Geometry of the heavens

A short essay on the beauty of fractals, complexity, Mandelbrot set and snowflakes.

---

Last week, it snowed in Ames again. For obvious reasons of lethargy and aversion to cold Iowan gusts, I had almost given up on picking up my lunch from a nearby restaurant. Instead, I wanted to order to get it delivered to my lab. To save myself a few bucks in extra fee and delivery charges, I wrapped myself in a bunch of layers, off to pick up my order of Tofu stir-fry. The snowfall was flaky and not heavy, characterized by small flakes of interesting patterns. Since the restaurant is a 10 minute walk from my lab, I had to do something to distract my mind from fixating on how terrible the weather was -- I started to observe the flakes depositing and melting on my jacket sleeve. Out of tens of those flakes, the one which motivated me to write this essay on fractals was perfectly symmetrical and hexagonal. To capture it before it would melt away, I fumbled my phone from my back-pocket the moment I saw it. Fig. 1 is what I managed to click with my crappy phone. Fig. 3 compares this snowflake with an artificial lab grown snow crystal. 

Figure 1: A perfectly symmetrical and hexagonal snowflake.

I. Fractals in nature

There is maybe nothing as beautiful as a perfectly symmetrical snowflake. The secret to its beauty is self similarity. The property of self-similarity, in simple terms, means that if you zoom in on a self-similar object, it will look similar to the original shape. This property of being similar at smaller, subdivided scales is called fractality, and the geometrical object displaying the structural/organizational similarity across the whole range of scales is called a fractal. If you zoom into a fractal structure, you will end up seeing similar patterns as the original one at all magnifications. The smallest unit you observe is similar to the whole. For idealized mathematical objects, the self similarity may be across an infinite range of scales (Fig. 2b), such as in the Koch snowflake which can be constructed by starting with an equilateral triangle, then recursively altering each line segment as shown in Fig. 2a. In real life and in nature the self similarity is only across a range of scales as evident in both the lab grown snow crystal as well as the one deposited on my jacket sleeve. In Fig. 3, the repeating hexagonal structure at the end of the diagonals is easily observed in the lab grown crystal. The similarity between the real snowflake and the artificial crystal is also clearly visible. 

Figure 2a: The Koch snowflake fractal geometry.	Figure 2b: Self similarity across an infinite range of scales in an ideal Koch snowflake.

Fractals are ubiquitous in nature. Apart from snowflakes, we see fractals in the organization of the most common objects we come across in daily life. Fractals appear in nature in trees (Fig. 4a), mountain ranges(Fig. 4b: a shot of mountain ranges surrounding Kashmir valley branching into mathematically and aesthetically beautiful patterns), our bodies (Fig. 4c and 4d: lungs, retina, complex network of arteries and veins), crystal structure of many elements (Fig. 4e: dendritic copper crystals forming), ferns, leaves (Fig. 4f: fractal structure of a leaf), broccoli (Fig. 4g: closeup photo of Romanesco broccoli), sea-shells (Fig. 4h: a nautilus shell), galaxies, and coastlines etc. 

 

Figure 3: Similarity between the lab grown snow crystal
and the real snowflake. The fractal nature of the hexagons
 is also clearly observed in the lab grown crystal. 
Artificial crystal from
 The Unbelievable World of Snowflakes by David DeFranza

Figure 4: Some fractal patterns found in nature. The references of the images used are listed at the end of the essay, in the references section. 

II. Simplicity: the origin of complexity

The most striking feature of all the complexity and beauty of fractals is that they can be constructed or governed by relatively simple mathematical rules. These mathematical rules involved are recursive in nature which means that a set of variables (geometric features) are updated repeatedly every iteration resulting in uniquely complex patterns. 

The Mandelbrot set

As an illustration, let me introduce the Mandelbrot set which is the set of complex numbers \(c\) for which the function \[f_c(z) = z^{2}+c\] does not diverge when iterated from \(z=0\) . We start with \(z=0\) and vary \(c\), both in the complex plane, and trace the orbit of the recursion every iteration. The values of \(c\) for which the resulting function does not escape to infinity forms the region known as the Mandelbrot set. It is named after Polish-born French-American mathematician Benoit Mandelbrot, who is considered as the pioneer of fractal geometry. The following function computes the Mandelbrot set in MATLAB®,

%--------------------------------------------------------------------------\
% Function to compute Mandelbrot set in Matlab                             |
% Shujaut H. Bader                                                         |
% Ph.D. Candidate, Aerospace Engineering Department, Iowa State University |
% -------------------------------------------------------------------------/
clc
clear
function mandelbrot(n, iter)
xleft = -2; xright = 1;
ybottom = -1.5; ytop = 1.5;

[x,y] = meshgrid(linspace(xleft, xright, n), linspace(ybottom, ytop, n));

c = x + 1i * y;
z = zeros(size(c));
k = zeros(size(c));

for ii = 1:iter
z = z.^2 + c;
k(abs(z) > 2 & k == 0) = iter - ii;
end

figure(1)
imagesc(k),
colormap jet
axis square

In Fig. 5a, the region colored in blue represents the Mandelbrot set -- the set of all \(c\) for which the function \(f_c(z)\) does not escape to infinity when \(z=0\). The Mandelbrot set is self-similar under magnification in the neighborhoods of some special points. In strict sense, it is quasi-self-similar as some of the arbitrarily chosen small scales reveal slightly different versions of the original set. Figure 5b shows the self-similarity of the Mandelbrot set upon zooming in on the series of blue circles attached to the main cardioid (the cleaved blue region) on the right side of the plot. It is astonishing how a simple equation gives rise to such level of beauty and complexity!        

Figure 5a: The Mandelbrot set. Plot generated as the output of the code listed above.	Figure 5b: Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction.

The Barnsley fern

Another fascinating example of simple recursive equations giving rise to intricate and complex patterns is the Barnsley fern named after British mathematician Michael Barnsley who first described it in his book Fractals Everywhere. The computer generation of the Barnsley fern uses a simple set of four transformations acting on the first point drawn at the origin (\(x_0 = 0, \ y_0 = 0\)),

\[x_{n + 1} = 0, \ y_{n + 1} = 0.16 y_n \ (Chosen \ 1\% \ of \ the \ time. \ Generates \ stem)\\ x_{n + 1} = 0.85 x_n + 0.04 y_n, \ y_{n + 1} = −0.04 x_n + 0.85 y_n + 1.6. \ (Chosen \ 85\% \ of \ the \ time. \ Generates \ successively \ smaller \ leaflets.) \\ x_{n + 1} = 0.2 x_n − 0.26 y_n, \ y_{n + 1} = 0.23 x_n + 0.22 y_n + 1.6. \ (Chosen \ 7\% \ of \ the \ time. \ Generates \ largest \ left-hand \ leaflet.)\\ x_{n + 1}= −0.15 x_n + 0.28 y_n, \ y_{n + 1} = 0.26 x_n + 0.24 y_n + 0.44. \ (Chosen \ 7\% \ of \ the \ time. \ Generates \ largest \ right-hand \ leaflet.) \]

Upon applying these simple transformations with the starting point \(x_0,y_0\) at the origin, the new points are iteratively computed resulting in a very complex structure of a fern. The resulting fractal-fern computed in MATLAB® is presented in Fig. 6. The computer code of the Barnsley fern is available upon request.

Figure 6: Barnsley fern generated by 4 recursive relations in MATLAB®.

III. Philosophical implications

As it has been established that complex things can emerge from simple rules, we can extrapolate this idea to conjecture about the origin of all the diversity and complexity present in the universe. Creationists often claim this diversity and complexity to be the result of a very complex deity who possesses all the anthropocentric qualities like consciousness, love and wrath etc. Some creationist groups even describe the physical state/appearance of this deity/entity. They fail to understand and appreciate the power of emergence and how complex forms emerge from simple laws. A reasonable argument would assume this initial entity/whatever-it-is to be immensely simple unlike a complex, conscious deity. The creationist argument simply begs the question. It should not be hard to understand that the events like Big-bang, appearance of the DNA, or the emergence of consciousness potentially do not necessitate a complex deity. An attempt to answer and explain the complexity in the universe with an even more complex entity, a God-like being, is simply changing your goalpost and does nothing to advance our scientific knowledge. The question of God's complexity evident from the attributes that religions ascribe to him, should not be a stage where our questions must cease. Invoking God to explain the complexity of the universe is analogous to invoking an immensely complex, supernatural explanation for any natural phenomenon. From the history of science and philosophy, we know such explanations did not stand the test of time. Eclipses are not the moments of copulation between the sun and the moon that spawns stars, we no longer need epicycles, epilepsy is not caused by demonic possession, fever is not caused by the heat of Hell etc. For all these examples, we have simple, naturalistic explanations. The presence of zero entropy state at the beginning of the universe implies that the complexity was non-existent at that moment. Any reasonable scientific hypothesis about the nature of the entities present at the moment of the Big-bang must, therefore, necessarily be written down with these constraints in mind -- zero entropy, zero complexity. Simple, unintelligent rules and entities can potentially create all the diversity of the physical universe and the complex life forms present within it. The point to stress upon is that the existence of complexity does not necessarily point to a complex designer. Complexity does not require a conscious programmer, or intelligence of any sort. All it requires is a bunch of simple rules and time. Intelligence and consciousness then emerge, not the other way round.  

IV. Implications in other scientific disciplines

In many physical phenomena like chaos, turbulence, and disorder, fractal geometry is at play. In turbulent flows, the experimental evidence points towards a strong dynamic self similarity in the inertial range i.e. the transfer of energy between scales (in this range) occurs in a self-similar manner, that is similarly at every scale. This observation suggests an astonishing conclusion that chaos and turbulence must be born out of the same underlying processes as complex fractal networks of arteries, fractal coastlines and mountains etc. How beautiful and revealing it is to realize that the complexity of flow in an ever-branching human lung can be understood as a mirror image of chaotic flow of a river. Fractal order lies at the core of both of these processes. 

We assume the complexity of human body as a manifestation of some sophisticated instructions but as I have stated before, all the involved instruction for growth and development must have a simple precursor, for all complexity emerges from simple, unintelligent laws. 

 

---

Cite as

Bader, Shujaut H., “Geometry of the heavens: A short essay on the beauty of fractals, complexity, Mandelbrot set and snowflakes." Backscatter, January 26, 2021. https://backscatterblog.blogspot.com/2021/01/geometry-of-heavens.html

References

Figure 2: 

a). António Miguel de Campos/Wikimedia Commons

b). PD-self/Wikimedia Commons

Figure 4:

a). silvergreen.deviantart.com

b). screenshot from Google Earth

c). https://fractalfoundation.org/

d). National Science Foundation - Culturally Situated Design Tools (CSDT) 

e). Paul/Wikimedia Commons

f). Curran Kelleher/flickr

g). Manuel Noah Angeja/flickr

h). istockphoto.com/flamingpumpkin

Figure 5:

b). Cuddlyable3 (Transferred by TeleComNasSprVen)/Wikimedia Commons