambling the order of a very small part of
the chain or (Figure 7s) changing of the L/S ratio. These results show that the fractal pattern, especially
the cardioid shape, is very sensitive to any modification except changing the L/S ratio, in which case
only the scaling of the fractal pattern changes. A linear transformation of the size of the two tiles
S1 → S2 and L1 → L2 expressed as u′r = f (ur) = aur + b provides by linearity of the Fourier transform
the scaled result as observed. In other words, the fractal pattern is a direct result of the substitution
rule (or inflation/deflation rule) which gives the Fibonacci ordering of the L and S. The exact value of
the L and S does not matter for the pattern. Even the smallest breakdown of the rule will result in a
non-prefect-closure of the cardioid and therefore result in a completely different pattern. It is similar to
the butterfly effect in chaos theory.
3.1. Variations by Cyclic Permutations
A cyclically permuted Fibonacci chain urp where rp = Mod[r + b np c, n], is created, based on
Equation (2). For example when n = Fm = F18 we get 100 possible value of p and corresponding
permutations of rp when b np c ∈ {1, 2, 3, ...646, 861, 1292, 2584}. Superposing these 100 Fourier spectra
in one image reveals the formation of rings of various density, separated by rings of zero density,
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as in Figure 7q. This shows that the cyclic permutation results in the changing of the phases in the
Fourier spectrum.
Figure 7. The Fourier space representation of a Fibonacci chain of 25th iteration, with (a) the first
segment removed; (b) the first two segments removed; (c) the first three segments removed; (d) the first
six segments removed, (e) the first seven segments removed, (f) the first segment length replaced with
0; (g) no modification; (h) the last segment removed, (i) the last two segments remove; (j) the last 46,367
segments removed; (k) the last 46,368 segments removed and the Fibonacci chain of 25th iteration is
truncated to the Fibonacci chain of 24th iteration; (l) the 1st segment L replaced with S; (m) the last two
segments flipped order; (n) the order of the last five segments scrambled; (o) the order of the last ten
segments scrambled; (p) the order of the last 100 segments scrambled, (q) the superposition of spectra
of cyclic permutations of the 17th iteration and the comparison between (r) the original Fibonacci chain
of 27th iteration and (s) the chain with modified L/S ratio where L/S = 2.
3.2. Variations and the Generalized Mandelbrot Set
Some variations reminds one the Mandelbrot sets of higher degree, named generalized
Mandelbrot set.
zn+1 = z
p
n + c
(6)
The exponent p equals to 2 for the Mandelbrot set and will give a cardioid (Figure 7g).
z(t) = x + iy =
eit
2
− e
2it
4
(7)
The power p can be 3 and give a Mandelbrot set-like fractal with two bulbs and the cardioid
will be replaced by a nephroid (like Figure 7e). And when p = 4 there will be 3 bulbs, (Figure 7a);
when p = 5 there will be 4 bulbs (Figure 7f). The symmetries observed in the derivative Fourier
spectrum are the same as the main feature of the generalised Mandelbrot set illustrated in Figure 8.
The mathematical relationship is a result of the fact that both these curves and the Fourier spectrum are
sum of imaginary exponentials. There is a subset of the discrete Fourier spectrum set which are very
close to the points from Equation (4) after rescaling. It is more hidden but similar for the higher order.
Its worth noting that derivative patterns resemble the cycloidal pattern drawn by a geometric
chuck. Figure 9 shows pictures taken from a book published 100 years ago by Bazley [15]. Figure 9a is
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a picture of a geometric chuck. Figure 9b–d are a few cycloids created by the geometric chuck with
different settings on the chuck. The fractal signature of a perfect Fibonacci chain may correspond to
some perfect quasiperiodic gearing between the plates of the geometric chuck.
Figure 8. Curves on the complex plane, from main features of the generalised Mandelbrot sets,
with their equations; (a–d) Epicycloid from the Mandelbrot set and its 3 first circular bulbs centered on
the real line (e–h) Nephroid and its higher generalisations from the generalized Mandelbrot set for
exponents 3 to 6.
Figure 9. (a) A picture of a geometric chuck; (b–d) Cycloids generated by the geometric chuck with
different settings. The following settings are used: P = 55,V1 = 111
55 =
2P+1
P
, (a): V2 = 55 out, Ex1 = 30,
Ex2 = 55, SR = 27, (b): V2 = 55 in, Ex1 = 35, Ex2 = 45, SR = 30, (c): V2 = 552 out, Ex1 = 30, Ex2 = 55,
SR = 27, The equation is not given but the information in Reference [15] should help with finding it.
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4. Summary
This paper reports a novel fractal analysis of a Fibonacci chain in Fourier space by computing
the pointwise dimensions and visual inspection of the Fourier pattern. The cardioid structure in the
Mandelbrot set is observed for the case of the one dimensional Fibonacci chain, with alternating mirror
symmetry according to odd or even instances of the substitution iteration that generates the Fibonacci
chains. The Fourier pattern is very sensitive to how strictly the chain follows the substitution rules.
Any small variation in that will completely change the pattern. In comparison, the change in the L/S
ratio has no influence in the Fourier pattern other than a change in scaling.
Author Contributions: Conceptualization, F.F. and K.I.; Investigation, F.F., R.A. and K.I.; Methodology and
Writing Manuscript, F.F. and R.A.; Software, F.F. and R.A.; Supervision, K.I.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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