The Romans worshipped the idea of order, the Greeks the idea of chthonic deities, and the allure of formless power living according to its forbidden rules. It was these rules and patterns that the brilliant scientist, theorist, and mathematician Benoît B. Mandelbrot found in the real world around us. Mandelbrot discovered to science the harmony of chaos. Thanks to him, for the last 40 years, we have been looking at trees and Romanesque cauliflower, cities, and the stock market crash with different eyes.
Unfortunately, there is no definition of fractals that is both simple and accurate. Like so many things in modern science and mathematics, discussions of “fractal geometry” can quickly go over the heads of the non-mathematically-minded. It is a real shame because there is profound beauty and power in the idea of fractals. Classical, or Euclidean, geometry is perfectly suited for the world that humans have created. But if one considers the structures present in nature, which are beyond the realm of smooth human construction, many of these rules disappear. Clouds are not perfect spheres, mountains are not symmetric cones, and lightning does not travel in a straight line. Nature is rough, and until very recently, this roughness was impossible to measure. The discovery of fractal geometry has made it possible to mathematically explore the kinds of rough irregularities existing in nature.
These shapes have something in common
Clouds, mountains, coastlines, cauliflowers, and ferns are all-natural fractals. These shapes have something in common – something intuitive, accessible, and aesthetic. They are all complicated and irregular: the sort of shape that mathematicians used to shy away from in favor of regular ones, like spheres, which they could tame with equations. The chaos and irregularity of the world – Mandelbrot referred to it as “roughness” – is something to be celebrated.
What are fractals, and why is there so much buzz about them? A Fractal is a type of mathematical/geometric shape that has the property of self-similarity. In other words, the shape can be divided into parts that are reduced-size copies (at least approximately) of the whole. The term Mandelbrot set refers both to a general class of fractal sets and to a particular instance of such a set. Plotting the Mandelbrot set is relatively simple. Mandelbrot set is one of the most famous fractals, including outside mathematics, thanks to its color visualizations. Its fragments are not strictly similar to the original set, but certain parts look more and more like each other when repeatedly zoomed in. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization, mathematical beauty, and motif. During the 1980s, people became familiar with fractals through those weird, colorful patterns made by computers. But few realize how the idea of fractals has revolutionized our understanding of the world and how many fractal-based systems we depend upon.
Look closely at a fractal, and you will find that the complexity is still present at a smaller scale. A small cloud is strikingly similar to the whole thing. A pine tree is composed of branches which in turn are composed of other branches. A tiny dune or a puddle in a mountain track has the same shapes as a giant dune and a lake in a mountain gully. The essence of fractals is in self-similarity: their geometrical structure remains approximately the same at any observation scale. Therefore, did fractals exist before?
Yes, as a principle of the world structure, they have always existed. It is worth mentioning that only recently, the concept of fractal appeared when it was realized that self-similarity is a universal property of nature. Since then, the rapid development of fractal geometry began. Fractals were found in almost all natural phenomena and processes. Fractal models are used, as mentioned above, in medicine both for heart rhythm tracking in cardiology and for early cancer diagnostics. In geology and soil science; in material science for studying destruction processes; in nuclear physics and astronomy for studying elementary particles and various processes on the Sun; in computer science for data compression and improving Internet traffic; for analyzing fluctuations of market prices in the economy, whether in meteorology; in chemistry, art history – the list can be continued endlessly.
Computing presentation of fractal structure
In 1961, IBM looked to Mandelbrot to provide a new perspective on one of their problems. The task was simple enough: IBM was involved in transmitting computer data over phone lines, but a kind of white noise kept disturbing the flow of information—breaking the signal. In other words, it was necessary to understand how to predict the occurrence of noise interference in electronic circuits when the statistical method proves ineffective. Since he was a boy, Mandelbrot had always thought visually, so instead of using the established analytical techniques, he instinctually looked at the white noise in terms of the shapes it generated—an early form of IBM’s now-renowned data visualization practices. A graph of the turbulence quickly revealed a peculiar characteristic. Regardless of the scale of the graph, whether it represented data over the course of one day or one hour or one second, the pattern of disturbance was surprisingly similar. There was a larger structure at work.
The problem was familiar to Mandelbrot. He recalled the advice his mathematician uncle, Szolem Mandelbrojt, had given him years ago in France—an attempt to make something of the obscure theories of iteration established by French mathematicians Pierre Fatou and Gaston Julia. Their work intrigued mathematicians worldwide and revolved around the simplest of equations: z = z² + c. With a variable of z and parameter of c, this equation maps values on the complex plane—where the x-axis measures the real part of a complex number and the y-axis measures the imaginary part (i) of a complex number. At the time of the advice, Mandelbrot couldn’t find any breakthrough, but the intellectual freedom he found at IBM allowed him to engage in this new project fully.
While working at IBM, Benoit Mandelbrot went far beyond the purely applied problems of the company. He worked in linguistics, economics, aeronautics, geography, physiology, astronomy, physics. He liked to switch from one subject to another, to study different directions. In his economics study, Mandelbrot discovered that apparently random fluctuations in price could follow a hidden mathematical order over time that is not described by standard curves. He took up the study of cotton price statistics over a long period of time (more than a hundred years). Price fluctuations over the course of a day seemed random, but Mandelbrot was able to figure out a trend in prices. He traced the symmetry in long-run price fluctuations and short-run fluctuations. Mandelbrot tried using fractal mathematics to describe the market – in terms of profits and losses traders made over time and found it worked well. This discovery came as a surprise to economists. In fact, he applied the rudiments of his (recursive) fractal method to solve this problem. Mandelbrot came up with the term fractal (from the Latin fractus, meaning “broken, shattered”) and first used the term in 1975.
In 1980, building on IBM’s technology, Mandelbrot used high-powered computers to iterate the equation or use the equation’s first output as its following input. Mandelbrot crunched and manipulated the numbers a thousand times over, a million times over, and graphed the outputs with these computers. The result was an awkwardly shaped bug-like formation, and it was perplexing, to say the least. But as Mandelbrot looked closer, he saw the detailed edges of this formation held smaller, repeating versions of the larger bug-like formation. What’s more, every smaller version held more complex detail than the previous version. These structures were not exactly alike, but the general shape was strikingly similar; only the details differed. The specificity of these details, it turned out, was limited only to the power of the machine computing the equation, and similar shapes could continue forever—revealing more and more detail on an infinite scale. This was a definite geometry, there were rules and parameters to this roughness, but it was a form of geometry previously unidentified by the scientific community. Instantly, Mandelbrot knew he was onto something. He saw unquestionably organic structures in the details of this shape and quickly published his findings.
Immediately after Mandelbrot’s theory of fractals was announced, there was an increased interest in its research. After attending Benoit Mandelbrot’s lecture in Budapest, Nathan Cohen was inspired by the idea of the practical application of his knowledge. However, he did it intuitively, and a happenstance helped his discovery. As an amateur radio operator, Nathan aimed to develop an antenna with the highest possible sensitivity. He realized that an antenna made according to a fractal pattern had high efficiency and covered a much wider frequency range than classical solutions. In addition, the shape of the antenna in the form of a fractal curve allows the geometric dimensions to be significantly reduced. Nathan Cohen even derived a theorem proving that it is sufficient to give it the shape of a self-similar fractal curve to create a broadband antenna.
The future co-founder of the legendary Pixar studio, Loren C. Carpenter, began working at Boeing Computer Services in 1967. His duties included developing images of projected airplanes. He had to create pictures of new models, showing future planes from different sides. At some point, the future founder of Pixar Animation Studios had the creative idea to use an image of mountains as a background. Today any schoolkid could do this, but in the late seventies, computers couldn’t handle such complicated calculations – there were no graphic editors, to say nothing of 3D applications. In 1978, Loren happened to see in a store a book by Benoit Mandelbrot, Fractals: Form, Hasard and Dimension. What caught his attention in this book was that the author gave many examples of fractal forms in real life and proved that a mathematical expression could describe them. It was Loren Carpenter who was one of the first to try out the fractal method in practice. Loren was able to visualize a realistic image of the mountain system on his computer. In other words, he used formulas to draw a pretty recognizable mountain landscape.
The principle Loren used to achieve the goal was very simple. It consisted of dividing a larger geometric figure into smaller elements, and those in turn divided into similar smaller figures. Using larger triangles, Loren broke them up into four smaller ones and then repeated this procedure over and over again until he had a realistic mountainous landscape. In this way, he succeeded in being the first artist to use the fractal algorithm in computer graphics to build images. As soon as the work became public, enthusiasts around the world picked up the idea and began to use the fractal algorithm to imitate realistic natural forms. Many significant advances in the science of fractals became possible only using computational mathematics methods using modern computers. One can say that the emergence and development of computers made it possible to obtain a relatively complete understanding of various fractal structures and the causes of their occurrence. At present, with the help of comparatively simple algorithms, it is possible to create three-dimensional images of fantastic landscapes and shapes, which can transform in time into even more fascinating pictures.
What emerged was a geometry of the cosmos—one that broke all Euclidean laws of the man-made world and deferred to the properties of the natural world. If one identified an essential structure in nature, Mandelbrot claimed, the concepts of fractal geometry could be applied to understand its component parts and make postulations about what it will become in the future. This new way of viewing our surroundings, this new perception of reality, has since led to a number of remarkable discoveries about the worlds of nature and man and has shown that they are not as disconnected as once thought.
The fractal mathematics Mandelbrot pioneered, together with the related field of chaos theory, lifts the veil on the world’s hidden beauty. It inspired scientists in many disciplines – including cosmology, medicine, engineering, and genetics – and artists and musicians, too. Fractal mathematics has many practical uses, such as producing stunning and realistic computer graphics, computer file compression systems, the architecture of the networks that make up the internet, and even diagnosing some diseases. Fractal mathematics cannot predict the significant events in chaotic systems – but it can tell us that such events will happen. As such, it reminds us that the world is complex – and delightfully unpredictable.
After Mandelbrot turned to the theory of iterations of rational mappings of the complex plane, he was the first person to discover the beautiful and mysterious set, which later got his name. The most famous visual fractal is the Mandelbrot Set, first described as an equation by French mathematician Pierre Fatoux in 1905 and visualized by a diagram in 1980. The Mandelbrot set now plays the role of a constant in the field of nonlinear complex iterative processes, just as the numbers π and e do in number mathematics. The Mandelbrot set is one of the most complicated objects in modern mathematics. An altered representation of the Mandelbrot Set, augmented with iterations and rotated at a certain angle, is called the “Buddhabrot”. It is sometimes called the “Thumbprint of God” because of its resemblance to the meditating Buddha. At the same time, many aspects of the human body are described by Eastern religions as the energy center (in the navel area), and the last two human chakras can be seen in the fractal figure.
Z<->z²+c. This apparently simple mathematical equation, made possible for viewing with the current day’s more powerful computers, demonstrated that infinity is, in fact, real and truly exists. Now, this is deep. It brings up questions of God and eternity and makes us take another look at our own reality and what we perceive it to be. The Mandelbrot Set is real, even if we can’t actually touch it. Because it reflects and demonstrates the physical world that is all around us, and maybe even the spiritual world that exists in the same place, yet somehow also elsewhere. Just like you and me, it exists, and it proves there is infinity. No matter where you explore it, it takes you to another possibility and a different outcome. It shows that we have freedom in this Universe and can take any paths in our lives out of Infinity possibilities. Their images take our minds to another dimension, familiar yet alien, and they invoke thoughts about our physical world as well as spiritual.
Why are fractals important? The application and use of fractals have been increasing with the increase of computer power. The timing and sizes of earthquakes and the variation in a person’s heartbeat, and the prevalence of diseases are just three cases in which fractal geometry can describe the unpredictable. Another is in the financial markets, where Mandelbrot first gained insight into the mathematics of complexity while working as a researcher for IBM during the 1960s. Fractals have shown their usability in various domains from biology and medicine, image processing, art.
Biology and healthcare are only some of the latest applications of fractal geometry. Fractal patterns have appeared in almost all of the physiological processes within our bodies. For ages, the human heart was believed to beat in a regular, linear fashion, but recent studies have shown that the proper rhythm of a healthy heart fluctuates radically in a distinctively fractal pattern. Blood is also distributed throughout the body in a fractal manner. Researchers are using ultrasound imaging to identify the fractal characteristics of blood flows in both healthy and diseased kidneys. The hope is to measure the fractal dimensions of these blood flows and use mathematical models to detect cancerous cell formations sooner than ever before.
Today, we have merely scratched the surface of what fractal geometry can teach us. Weather patterns, stock market price variations, and galaxy clusters have all proven to be fractal in nature, but what will we do with this insight? The possibilities, like the Mandelbrot set, are infinite.
Mandelbrot’s main merit, however, is something else: He took the liberty of looking around and discovering such structures everywhere. He discovered what was always in plain sight. And such discoveries are the most challenging.
While Mandelbrot will always be known for discovering fractal geometry, he should also be recognized to bridge the gap between art and mathematics and show that these two worlds are not mutually exclusive. His creative approach to complex problem solving inspires humans with a strong belief in the power of perspective. Decades after his discovery of the Mandelbrot set, data visualization continues to provide fresh and unexpected insights into some of the world’s most challenging problems by altering our perspective, challenging our preconceptions, and revealing connections previously invisible to the eye. Mandelbrot’s cosmological and philosophical views from a historical perspective are well reflected in his unpublished note “Two heirs to the Great Chain of Being”: “We see the first members of a progressive relationship of worlds and systems, and the first part of this infinite progression enables us already to recognize what must be conjectured of the whole. There is no end but an abyss … without bounds.”
Fractal forms are unconsciously or intentionally used in an embodiment of the sacral idea in material and mentally created objects both in ancient times and in modern culture. Hindu, Buddhist, Christian, and some other religious and world perceptions are based on the fractal model of Being, which reflects fractal structures of nature and society. In the XXI century, with the help of computer modeling and high-tech research of macro and microcosmos, man has found an opportunity to learn and construct the fractal codes of the Universe. However, long before digital innovations came into our lives, the fractal code of Genesis was somehow displayed in various sacred texts and sacred images of ancient mythologies and religious teachings.
The most ancient fractals in human history are ideas and artifacts of ontological order, religious and mythological formulas of being. In theological pictures of the world, fractality refers, first of all, to the essence of God himself. For example, Krishna, the main incarnation of the divine in Hinduism, contains all worlds and the entire Universe. In Tibetan Buddhism, an important place is occupied by the Kalachakra, one of the ideas of which is the unity of being, the identity of macrocosm and microcosm, the Universe and man. In this, the Buddha embodies the highest principle of unity of all things. Buddha is everywhere and in everything, in each of the multitudes and multitudes of beings walking the long path of rebirth to spiritual perfection (every “enlightened” person is a Buddha). At the same time, the whole Universe is the Buddha’s Body, and the Buddha is the law of the world’s development.
In this context, we recall again one of the most famous esoteric symbols of the duality of the Universe, Yin-Yang or Taiji “Supreme Pole/goal” – the ancient Chinese principle of the Supreme Element that generates five primary elements from which “ten thousand things” arise, including man. It is worth noting that Yin-Yang not only possesses fractality but, as mathematical calculations show, its graphic image corresponds to the Fibonacci Golden Proportion and its fractal numerical series, which some researchers regard as characteristics of male and female life cycles. Hindu samskaras, as the beginning of previous births and past experiences, has the process’s recursive nature within this process itself and forms the conceptual fractal of human existence in this world. And man’s earthly paths, in the transmigration of souls and reincarnation cycles, create fractal trajectories of a special kind.
In the Christian Trinity, One in Essence and Undivided, the fractal unity of the Divine is also affirmed. The hypostases of God the Father, God the Son, and God the Holy Spirit both together and separately comprise a single divine essence. Man, according to Holy Scripture, was created in “the image and likeness of God”. In other words, each person is a fractal pattern of the conceptual fractal of the One God. According to some researchers, the Bible’s text itself has fractal properties, in the content of which we can distinguish such fractals as the Tree of Life, the Image of God, the Likeness of God. A vivid example of fractal repetition is the Christian maxim: All those things, then, which you would have men do to you, even so, do you to them. Matthew 7:12
In turn, symbols that are part of a ritual, a metaphysical experience, or a religious idea represent at the same time a common doctrinal meaning. So, for example, geometric fractals are noticeable in mandala, icons, sacred labyrinth, multi-tiered Chinese pagoda, multi-headed Orthodox cathedral. And the Christian cross is not only a geometric or spatial fractal, repeated in the layout of cathedrals, on domes, in church interiors and utensils, in personal objects of worship, and a conceptual fractal encapsulates the entire content of the Christian faith.
The whole Universe is fractal, and so there is something joyfully quintessential about Mandelbrot’s insights. Man can be imagined as an integral fractal object of the Universe, which is in continuous cyclic development within the transition “material world (Yang) – spiritual or informational state (Yin)”.
In everything that surrounds us, we often see chaos, but it is not an accident, but a perfect form, which fractals help us discern. Nature is the best architect, the perfect builder, and engineer. It is arranged very logically, and if somewhere we do not see a pattern, it means that we need to look for it on a different scale.