The simplest common denominator of Reality is motion.
...................When nothing moves
.............................There is something,
...................Which evolves to
.............................Everything.
Genius lies in the
originality
and simplicity of
salient ideas.
The pluperfect beauty of powerful simplicity,
in the formulae of Unimetry's complexity,
reinforces the fundamental concepts of
Conceptualism.
Unimetry is the geometry of the Universe
Conceptualism is a total,
belief system that requires
only a minuscule of faith that
there is hyper-relativistic,
complex, oscillating motion
between the dualities of Infinity,
that manifests as all phenomena
I have deep faith that
the principle of the
universe will be
beautiful and simple.
.....-Albert Einstein
Any intelligent fool can
make things bigger and
more complex...
It takes a touch of genius
--- and a lot of courage
to move in the
opposite direction.
.....-Albert Einstein
Views of Fibonacci Dynamics
library.thinkquest.org
Clifford A. Reiter
Department of Mathematics, Lafayette College, Easton, PA 18042 USA
Preprint: to appear in Computers & Graphics
Abstract
The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function
of a complex variable. We experimentally study the complex dynamics of the Fibonacci
numbers viewed in that manner. Attracting and repelling fixed points are related to the
filled Julia set and to regions of escape time images with fascinating behavior.
Introduction
The Fibonacci numbers are traditionally described as a sequence Fn defined by F0 = 0 ,
1 1 F = , and -1 -2 = + n n n F F F . The sequence begins
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,....
The Fibonacci sequence has many remarkable properties, ranging from routine to
startling [1-4]. Moreover, the numbers arise in nature, for example, as the number of
spirals of pinecone petals. They may also be used to construct mathematical quasicrystals
[5].
One of the beautiful formulas of Fibonacci numbers is the Binet formula. Binet
described a version of the formula in 1843 [6-7]. Its beauty arises from the fact that the
formula gives a closed form solution to a recursive definition, and from the symmetry of
the formula itself. The Binet formula my be derived from the theory of difference
equations, it can be derived by diagonalizing a suitable matrix, or it can be proven by
induction [1-3]. The Fibonacci recursion has characteristic equation x2 - x -1 = 0 which
has roots 1.618
2
t = 1+ 5 ˜ and -0.618
2
t = 1- 5 ˜ where t is the golden ratio and t
is the conjugate of t . Choosing constants to satisfy the initial conditions 0 0 F = and
1 1 F = gives the Binet formula:
5
n n
n F = t - t . To obtain the Fibonacci numbers as a
function of a complex variable, instead of viewing the index n in the Binet formula as an
integer, we view it as a complex variable z. Thus we define the following complex
Fibonacci function.
5
( )
z z
F z = t - t
The number t is negative and t appears as the base of an exponential in the Binet
formula. Thus, complex numbers will result for fractional real arguments. Nonetheless,
the Binet form gives a natural generalization of the Fibonacci sequence. It satisfies the
initial conditions F(0) = 0 and F(1) = 1. It also satisfies the recursion
F(z) = F(z -1) + F(z - 2) and it is defined for all complex values z.
Thus, we can ask questions about the complex dynamics of this function. What are its
fixed points? Are they attracting or repelling? What happens upon iteration of the
2
function? In this note we take a visual look at those questions and see that the Fibonacci
numbers have interesting and beautiful complex dynamics.
Fixed Points
The fixed points of a function F(z) are the values of z such that F(z) = z . Table I shows
the values of the Fibonacci numbers at several integer values of z.
z -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
F(z) -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21
Table I. Values of F(z) at some integer points.
Notice that z = 0,1,5 are all fixed points. It might seem as though there ought to be
another fixed point between -2 and -1 since F(z) changes from negative to positive, but
remember that since the definition of F(z) involves an exponential with a negative base,
we get complex values for F(z) at intermediate values. For example,
F(-1.5) ˜ 0.217287 - 0.920442i . There appear to be many complex fixed points. For
example, there is a fixed point near - 2.00376 - 0.197445i .
The fixed points of F(z) correspond to the zeros of F(z) - z . If we look at the
magnitude of F(z) - z along the real axis, we get the function shown in Figure 1. Note
that the figure shows z = 0,1,5 are zeros and hence fixed points of F(z) and that it
appears that there are no other real fixed points.
Figure 1. The magnitude of F(z) - z along the real axis.
The situation off the real axis can be examined by looking at a false colored contour
plot of the magnitude of F(z) - z . Figure 2 shows such a plot where - 36 = Re(z) = 36
and - 36 = Im(z) = 36 . The lowest points are shown in black and higher points via hues
running from red to magenta (highest). Notice there is a large black region near the
center. There are some black regions appearing in a sequence above the center and others
in a sequence mostly running to the upper left. This suggests that there are infinitely
many fixed points in the complex plane in the upper left quadrant. Table II gives the
values of F(z), its derivative, and the magnitude of that, at the fixed points z = 0,1,5.
3
Figure 2. The magnitude of F(z) - z in the complex plane for - 36 = Re(z), Im(z) = 36 .
z F'(z) | F'(z) |
0 0.430409-1.40496i 1.46941
1 0.215204+0.868315i 0.894586
5 2.36725+0.126685i 2.37064
Table II. Derivatives at Some Fixed Points.
The magnitude of the derivative at z = 0 and z = 5 is greater than 1. That implies
those are repelling fixed points. However, the magnitude of the derivative at z = 1 is less
than 1, so this is an attracting fixed point. Thus, we expect some region around z = 1 to
not diverge to infinity, but instead, remain finite. The points in the complex plane that are
eventually attracted to z = 1 are called the basin of attraction of z = 1. The set of points
that do not diverge to infinity are the filled Julia set. When the Julia set is nontrivial, it
has become common view such sets with an escape time image showing how quickly
points outside the filled Julia set get large.
Escape Time
In particular, an escape time image corresponds to some region in the complex plane and
typically color is used to indicate the number of iterations required before iterates get
large. Perhaps the most famous illustrations of those occur for the famous quadratic Julia
and Mandlebrot sets, but escape time images and basins of attraction have been utilized
to visualize the dynamics of many processes [8-13].
In order to create an escape time image of a function f (z) , one uses an algorithm of
the following type.
• Select a maximum iteration bound, N, and a sense of unbounded, M.
• For all pixels ( j, k) corresponding to points z in a rectangular portion of the
complex plane, do the following:
• let i = 0
4
• While z < M and i < N do
• z = f (z)
• i = i +1
end while
• If i = N , mark the pixel ( j, k) black, otherwise, mark the pixel a hue that
corresponds to i.
end for all.
We apply this algorithm to F(z) with N = 512 and M = 1010 . Figure 3 shows the
escape time where - 6 = Re(z) = 6 and - 6 = Im(z) = 6. Red corresponds to rapid escape
and other hues, running to magenta, correspond to slow escape. Notice the large black
region on the right of center and many smaller regions. There are also fans of black
regions, for example, a sequence of six of them appear to be marching across the red
region in the lower half of that figure. A J [14] script that duplicates the image shown in
Figure 3 is available at [15].
Figure 3. Escape time of F(z) for - 6 = Re(z), Im(z) = 6 .
Figure 4 shows the escape time where - 36 = Re(z) = 36 and - 36 = Im(z) = 36.
Notice the huge fans in a vertical sequence and the complex array of black regions in the
upper left portion of the image. Figure 5 gives an image centered on the origin with width
1. Notice that there appears to be a spiral of fans, five fans per spiral, approaching the
origin. An animation zooming toward the origin may be viewed at [15]. It reinforces that
perception of the spiral. An animation zooming toward z = 5 may also be viewed at [15];
it shows that the repelling fixed point appears to be on the lower right edge of the large
fractal black region that contains z = 1.
5
Figure 4. Escape time of F(z) for - 36 = Re(z), Im(z) = 36 .
Figure 5. Escape time of F(z) for - 0.5 = Re(z), Im(z) = 0.5.
Figure 6 shows more detail of the large fan above and to the right of the origin.
Notice the fan is a fractal array of fans and black regions. The Julia set for this function
seems quite complicated.
6
Figure 6. Escape time of F(z) near 12 + 5i .
Conclusions
By using the Binet formula we have been able to investigate the complex dynamics of the
Fibonacci numbers. There are integer fixed points that are associated with a large basin of
attraction, an edge of that basin, and a spiral of fans. There are additional complex fixed
points, and the escape time images show the Fibonacci numbers have rich complex
dynamics.
References
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[2] Rosen KH. Elementary Number Theory and its Applications, 4th edition. Reading: Addison-Wesley,
1999.
[3] Strayer JK. Elementary Number Theory. Boston: PWS Publishing, 1993.
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quelconque, á coefficients variables. Comptes Rendus des Séances de L'académie des Sciences, 17 (1843)
559-565.
[7] Dickson LE. History of the Theory of Numbers, vol. I, New York: Chessea Publishing Company,
reprint 1971.
[8] Devaney R. Chaos, Fractals and Dynamics, Computer Experiments in Mathematics. Menlo Park:
Addison-Wesley, 1990.
[9] Peitgen H-O, Jürgens H, Saupe D. Chaos and Fractals, New Frontiers of Science. New York: Springer-
Verlag, 1992.
[10] Peitgen H-O, Richter PH. The Beauty of Fractals. Berlin: Springer-Verlag, 1986.
[11] Pickover CA. Computers, Pattern, Chaos and Beauty. Phoenix Mill: Alan Sutton Publishing, 1990.
[12] Reiter CA. Fractals, Visualization and J, 2nd ed. Toronto: Jsoftware Inc, 2000.
[13] Sprott JC, Pickover CA. Automatic generation of general quadratic map basins. Computers &
Graphics 19 (1995) 309-313.
[14] Jsoftware, jsoftware.com
[15] Reiter CA. Escape Time Zooms of the Fibonacci Numbers.
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