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The Unexpected Fractal Signatures in Fibonacci
chain
Fang Fang, Raymond Aschheim and Klee Irwin
Quantum Gravity Research, Los Angeles, CA, U.S.A.
E-mail: fang@QuantumGravityResearch.org
Abstract.
Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a
new cycloidal 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 0.7. Various variations such as truncation from the head
or tail, scrambling the orders of the sequence, changeing 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 the change in the Fibonacci order but not to the L/S ratio.
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 makes them fractal. Historically,
research on this aspect of quasicrystalline property has revolved around spectral and wave
function analysis [2, 3, 4]. Mathematical investigation [5, 6] of the geometric structure of
quasicrystals is less well 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 derivatives 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 [7].
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 [8]. It is an important 1D
quasicrystal that uniquely removes the arbitrary closeness in the non-quasicrystalline grid space
of a quasicrystal and convert it into a quasicrystal as well [7]. The Fourier representation of a
Fibonacci chain is given as follows:
zs =
1
√
n
n
r=1
ure
2πi(r−1)(s−1)
n , (1)
where
ur = (φ − 1)( (r + 1)φ − rφ − 2) + φ = (φ − 1)( (r + 1)φ − rφ ) + (2 − φ), r = 1, 2, ...n (2)
is a Fibonacci chain of length n with units of length φ and 1. Here φ =
√
5+1
2 is the golden ratio.
Note that the Fibonacci chain can also be generated using substitution rules [8] where m
iterations of the substitution method produces a Fibonacci chain of length equal to the mth
Fibonacci number.
Consecutive iterations of the Fibonacci chain are plotted in the Fourier space in Fig. 1. The
main cardioid (heart shape) in the Mandelbrot set [9] 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 [10]. The plots reveal that the size of
the cardioid decreases with successive iterations, and the scaling factor for the size change is
approximately one half of the golden ratio. Moreover, 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. The fractal dimension1 of the complex Fourier
representation of the Fibonacci chain is approximately 0.7.
Figure 1. The 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.
3. The derivatives 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 5 shows the results of the
Fourier space of the Fibonacci chain of 26th iteration with various modifications, mostly minor
modifications, for example, small truncation from the head or tail of the chain, scrambling the
order of a very small part of the chain or changing of the L/S ratio. These results show that the
1
pointwise dimension [11]
Figure 2. a) The Mandelbrot set, c) the fractal structure in the Fourier space of a Fibonacci
chain with 25 iterations and b) overlays a and c together.
Figure 3. Appearance of the fractal pattern at the 34th iteration of the Fibonacci chain.
fractal pattern, especially the cardioid shape, is very sensitive to any modification except the
changing the L/S ratio, in which case only the scaling of the fractal pattern changes. 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. Any little breakdown of the rule will result in a non-prefect-closure of
the cardioid therefore a complete different pattern. It is similar to the butterfly effect in chaos
theory.
Its worth noting that derivative patterns resemble the cycloidal pattern drawn by a geometric
chuck. Figure 6 show pictures taken from a book published 100 years ago by Bazley [12]. Figure
6a is a picture of a geometric chuck Figure 6b-d are a few cycloids created by the geometric chuck
with different settings on the chuck. The fractal signature of the perfect Fibonacci chain may
correspond to some perfect quasiperioidic gearing between the plates of the geometric chuck.
Figure 4. Pointwise dimension of the of the Fibonacci chain in Fourier space.
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.
Figure 5. The Fourier space representation of a Fibonacci chain of 26th 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 46367 segments removed; k) the last 46368 segments removed and the Fibonacci chain
of 26th iteration is truncated to the Fibonacci chain of 25th 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, and the comparison between q) the original Fibonacci chain and r) the chain with
modified L/S ratio where L/S = 2.
Figure 6. a) A picture of a geometric chuck; b-d) Cycloids generated by the geometric chuck
with different settings.
5. References
[1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order
and no translational symmetry,” Physical Review Letters, vol. 53, no. 20, p. 1951, 1984.
[2] U. Grimm and M. Schreiber, “Energy spectra and eigenstates of quasiperiodic tight-binding hamiltonians,”
arXiv:cond-mat/0212140, 2002.
[3] P. Repetowicz, U. Grimm, and M. Schreiber, “Exact eigenstates of tight-binding hamiltonians on the penrose
tiling,” arXiv:cond-mat/9805321, 1998.
[4] N. Mac´e, A. Jagannathan, and F. Pi´echon, “Fractal dimensions of the wavefunctions and local spectral
measures on the fibonacci chain,” arXiv:1601.00532, 2016.
[5] P. Ramachandrarao, A. Sina, and D. Sanyal, “On the fractal nature of Penrose tiling,” Current Science,
vol. 79, no. 3, pp. 364–367, 2000.
[6] V. Yudin and Y. Karygina, “Fractal images of quasicrystals as an example of Penrose lattice,” Crystallogrphy
reports, vol. 46, no. 6, pp. 922–926, 2001.
[7] F. Fang and K. Irwin, “An icosahedral quasicrystal as a golden modification of the icosagrid and its connection
to the e8 lattice,” arXiv:1511.07786, 2015.
[8] D. Levine and P. J. Steinhardt, “Quasicrystals. i. definition and structure,” Physical Review B, vol. 34, no. 2,
p. 596, 1986.
[9] A. Douady, J. H. Hubbard, and P. Lavaurs, “Etude dynamique des polynˆomes complexes,” Universit´e de
Paris-Sud, D´ep. de Math´ematique, 1984.
[10] R. L. Devaney, http://math.bu.edu/DYSYS/FRACGEOM2, 2016.
[11] S. H. Strogatz, Nonlinear Dynamics and Chaos with applications to Physics, Biology, Chemistry, and
Engineering. Perseus Books, 1994.
[12] T. S. Bazley, Index to Geometric Chuck. Waterlow and Sons, 1875.

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The unexpected fractal signatures in fibonacci chains 10.06.16 - final - ff

  • 1. The Unexpected Fractal Signatures in Fibonacci chain Fang Fang, Raymond Aschheim and Klee Irwin Quantum Gravity Research, Los Angeles, CA, U.S.A. E-mail: fang@QuantumGravityResearch.org Abstract. Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a new cycloidal 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 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing 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 the change in the Fibonacci order but not to the L/S ratio. 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 makes them fractal. Historically, research on this aspect of quasicrystalline property has revolved around spectral and wave function analysis [2, 3, 4]. Mathematical investigation [5, 6] of the geometric structure of quasicrystals is less well 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 derivatives 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 [7]. 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 [8]. It is an important 1D quasicrystal that uniquely removes the arbitrary closeness in the non-quasicrystalline grid space of a quasicrystal and convert it into a quasicrystal as well [7]. The Fourier representation of a Fibonacci chain is given as follows: zs = 1 √ n n r=1 ure 2πi(r−1)(s−1) n , (1) where ur = (φ − 1)( (r + 1)φ − rφ − 2) + φ = (φ − 1)( (r + 1)φ − rφ ) + (2 − φ), r = 1, 2, ...n (2)
  • 2. is a Fibonacci chain of length n with units of length φ and 1. Here φ = √ 5+1 2 is the golden ratio. Note that the Fibonacci chain can also be generated using substitution rules [8] where m iterations of the substitution method produces a Fibonacci chain of length equal to the mth Fibonacci number. Consecutive iterations of the Fibonacci chain are plotted in the Fourier space in Fig. 1. The main cardioid (heart shape) in the Mandelbrot set [9] 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 [10]. The plots reveal that the size of the cardioid decreases with successive iterations, and the scaling factor for the size change is approximately one half of the golden ratio. Moreover, 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. The fractal dimension1 of the complex Fourier representation of the Fibonacci chain is approximately 0.7. Figure 1. The 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. 3. The derivatives 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 5 shows the results of the Fourier space of the Fibonacci chain of 26th iteration with various modifications, mostly minor modifications, for example, small truncation from the head or tail of the chain, scrambling the order of a very small part of the chain or changing of the L/S ratio. These results show that the 1 pointwise dimension [11]
  • 3. Figure 2. a) The Mandelbrot set, c) the fractal structure in the Fourier space of a Fibonacci chain with 25 iterations and b) overlays a and c together. Figure 3. Appearance of the fractal pattern at the 34th iteration of the Fibonacci chain. fractal pattern, especially the cardioid shape, is very sensitive to any modification except the changing the L/S ratio, in which case only the scaling of the fractal pattern changes. 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. Any little breakdown of the rule will result in a non-prefect-closure of the cardioid therefore a complete different pattern. It is similar to the butterfly effect in chaos theory. Its worth noting that derivative patterns resemble the cycloidal pattern drawn by a geometric chuck. Figure 6 show pictures taken from a book published 100 years ago by Bazley [12]. Figure 6a is a picture of a geometric chuck Figure 6b-d are a few cycloids created by the geometric chuck with different settings on the chuck. The fractal signature of the perfect Fibonacci chain may correspond to some perfect quasiperioidic gearing between the plates of the geometric chuck.
  • 4. Figure 4. Pointwise dimension of the of the Fibonacci chain in Fourier space. 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.
  • 5. Figure 5. The Fourier space representation of a Fibonacci chain of 26th 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 46367 segments removed; k) the last 46368 segments removed and the Fibonacci chain of 26th iteration is truncated to the Fibonacci chain of 25th 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, and the comparison between q) the original Fibonacci chain and r) the chain with modified L/S ratio where L/S = 2.
  • 6. Figure 6. a) A picture of a geometric chuck; b-d) Cycloids generated by the geometric chuck with different settings.
  • 7. 5. References [1] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Physical Review Letters, vol. 53, no. 20, p. 1951, 1984. [2] U. Grimm and M. Schreiber, “Energy spectra and eigenstates of quasiperiodic tight-binding hamiltonians,” arXiv:cond-mat/0212140, 2002. [3] P. Repetowicz, U. Grimm, and M. Schreiber, “Exact eigenstates of tight-binding hamiltonians on the penrose tiling,” arXiv:cond-mat/9805321, 1998. [4] N. Mac´e, A. Jagannathan, and F. Pi´echon, “Fractal dimensions of the wavefunctions and local spectral measures on the fibonacci chain,” arXiv:1601.00532, 2016. [5] P. Ramachandrarao, A. Sina, and D. Sanyal, “On the fractal nature of Penrose tiling,” Current Science, vol. 79, no. 3, pp. 364–367, 2000. [6] V. Yudin and Y. Karygina, “Fractal images of quasicrystals as an example of Penrose lattice,” Crystallogrphy reports, vol. 46, no. 6, pp. 922–926, 2001. [7] F. Fang and K. Irwin, “An icosahedral quasicrystal as a golden modification of the icosagrid and its connection to the e8 lattice,” arXiv:1511.07786, 2015. [8] D. Levine and P. J. Steinhardt, “Quasicrystals. i. definition and structure,” Physical Review B, vol. 34, no. 2, p. 596, 1986. [9] A. Douady, J. H. Hubbard, and P. Lavaurs, “Etude dynamique des polynˆomes complexes,” Universit´e de Paris-Sud, D´ep. de Math´ematique, 1984. [10] R. L. Devaney, http://math.bu.edu/DYSYS/FRACGEOM2, 2016. [11] S. H. Strogatz, Nonlinear Dynamics and Chaos with applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, 1994. [12] T. S. Bazley, Index to Geometric Chuck. Waterlow and Sons, 1875.