The Thumbprint of God: The Mandelbrot Set

The Thumbprint of God: The Mandelbrot Set

by

Bradly Gassner

Calculus III

Dr. Barry Bookout

October 14, 2012

Bradly Gassner

Calculus II

Fall 2012

Dr. Barry Bookout

The Thumbprint of God: The Mandelbrot Set

Mathematics is the only tool scientists and engineers have to describe the world in which

we live. We use the geometric axioms of Pythagoras, Euclid, and their successors to model our

surroundings. Triangles, circles, squares, and their higher dimensional counterparts together

make a full complement of tools with which we can analyze structures we encounter. However

for years mathematicians viewed the irregularity of natural systems such as clouds, mountains,

and coastlines as indescribable with their tools. At best, clouds were modeled by spheres,

mountains by cones, and coastlines as circles (Mandelbrot 1). This all changed with the

development of fractal geometry in the 1970s, pioneered by the French mathematician Benoit

Mandelbrot.

Mandelbrot’s fractal geometry centers on the very basic idea of self-similarity. Many

structures in the previously mathematically indescribable realm of nature exhibit this

feature. See figure 1 for a representation of a fern leaf. Notice the self-

similarity the structure exhibits on many levels. Smaller fronds

resemble the structure of the leaf as a whole. Still smaller fronds

upon the fronds resemble the leaf as a whole. The structure of

leaves upon fronds upon leaves continues for several “levels” down

into the structure of the fern. Mandelbrot’s revolution in geometry allows for a quantification of

Figure 1: A Fern Leaf

the structures that we see in these objects. We have discovered other, purely mathematical things

which are created by continuing this sequence ad infinitum.

Repetition of this kind is what a mathematician would call iteration. Iteration is the

process of taking the output of a function and feeding it into the function a second time, taking

that output and inputting it again. Stepping aside from the implications of this new geometry to

the potential mathematical description of nature, we turn our discussion to a more purely

theoretical mathematical result.

Gaston Julia was a French mathematician whose ideas came before the time of

Mandelbrot, but whose work on iterated rational functions laid the path upon which Mandelbrot

built his new geometry. The most popular type of Julia’s iterated rational functions is given by

the family of quadratic polynomials of the formf c ( zn )=zn−12+c, where the constant c is a

complex number. The famous Julia Sets are sets of complex numbers that remain bounded as the

function is iterated, or as the outputf c ( z ) is repeatedly input into the equation asz. These are

simply maps existing in the complex plane of all the points

whose coordinates( z ) when input into the function keep the

output values of the function bounded, or those that do not tend

to infinity. As is the case with many iterated fractal systems,

very simple rules can lead to wonderfully complex results.

In figure 2, we see a beautiful Julia Set corresponding to

the constant as shown. The dark areas are the set of points for which the iterated function stays

bounded. These are the values belonging to this particular set. An infinite amount of these sets

exist, corresponding to every complex numberc. The most interesting of these sets, many similar

to the one shown, havec values lying within or about the unit circle on the complex plane. These

Figure 2: A Julia Set with c = -0.391-0.587i

structures have a boundary of infinite length and infinitely complex structure on every scale of

magnification. No matter how far we zoom in, we will continually see new detail. The

boundaries of these areas are very different from the curves of normal rational functions. They

are continuous and yet non-differentiable everywhere (Edgar 27). Calculating whether the value

of the function at every point in the complex plane is bounded or not is a very computationally

intensive task.

Early attempts were made to plot these points on graph paper, with limited results

(NOVA). At points near the boundary, the function may require hundreds of iterations before the

true nature of the function can be determined. As a result, images of these beautiful sets were not

available until the advent of the computer (NOVA). Benoit Mandelbrot was working as a

research fellow at IBM’s Thomas J. Watson Research Center while researching these fractal

forms in 1977 (Mandelbrot iii). It was his close proximity to the development of the computer

which allowed him to plot these immensely complex functions, and as such, he was among the

first to see them.

Mandelbrot imagined a set of complex numbers similar in definition to the Julia sets, but

with one distinction: he varied the constantc of Julia’s

function as input into the equation by lettingc equal the

complex coordinates of the point in question in the

complex plane. Plotting the resulting function Z=Z2+c for

every point in the complex plane yields a set that is truly

beautiful. The result has become the poster child for fractal

geometry: the famous Mandelbrot Set, or M-set.

Figure 3: The Mandelbrot Set

This Mandelbrot Set is defined as the set of complex numbers c∈C for which the

iterated sequence c , c2+c ,(c2+c)2+c ,… does not tend to infinity (Branner 75).

M= {c∈C|c , c2+c , ( c2+c)2+c , …↛� ∞ }

Because of the fact that thec values in the function of the Mandelbrot Set are

continuously changing as the coordinates of the corresponding points in the complex plane, the

fractal has a close connection to the previously discussed Julia sets. Consider several of them,

figures 4-7:

One distinguishing feature that mathematicians call “connectedness” can be obtained

from looking at the very center of each set. The center of symmetry is the origin of the complex

plane. If the point there is part of the set, the set is deemed “connected.” If it is not part of the set,

it is termed “disconnected.” Instead of visually checking if the point at the origin is part of the

set, we may evaluate the fate of the coordinate at that point0+0i as we evaluate the Julia set’s

function. Once again, if the function there tends to infinity, it is not part of the set. If it stays

bounded, it is part of the set. Imagine evaluating the connectedness of every single Julia set, and

plotting the result on the complex plane. The result is, again, the M-set. It is an “atlas” to the

Julia sets. Every point that is a member of the set, every coordinate colored black in figure 8,

points to a corresponding connected Julia set. These points are indeed the values ofc that are

used in the Julia set’s function. In figure 8, several values ofc have been selected and their Julia

Figure 4 Figure 5 Figure 6 Figure 7

sets plotted. Notice the connected Julia sets that originate from within the set, and the

disconnected Julia set that comes from the region just outside the M-set.

The Mandelbrot set, like other fractals, is self-similar, or

in this particular case, quasi-self-similar. The M-set contains

small copies of itself which in turn contain smaller copies of the

set, and so on. In actuality, however, every mini Mandelbrot has

its very own pattern of external decorations, every one different

from every other (Branner 76). The particular portion of the M-

set on display in figure 9 can be found in the region of thec value

given in the caption. It is located near the spindly projection

emanating from the left side of the set. Many small-scale

features in this region have spindly characteristics. In addition

to admiring beautiful pictures of the Mandelbrot, there are

quantitative ways of measuring it. One of these measures is the

Hausdorff dimension, a quantity that describes the “crinkliness”

of fractals.

The Hausdorff dimension is not a spatial dimension. Surely, the M-set resides in a two

dimensional space. This is a quantity which describes something different entirely, but can be

derived from simpler examples. The idea behind the dimension calculation is rather simple: by

what factor do the smaller parts fit into the original whole when the dimensions of the original

are modified? For example, when the dimensions of a cube are increased by a factor of two,

there are eight of the original cubes which can be placed in the new cube. We can quantify this

by the formula for the Hausdorff dimensionX :

Figure 9: Miniature Mandelbrot atc = -1.62917,-0.0203968

Figure 8: The M-set and Julia Sets

X=log(N )log(P)

WhereN is the number of increase in units andP is the increase in length. Applying the formula

for the Hausdorff dimension, N = 8, P = 2. The number of increase in units is eight when the size

is doubled. So, the dimension of the cube is log (8)log (2)

=3. This agrees with our understanding of

the cube occupying three spatial dimensions (Praught). We can also apply this to our fractal, and

get a measure of its dimension.

When we apply this formula to our M-set, it has been shown that it has a Hausdorff

dimension of exactly two (Shishikura). This is not common among the dimensions of many other

fractals. Most fractals sport Hausdorff dimensions of non-integer values. The M-set is truly

unique in this regard.

We have discussed the origins of fractal geometry and the creation of Julia sets and the

venerated Mandelbrot set. These self-similar fractals of enormous complexity arise from very

simple rules. In our case, the repeated iteration of a simple quadratic polynomial leads to a

visually and mathematically striking outcome. The idea that such a beautiful and marvellously

complex set can exist within the domain of complex numbers is breathtaking. If one is feeling

hard-pressed to find an example of beauty in mathematics, look no further. These fractals,

however lovely, simply existed before mankind stumbled upon them. This is an unavoidable

result of a sufficiently powerful mathematics. The feeling of joy and helplessness people

experience when beholding the absolute beauty of the set compels some to refer to the

Mandelbrot Set as simply “The Thumbprint of God.”

Works Cited

Branner, Brodil. “The Mandelbrot Set.” Proceedings of Symposia in Applied Mathematics. Volume 39, 75-105. American Mathematical Society, 1989. Print.

Edgar, Gerald. Classics on Fractals. Boulder: Westview Press, 2004. Print.

“Fractals: Hunting the Hidden Dimension” NOVA: PBS Home Video, 2011. DVD.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1977. Print.

Praught, Jeff. “Fractal Dimension.” <http://www.upei.ca/~phys221/jcp/Fractal_Dimension/fractal_dimension.html> Accessed 11 October 2012

Shishikura, Mitsuhiro. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Cornell University Library Online: 1991.< http://arxiv.org/abs/math/9201282>, Accessed 11 October 2012

Figure 1 obtained from <http://www.home.aone.net.au/~byzantium/ferns/fractal.html>, Accessed 11 October 2012

Figure 2 obtained from <http://yozh.org/2011/03/14/mset006/>, Accessed 11 October 2012

Figure 3 obtained from <http://warp.povusers.org/Mandelbrot/pic2.png>, Accessed 11 October 2012

Figure 4 obtained from <http://math.fullerton.edu/mathews/c2003/juliamandelbrotset/ JuliaMandelbrotPlates/Images/ColorPlate5.small.gif>, Accessed 11 October 2012

Figure 5 obtained from <http://puzzlezapper.com/aom/mathrec/julia5.png>, Accessed 11 October 2012



Figure 8 obtained from <http://paulbourke.net/fractals/juliaset/julia_mandel.gif>, Accessed 11 October 2012

Figure 9 obtained from <http://paulbourke.net/fractals/mandelbrot/b3125.gif>, Accessed 11 October 2012

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The Thumbprint of God: The Mandelbrot Set