History

The first person, that we know of, who researched the topic of fractal geometry was Gottfried W. Leibniz. He was a German philosopher and mathematician of the 17th century, also known for defining the binary numeral system and mathematical analysis. His notion on self-similarity and the principle of continuity declares that nature does not make leaps, i.e. natura non facit saltus. He had a thought on self similarity, which stated that: "the straight line is a curve, any part of which is similar to the whole". He also had an idea about drawing three concentric circles with maximal radii. This action can be repeated an infinite number of times with yet another three circles. Thus he came up with the idea of self-similarity, a discovery that was two centuries ahead of his time.

Two centuries later, the first approximate definition of the fractal was created by Karl Weierstrass in mathematics: "the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere". This was a huge discovery at the time, as, in the 19th century, it was believed that all continuous functions must be differentiable or "smooth" at one point at least. Weierstrass's function showed that the gradient of this function's curve can never be found since it consist of an infinite number of cosine curves. So by increasing the number of terms, you get a really complex and "fuzzy picture".

At the time, this idea got a lot of backlash from the mathematicians, who called it a "dreadful plague" or "pathological monster". However, the negative reception didn't stop mathematicians from creating new monsters. One of the most famous ones was David Hilbert's function, which was defined as a space filling curve. "This curve is one-dimensional but has the property that it completely fills a two-dimensional space." However, this is a great example of a infinitely continuous function that doesn't have a continuous inverse. This is where fractals come in play.

The term fractal was first defined by Benoît B. Mandelbrot a mathematician, also know as the father of fractal geometry. He described it as: "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole". The word comes from the Latin "fractus", which means broken. Mandelbrot is also know for his famous fractal set, also named after him called the Mandelbrot set. He used a computer to arrange images of Julia sets and by analyzing the topology of it, he introduced the Mandelbrot set in 1979. He never felt like he invented anything or, as he put it: "Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.". The equation is the following:

The equation is often written as z= z²+c. The equation follows a chaotic dynamic (more about this later). It is calculated in the number system of complex numbers, which is two dimensional. This is also the equation for Julia Sets as well, which is not a surprise because it's like viewing the set in two different ways.

To get the iconic shape, it's iterated (the repetition of a the function in order to get a sequence of outcomes): z²+c is calculated and then z is assigned to the value of z²+c, and it's "fed" back into equation repetitively.

Mandelbrot also stated that what fractals can help with is the understanding of nature and understanding the geometry of nature.

What is a set of fractals? Sets that show an infinite range of fractalness are defined as mathematical fractals. It's basically a pattern that has no end. No matter the scale of it, it's self-similar and very complex. There are 4 common characteristics of fractals: infinite complexity, zoom symmetry, complexity from simplicity and fractional dimensions. So the following applies to all fractals: "The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes. Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with. Tiny bits of the fractal can look exactly the same as the whole. No matter how many times the shape was magnified, any glimmer of a smooth line would invariably dissolve into a never-ending cascade of corners, packed ever-more tightly together. Weierstrass’ shape had irregular details at every possible scale - the first key feature of a fractal shape. The shape of a fractal can be completely captured by a small list of mathematical mappings that describe exactly how the smaller copies are arranged to form the whole fractal. Fractals are not one-, two- or three-dimensional, but somewhere in-between. Nature seems perfectly happy to use fractional dimensions, so we should be too."

So, simply put, you can find fractals are everywhere, and, if you look carefully, you might be able discover lots of them in your surroundings!

Fractals and resilience

So, what do fractals have to do with resilience? Mandelbrot basically showed how you can describe this chaotic roughness of fractals mathematically. This chaos can be observed everywhere in nature (and I didn't even mention entropy yet, which is partially analogous with this chaotic concept).

In the 1970s, Edward Lorenz, a meteorologist and mathematician, was trying to model weather patterns with computers. He discovered that even a slightest change in variables, even in a simplified version of the model, caused a big change in the results. He called this the butterfly effect. The function that Mandelbrot came up with has the same dynamics.

Chaos and entropy, in a way, are nature's order. In thermodynamics, entropy is a measure of the disorder in a closed system.

He used computer modelling to generate mathematical functions depicting fractals. His work had many critics since other scientists claimed that these patterns can only be observed in nature. At the same time, it must be noted that fractal patterns also appear in architecture, textiles, population shifts and movements of investment and the market.

So, this pattern is not only found in forests and river branches but also when it comes to the spread of lightning and clouds. We are also made up of several fractals such as our lungs, venous system and brain. It is said that even our behaviors and perception show fractal patterns. According to one theory, the reason why people like to spend time in forests is that we are also made up of fractals.

There are examples of other mathematical patterns in nature. For example, pine cones, shells and the iris of our eyes follow the Fibonacci sequence. Fourier transformations help to understand how frequency is translated into perception. So we can say that there is something profoundly rhythmic in the nature of growth and evolution—and in the nature of humans. It seems that, through fractals, everything is interconnected, existing in a common ecosystem. These complex natural systems are balancing between chaos and order, which creates diversity and makes interconnected functioning robust and resilient.

One of my friends told a story about how forests stay resilient at time of forest fire. Every year, researchers analyze how many trees have to be cut down, so that the forest fire doesn't spread throughout the entire forest area. There have to be several roads or hiking trails, so that if a forest fire or some other natural disaster takes place, the forest can survive. This intervention in the forest makes it resilient.

This thought process can be applied to fractals as a whole. The resilience of fractals can be portrayed as a dilemma. If either the randomness or the orderliness is changed, it can make the system resilient. It is also interesting to apply a trilemma to the creation of the fractals from Mandelbrot. "A trilemma can be expressed as a choice among three unfavorable options, one of which must be chosen, or as a choice among three favorable options, only two of which are possible at the same time". Here, the three components are: input, process, output. Without one of these components, the functions can't be generated, can't be modeled by a computer.

In terms of defining a fractal, self-similarity, infinite complexity and dimensionality are what makes fractals what they are. Changing a single one of these variables, will result in a significant change to the whole system, as mentioned above, this is the butterfly effect.

Contradicting thought

Despite all this, I did find an interesting study by Orly R. Shenker writing about fractal geometry not being the geometry of nature. In the study he is strongly critical about Mandelbrot's hypothesis and says "It first analyzes the concept of a fractal. The analysis reveals that fractals are endless geometrical processes, and not geometrical forms. A comparison between fractals and irrational numbers shows that the former are ontologically and epistemologically even more problematic than the latter. Therefore, it is argued, a proper understanding of the concept of fractal is inconsistent with ascribing a fractal structure to natural objects. Moreover, it is shown that, empirically, the so-called fractal images disconfirm Mandelbrot's hypothesis. It is conceded that the fractal geometry can be used as a useful rough approximation, but this fact has no bearing on the physical theory of natural forms." His four main arguments are: 1. fractals are geometrically abnormal and should be viewed as geometric processes instead of objects, 2. Mandelbrot's hypothesis is empirically mistaken, relying on fractal images which are generated by functions or just unnatural due to them being "embedded in essentially non-isotopic abstract spaces", 3. fractal geometry can be used only in some approximation of natural spatial forms, however these systems could be described by other scientific principles, 4. Mandelbrot's hypothesis regarding fractals is a collection of abstract functions, lacking interpretation therefor they can't be taken seriously as a part of science. I believe, to get a deeper understanding of Shenker's point of it's important to read his whole study to decide for yourself who's thought you can resonate with more.

Hallucinogens and fractals

People's consciousness is controlled by several molecules and receptors in the brain which are being studied while hallucinating with an fMRI machine. It turned out that, the brain has a decreased usage in the default mode network during hallucinations. The default mode network in the brain is responsible for worrying, reason, motivation, reflecting on your past and planning your future.

Let's see the process: When you take in a mushroom with the compound of psilocybin for example, your consciousness, fear of mortality, obsession with the ego starts to dissolve, and your brain starts to add information and use the neurology of your brain in different ways to cope leading to a hallucination. This can happen because of the tryptamine core scaffold of LSD is similar to serotonin (the "feel-good hormone") and due to this LSD molecule can bind with the receptor that binds with serotonin and even longer because of it's chemical composition. When the binding happens, the default mode network decreases. This could cause the feeling of ego-dissolution which makes the person feel less alone/not feeling like a stranger and more close to nature. Experienced mediators can achieve the same thing by consciously turning off the default mode network and temporarily vanish their ego. So by the decreased default mode network activity, the brain doesn't work efficiently, and has to make sense of the increased incoming information. The brain "fills in the missing information", and that creates hallucinations. In this state, it uses parts of the brain that haven't been used before.

"Drugs alter our perception by influencing the interaction between predictions and sensory inputs."- So to put it: we could understand hallucinations as our brains pattern matching system going in overdrive: basically it combines our brain's prior expectations to reality. When seeing certain patterns in hallucinations, that is your brain finding similarities in the real world and combining the experience. People can see the underlying geometric patterns that occur in nature. So what we see is that we take the implicit theory geometry that are built into the brain and watch it apply to different patterns of the world.

As I mentioned above, hallucinations can occur during a deep meditative state. In Zen Meditation it's called Makyō. The word has several meanings: realm of demons or uncanny realm (ma, meaning the devil, kyo meaning the objective world). Ma possibly symbolizes Mara, a tempter figure in Buddhist religion. It's also interesting how they interpret illusions: it's said to be a delusion that originates from attaching ourselves to our experiences. Zen meditations warn practicers that these hallucinations can be easily mixed with "seeing the true nature", kenshō, and these distortions should be ignored. Hakuun Yasutani (" fiery and controversial figure in 20th century Zen Buddhism") said it this way: "Makyō are the phenomena–visions, hallucinations, fantasies, revelations, illusory sensations–which one practicing zazen is apt to experience at a particular stage in his sitting. ...Never be tempted into thinking that these phenomena are real or that the visions themselves have any meaning. To have a beautiful vision of Buddha does not mean that you are any nearer becoming one yourself, any more than a dream of being a millionaire means you are any richer when you awake."

In Hindu tradition, this state of hallucination is said to be a very fragile and unstable state, or as they call it: sukshma sharira, or "experience body". They say that this is a form of Māyā meaning pretense or deceit, so a form of illusion of the world. Nyam is the analogous term used in Tibetan, meaning "temporary experience". They say that in this state the energy in your body reorganizes.

"Nyam is set in contrast to tokpa, which means “realization.” Nyam is like pleasant vapor. No matter how good it feels, it always evaporates. Tokpa is like a mountain. It stays. A nyam always has a beginning and an end. One day you soar into the most heavenly meditation, but eventually you drop back to Earth. There are no dropouts with authentic realization."

Summary

In conclusion, it seems like everything is interconnected in some way.

In one podcast of Paul Chek (world-renowned expert in the fields of corrective and high-performance exercise kinesiology, stress management and holistic wellness) he talks about ego-dissolution, intuitive eating and all, and he brings up a really interesting thought. Consciousness is the reference point from which information flows. You are conscious of what is happening within the range of your sensory perception: the reference range of humans goes down deep to an atomic level and expends to vibrational levels, which can be understood as a fractal structure of our understanding of the world.

According to another article the fractal nature of everything can be understood as follows:

"Let’s start with imagining yourself in a park and that in some way you can go out of your body and fly up. After a few meters, you will see yourself as a dot. Flying more up, you will see a park as a dot. Then a city you are in look like a dot, then a county, then you will see the Earth as a dot. After you rose far enough, the whole solar systems and galaxies will become a dot. — That is telling us that everything is a dot from the way up.

Now go back to your body and let’s dive into it deeper. You will find out that you are made of millions of dots, cells. Then go deeper onto the surface of the cell and look around. You will see that it is made out of millions of smaller dots, if you go into one of those dots, you will see atoms. — And yes, that exactly means that there are dots all way down."

This shows that, even in this thought experiment, everything is a fractal in a sense. And all of these are interconnected with each other.

Resources

Web:

1. 2022. 11. 23. https://www.resilience.org/stories/2022-02-01/the-shape-of-things-to-come/

2. 2022. 11. 23. https://news.globallandscapesforum.org/43195/fractals-nature-almost-all-things/

3. 2022. 11. 23. https://plus.maths.org/content/uncoiling-spiral-maths-and-hallucinations

4. 2022. 11. 23. https://www.quantamagazine.org/a-math-theory-for-why-people-hallucinate-20180730/

5. 2022. 11. 23. https://slatestarcodex.com/2015/08/28/mysticism-and-pattern-matching/

6. 2022. 11. 23. https://fractalfoundation.org/resources/what-are-fractals/

7. 2022. 11. 23. https://www.youtube.com/watch?v=xliky7YfyTc&t=187s

8. 2022.11.26. https://www.lionsroar.com/just-when-you-think-youre-enlightened/

9. https://www.youtube.com/watch?v=7MotVcGvFMg

Articles:

1. Oldrich Zmeskal, Petr Dzik, Michal Vesely, Entropy of fractal systems, Computers & Mathematics with Applications, Volume 66, Issue 2, 2013, ISSN 0898-1221,

2. Orly R. Shenker, Fractal geometry is not the geometry of nature, Studies in History and Philosophy of Science Part A, Volume 25, Issue 6, 1994, ISSN 0039-3681

Books:

Mandelbrot BB. The Fractal Geometry of Nature. Updated and augmented ed. New York: W.H. Freeman; 1983.

Duarte, German A. (2014). Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces. Transcript-Verlag. ISBN 9783837628296.

Podcast:

https://open.spotify.com/episode/3GVeHaPyypmXb0NOuY8OAX?si=c5e27d539cd24ffb

Fun stuff

three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis: https://en.wikipedia.org/wiki/Mandelbulb

fractal generator: http://usefuljs.net/fractals/