Fractal Magic - an essay about fractal equations, their present applications and future implications
The Beauty of Fractals
Fractal Art has become very popular on the Internet, with a host of new fractal-making software programs, many of them freeware, including Ultra Fractal, Mandelbox and Mandelbulb, Fractal Explorer, and others. A recent search on the online art website Deviantart.com yielded almost 800,000 results for fractal art and animations.
Fractal geometry is being used in many different novel ways. In cinema, the founder of Pixar Animation Studios began experimenting with fractals to make his computer graphics appear more realistic, and this technique gave rise to many of the software packages now being used in the computer graphics industry to create more realistic special effects, such as the Genesis planet in Star Trek II and the damaged Death Star in Return of the Jedi.
Fractal geometry is being used for a new kind of image compression, converting images consisting of random information into fractal code, saving only a small,representative amount of information which is used later to re-created the original image. Because the fractal image is now computer code instead of pixels, the file size is reduced drastically and the image can be scaled to any size without losing its sharpness.
In biology, fractals are being used to accurately model the human lung, heart-beats and blood vessels, the brain, and many other physiological processes. Researchers are using fractal geometry to build models which they hope will identify microscopic patterns of diseases.
Fractal lungs
In the financial markets, after the financial crisis of 2007, many theorists started turning to the work of Benoit Mandelbrot, whose fractal approach to price variations revealed that the crash was not as unlikely as conventional forecasts had predicted. Mandelbrot dismissed the theory of efficient markets as too generalized and simplistic, and commented that the real world is not tidy or infinitely stable, that turbulence is a natural and unavoidable force. In 2004, Mandelbrot had discussed these views in his book "The (Mis)behavior of Markets: a Fractal View of Risk, Ruin, and Reward", co-written by Richard Hudson.
Fractal patterns in the markets (fractals are contained in the rectangular areas)
In the realm of climate science, scientists have recently demonstrated that the distribution of large branches to smaller branches in a single tree exactly replicates the distribution of large trees to small trees in an entire forest. Current research is underway to use this information to measure how much carbon dioxide a single forest is capable of processing. Then scientists will be able to apply their findings to every forest on earth, calculating how much carbon dioxide the entire earth can safely absorb.
Fractals can be found in the branching of tracheal tubes, the leaves in trees, the veins in the human hand, water swirling out of a tap, cumulus clouds, an oxygen molecule or DNA molecule, and the patterns of the stock market. In astronomy, most astronomers believe that the universe is smooth at very large scales, but some dissident scientists are starting to argue that the structure of the universe is fractal at all scales. With the proliferation of space probes into our solar system, it has been found that the outer boundary of our Oort cloud of icy planetesimals exists about half-way to Alpha Centauri, the nearest star to us. and it has been postulated by some astronomers that the edge of our Oort cloud at the outer limits of our solar system might interact with a similar Oort cloud surrounding Alpha Centauri, and that this interaction might be the cause of many new comets entering our solar system. If this relationship between the two stars is true, then our two planets might be part of a celestial fractal branch of a great cosmic tree. Astronomers have already found strings or filaments of gravitationally bound galaxies that appear to be separated by voids of space. Recently, these "voids" have even been found to contain shorter strings of faint galaxies, called "tendrils", by a team of astronomers at the University of Western Australia, who have stated: "The universe is full of vast collections of galaxies that are arranged into an intricate web of clusters and nodes connected by long strings. This remarkably organized structure could be called the 'cosmic web'." It would not be surprising to eventually find the hallmarks of fractal geometry in celestial structures: repeating patterns or self-similarity at every scale, and expanding or evolving symmetry as the space of our universe inflated after the big bang. There might also be an expanding self-similarity from the motions of particles at the very small quantum level, such as the orbits of electrons around nuclei, up to the larger levels of planetary systems and revolving galaxies, and beyond, all the way up to the higher dimensional "multiverse" which is conjectured to exist by modern Superstring theorists.
Fractals can be found in the branching of tracheal tubes, the leaves in trees, the veins in the human hand, water swirling out of a tap, cumulus clouds, an oxygen molecule or DNA molecule, and the patterns of the stock market. In astronomy, most astronomers believe that the universe is smooth at very large scales, but some dissident scientists are starting to argue that the structure of the universe is fractal at all scales. With the proliferation of space probes into our solar system, it has been found that the outer boundary of our Oort cloud of icy planetesimals exists about half-way to Alpha Centauri, the nearest star to us. and it has been postulated by some astronomers that the edge of our Oort cloud at the outer limits of our solar system might interact with a similar Oort cloud surrounding Alpha Centauri, and that this interaction might be the cause of many new comets entering our solar system. If this relationship between the two stars is true, then our two planets might be part of a celestial fractal branch of a great cosmic tree. Astronomers have already found strings or filaments of gravitationally bound galaxies that appear to be separated by voids of space. Recently, these "voids" have even been found to contain shorter strings of faint galaxies, called "tendrils", by a team of astronomers at the University of Western Australia, who have stated: "The universe is full of vast collections of galaxies that are arranged into an intricate web of clusters and nodes connected by long strings. This remarkably organized structure could be called the 'cosmic web'." It would not be surprising to eventually find the hallmarks of fractal geometry in celestial structures: repeating patterns or self-similarity at every scale, and expanding or evolving symmetry as the space of our universe inflated after the big bang. There might also be an expanding self-similarity from the motions of particles at the very small quantum level, such as the orbits of electrons around nuclei, up to the larger levels of planetary systems and revolving galaxies, and beyond, all the way up to the higher dimensional "multiverse" which is conjectured to exist by modern Superstring theorists.
A simulation of the "cosmic web"
In 2004, wide-field telescope observations (made with the Blanco Telescope in Chile) of a region of space that contains galaxies from a period when the universe was only a fifth of its present age, have revealed an enormous string of galaxies about 300 million light years long. This structure defies current models of how the universe evolved, which can't explain how a string this big could have formed so early, and shows that our current theories of how Dark Matter operates are lacking.
A cluster of galaxies: twisted, helical pairs like DNA/RNA strings
Video of the ancient galaxy string
With the discovery of galactic strings, another question which might be asked is this: Is there a self-similarity between the smallest level of our universe, where the tiny "strings" of String theory, which theoretically make up the quantum world, exist, and the galactic strings we observe in the vastness of space? If the current universe expanded from a tiny "zero point energy" (vacuum energy or Dark Energy) at the moment of the big bang, was there already a pattern in this small field that foreshadowed the structures of quantum and galactic strings in its own inner energy structure, similar or analogous to a DNA string? Just as galactic strings have been found to resemble DNA strings in their helical structure, quantum strings, which have never been found experimentally because they are so small, might be helical strings too. Perhaps this helix was present in the "cosmic egg" of vacuum energy at the "big bang" birth of our universe (see my essay on the theory of how black holes create new universes for more of my theory on this subject).
In a different essay, I will propose an idea for what the "fourth dimension" of hyperspace (or the "bulk" which contains various three-dimensional universes or "branes" such as ours in string theory) actually is and how it functions. This theory also relates to a fractal concept, in which the fourth dimension is similar to the trunk of a great tree, which connects all of the 3-D universes within the 4-dimensional hyperspace, through which the mysterious force of gravity ascends and descends through worlds of different densities (different fractal levels in universes created by continual iterations of black holes creating offspring "baby" universes). In my theory, the fourth dimension is basically a dimension of consciousness which connects the various levels of the "Great Mind" of the Cosmos. I know this sounds very metaphysical, but I believe I have some scientific evidence for such a theory; it is different from other modern theories about the "personified" or conscious universe, as far as I can ascertain. Perhaps the hypothetical graviton is the unifying unit or stringy tendril of conscious love, and orbiting celestial bodies are wedded in blissful gravitational love affairs. That would be elegant.
In a different essay, I will propose an idea for what the "fourth dimension" of hyperspace (or the "bulk" which contains various three-dimensional universes or "branes" such as ours in string theory) actually is and how it functions. This theory also relates to a fractal concept, in which the fourth dimension is similar to the trunk of a great tree, which connects all of the 3-D universes within the 4-dimensional hyperspace, through which the mysterious force of gravity ascends and descends through worlds of different densities (different fractal levels in universes created by continual iterations of black holes creating offspring "baby" universes). In my theory, the fourth dimension is basically a dimension of consciousness which connects the various levels of the "Great Mind" of the Cosmos. I know this sounds very metaphysical, but I believe I have some scientific evidence for such a theory; it is different from other modern theories about the "personified" or conscious universe, as far as I can ascertain. Perhaps the hypothetical graviton is the unifying unit or stringy tendril of conscious love, and orbiting celestial bodies are wedded in blissful gravitational love affairs. That would be elegant.
The equation below the fractal was used to create it
The two most important features of fractal geometry are self-similarity on different levels and non-integer dimensions. If you magnify or zoom in on a fractal image, you will see the same shape at every magnification; this is the self-similarity feature of fractals. If you zoom in on the classic Mandelbrot fractal below, you will see the same shape appearing over and over.
Deep Mandelbrot zoom video
The second feature of fractals, the non-integer dimensions, is a little harder to understand. Classical geometry deals with objects of integer dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. But many natural phenomena are better described mathematically using a dimension between two whole numbers. While a straight line has a dimension of one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves. The more the flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a hilly fractal scenario will reach a dimension somewhere between two and three. So a fractal landscape made up of a large hill covered with tiny mounds would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension. The video below does a good job of explaining how fractal dimensions are calculated.
Calculating fractal dimensions
In the first type, complex numbers are used to produce the fractals. A complex number consists of a real number, which is a value that represents a quantity along a continuous line, added to an imaginary number. A complex number is commonly called a "point" on a complex plane. If the complex number is Z= (a + b*i), the coordinates of the point are a (horizontal - real axis) and b (vertical - imaginary axis). The i is the unit of imaginary numbers,
, the square root of a negative one, which is an imaginary number, because any number squared is a positive number, and there is no such thing in the "real" mathematical world as the square root of a negative number.
Why imaginary number are real and their usage
The two leading researchers in the field of complex number fractals are Benoit Mandelbrot and Gaston Julia. and both have fractal "sets" named after them.
The Mandelbrot set is the set of points on a complex plain. To build such a set, you have to use an algorithm based upon the recursive formula: 
, separating the points in the complex plain into two categories, points inside the Mandelbrot set, and points outside the Mandelbrot set. The image below shows a portion of the complex plane, where the points of the Mandelbrot set have been colored black.
, separating the points in the complex plain into two categories, points inside the Mandelbrot set, and points outside the Mandelbrot set. The image below shows a portion of the complex plane, where the points of the Mandelbrot set have been colored black.
It is also possible to assign a color to the points outside the Mandelbrot set. Their colors depend on how many iterations have been required to determine that they are outside the Mandelbrot set. Below is a colored version of the Mandelbrot set.
The Mandelbrot set is determined by iterating or repeating the equation over and over again, plugging the new values back into the equation 
. You start with an initial value for Z and C (C is a complex number and its value will determine the Mandelbrot set), and after plugging these numbers into the equation, you get a new value for Z. Then you plug that value for Z into the equation and get a new value for Z. This process is repeated over and over again; that is why it is called an iterative equation.
. You start with an initial value for Z and C (C is a complex number and its value will determine the Mandelbrot set), and after plugging these numbers into the equation, you get a new value for Z. Then you plug that value for Z into the equation and get a new value for Z. This process is repeated over and over again; that is why it is called an iterative equation.
Computers are used to filter through a large sampling of the infinite number of points on the complex plane, and to determine which of these points fit the Mandelbrot set criteria for C. The computer performs this task by running iterations of the equation until it can verify that for a given value of C, Z never gets larger than 2. If the computer is assured of that fact, then it determines that C is in the Mandelbrot set and it plots that value of C on the complex plane by placing a dot of an arbitrary color (which can be set by the user) on the computer monitor. The only other possibility is that for the value of C, Z eventually does exceed 2, and the computer does not plot it.
By plotting the millions of C values that are a part of the Mandelbrot set, the computer creates the unique and distinctive Mandelbrot geometry. This process can be described as the "migration" of the initial point C across the plane. Different colors can be chosen for the points in the complex plain which are not a part of the Mandelbrot set, based upon how many iterations of the equation it takes before it is shown that the value of C is outside the set (or how quickly the value of Z heads toward infinity), and this is what gives many versions of the Mandelbrot set image their multi-colored features and interesting effects.
Julia sets are related to the Mandelbrot set in that they are created by iterations of the Mandelbrot equation with one exception: the value of C is constant throughout the process, while the value of Z varies. The value of C determines the shape of the Julia set and each point of the complex plane is related to a particular Julia set. After a point C is chosen on the complex plane, the equation is iterated with the value of the initial point Z to see how far it will migrate from its origin. If it remains close to its origin, it is a part of the Julia set; otherwise its value will go toward infinity and a color is assigned to Z depending upon how quickly the point escapes from its origin. To produce an image of the whole Julia set associated with the value of C, the process must be repeated for all the points Z in the complex plane whose coordinates are included within a distinct range of numbers.
A Julia set fractal image
First four steps of the Koch Snowflake construction
Many other examples of IFS fractals can be found on the web, such as fern leaves and spirals.
An IFS fern leaf fractal
Fractal design is still in its infancy, but already many interesting fractal images and animations have been created which bear an eerie resemblance to known designs in our universe. Recently, a fractal animation by an artist who calls himself FractKali was banned on many Internet websites (perhaps one of the reasons is that he gave it the title: "Fractal Orgy") because it appears to contain sexual content, even though it was created with a fractal computer program using a mathematical formula (a raw fractal).
"Fractal Orgy" by FractKali
Fractal art: "Spirally Born" by Fractali: Iteration of complex sin function and Julia Set circle inversion, colored by exponential smoothing technique
Nova's special on Benoit Mandelbrot and Fractal Geometry
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