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Complex non-linear systems:
"These systems are apt analogues for dependent co-arising (paticca-samuppada) in that, despite the fact that their behavior sometimes seems chaotic, their behavior actually contains deep, regular patterns. Furthermore, in some cases, these patterns can be utilized so that the system will behave in a desired way." (Thanissaro Bhikkhu)
The dictionary definition of chaos is turmoil, turbulence, primordial abyss, and undesired randomness, but scientists will tell you that chaos is something extremely sensitive to initial conditions. Chaos also refers to the question of whether or not it is possible to make good long-term predictions about how a system will act. A chaotic system can actually develop in a way that appears very smooth and ordered. (Chaos Theory and Fractals)
BUDDHABROT FRACTALS
Buddhabrot fractal exploration (based on Mandelbrot set)
Exploration of the 4D mandelbrot set on the Z-plane (Buddhabrot)
Mandelbrot set
Slow deep Mandelbrot zoom
Mandelbrot to Mandelbulb 3D
Fractal Definition: A geometric pattern that is repeated (iterated) at ever smaller (or larger) scales to produce (self similar) irregular shapes and surfaces that cannot be represented by classical (Euclidian) geometry. Fractals are used especially in computer modeling of irregular patterns and structures found in nature.(1)
What is unique about fractals? Many things! But one amazing, characteristic aspect of fractals, is that they are infinite! A fractal is infinite in two distinct senses, the macro level, and the micro level. Fractals extend to infinitely large values of their co-ordinates, outwards in all directions from the center. It would be like starting at the very center of the universe and then moving outward to try and find the end of the universe. Fractals also have infinite detail, in that one can zoom (magnify) in or out without limit (at least in theory) to show ever increasing detail within the image. Another distinguishing feature of fractals is it's property of self-similarity; an arbitrary region of a fractal looks very similar (but not necessarily identical) to the entire fractal. Just as DNA stores all the information that for each of us, all the information for a fractal is contained in its "parent" image. Fractals, in art and nature, can range from beautiful symmetry to the chaotic, from a perfectly symmetrical snowflake to a massive chaotic thunderhead cloud. But regardless of their symmetry, or level of chaos, fractals are everywhere we look and I hope the beauty of fractal art captivates your imagination, as they have mine!
Very good explanations with images also here:
http://www.miqel.com/fractals_
Buddhabrot technique
Fractal zoom 2 "The Blue Universe"
Mandelbrot set - from order to chaos
Showing the graph: Occasionally, the plot will become orderly again for a moment, whenever the pink dot passes over a baby Mandelbrot set, but it'll quickly return to the chaotic behaviour. The amazing thing is that, although this chaotic behaviour looks very different from the orderly behaviour, it's created by exactly the same equation, with just a small change to the input value.
A Journey in The Mandelbrot set
This film is based on the exhibition "Seduction" ( http://mandelbrot-set.elica.
2010: A Mandelbrot Odyssey (FractalNet HD)
An epic journey across the Mandelbrot set, exploring many different embedded Julia sets.
Fractal food: Broccoli Romanesco
Fractal forms—complex shapes which look more or less the same at a wide variety of scale factors, are everywhere in nature. However, some of the most pleasing patterns in geometric art exhibit exact or almost exact self-similarity. These are patterns which are composed of smaller copies of themselves ad infinitum, or at least until some limit where the similarity breaks down due to the granularity of the underlying material.
http://www.fourmilab.ch/
Mandelbulb
The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years with its cascading patterns and hypnotically colourful detail. It's known as a 'fractal' - a type of shape that yields (sometimes elaborate) detail forever, no matter how far you 'zoom' into it (think of the trunk of a tree sprouting branches, which in turn split off into smaller branches, which themselves yield twigs etc.).
It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).
It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal. For the impatient, you can skip to the nice pics, but the below makes an interesting read (with a little math as well for the curious).
Amazing Mandelbulb Gallery with Dhammacakka wheels in a fractal universe:
THE MANDELBROT MONK (who really discovered this? just for fun ;-)
On a holiday visit to Aachen cathedral, the burial place of Charlemagne, Schipke saw something that amazed him. In a tiny nativity scene illuminating the manuscript of a 13th century carol, O froehliche Weihnacht, he noticed that the Star of Bethlehem looked odd. On examining it in detail, he saw that the gilded image seemed to be a representation of the Mandelbrot set, one of the icons of the computer age. [*3]
Discovered in 1976 by IBM researcher Benoit Mandelbrot, the Mandelbrot set is the most famous fractal (a mathematical object with the property of infinite detail). Only the advent of fast computers made feasible the repeated calculations involved - or so it was thought. [*4] "I was stunned," Schipke says. "It was like finding a picture of Bill Gates in the Dead Sea Scrolls. The colophon [the title page] named the copyist as Udo of Aachen, and I just had to find out more about this guy." The first chapter, Astragali (Dice) was originally thought to be a discourse on the evils of gambling. It turned out to be Udo's research into what we now would call probability theory. He derived simple rules to add and multiply probabilities, and thus devised strategies for several card and dice games. More was to come. In the final and longest chapter, Salus (Salvation), Schipke uncovered the most radical work. Udo had, it seemed, investigated the Mandelbrot set, seven centuries before Mandelbrot.
Initially, Udo's aim was to devise a method for determining who would reach heaven. He assumed each person's soul was composed of independent parts he called "profanus" (profane) and "animi" (spiritual), and represented these parts by a pair of numbers. Then he devised rules for drawing and manipulating these number pairs. In effect, he devised the rules for complex arithmetic, the spiritual and profane parts corresponding to the real and imaginary numbers of modern mathematics. In Salus, Udo describes how he used these numbers: "Each person's soul undergoes trials through each of the threescore years and ten of allotted life, [encompassing?] its own nature and diminished or elevated in stature by others [it] encounters, wavering between good and evil until [it is] either cast into outer darkness or drawn forever to God." Below the description was drawn the first crude plot of the Mandelbrot, which Udo called the "Divinitas" ("Godhead"). He set it out in a 120x120 frame he termed a "columbarium" (i.e. a dovecote, which has a similar grid of niches) and records that it took him nine years to calculate, even with the newly imported technique of ‘algorism', calculation with Arabic numerals rather than abacus. "It tends to be taken for granted," Schipke says, "That the Mandelbrot is too calculation-intensive to be done without computers. What we have to remember is the sheer devotion of the monastic life. This was a labour of faith, and Udo was prepared to work for years. Some slowly-converging pixels must have taken weeks." "But there were also contemporary reasons why Udo's knowledge didn't make it into the mainstream. His basic belief - that salvation and damnation could be determined in advance - was heretical, and his use of Arabic numerals was thought a bit of a black art. And there was the disagreement with Thelonius." Despite the borderline nature of his work, Udo impressed his abbot at the monastery of Sankt Umbertus near Aachen. Life for a 13th century monk wasn't necessarily austere: the scurrilous Cantiones profanae poems record the delights of sex, eating, drinking and gambling. In a footnote to Astragali, Udo writes: "My enumeration of the ways [of dice] helped my lord abbot to win thirty-two florins and a fine new cloak from the Burgermeister at Irrendorf, and he has promised me a helper for my work". But Udo and his helper, Thelonius, ran into instant disagreement. Udo had always interpreted the Mandelbrot as signifying God. Thelonius took the opposite view: that it represented the Devil. Numbers that escaped to infinity, he argued, were souls flying free to heaven, and those caught in an orbit had fallen into the pit of Hell. Like many theological collaborations, they had a schism on their hands. Udo noted that their differences brought all work to a halt, and finally the two were reprimanded by the abbot for coming to blows in the refectory. "Sadly I write," says Udo on the last page of the Codex Udolphus, "that on pain of excommunication I must lay down my dice and my numbers. I have seen into a realm of heavenly complexity, and my heart is heavy that the door is now closed."
Mathematical explanation:
Fractals and math.
In the beginning, God created Math in his/her image. And God said, 'Be fruitful and multiply - and multiply - and multiply...'
Fractals are created using very simple mathematical equations. The fractal I am most interested in is called the Mandelbrot Set and its equation is: z = z * z + c which is about as simple as you get.
These equations are usually referred to as dynamic equations where the output of the equation is fed back into the equation. This is more commonly known as a feedback loop. Nature works very much like this in that it is one big giant feedback loop. For example the leaves from the trees fall to the ground and get fed back into the earth for the next generation of trees and plants. All things in nature work this way, so it's no surprise that mathematical equations using feedback can generate images that look so much like things in nature.
Self-similarity.
Despite its apparent solidity the Universe is a fractal. A fractal is a self-similar pattern or series of patterns with infinite detail. Self-similarity refers to the fact that the patterns repeat themselves within the system but they never repeat exactly. For example, there are many, many galaxies in our Universe but no two are exactly the same. Some look similar to each other (self-similarity) but you could search the whole Universe and definitely not find two with exactly the same details. Our bodies are also self-similar with patterns repeating themselves on different levels. The Chinese and Indian cultures have known this for years, mapping the whole body onto different parts of the body like feet, ears and eyes. Acupuncture and reflexology (to name a few) are based on this theory.
The key is in the details.
The key to the fractal is in the details. You can search the universe of a fractal (each fractal is a universe unto itself since it is infinite) and find the same pattern over and over again but when you look closely at the details you can see that they are quite different. There are over 6 billion people on our planet and again, you will not find two people that are exactly the same. Even identical twins and clones are not exactly the same when you look more closely at the details. It is important to realize this because modern medicine treats everyone as if they are exactly the same instead of realizing that each individual is unique and must be treated differently. The practitioner must take the time to know the patient in all the details before deciding on a treatment for an ailment. This is the biggest mistake of modern medicine.
Fractal dimension.
Another important fact about fractals is that they exist between dimensions. We are use to describing the universe in 2, 3 and even 4 dimensions (including time) but what appears to be solid matter in the universe (galaxies, stars, planets, trees and animals etc�) actually form and exist in the space referred to as the fractal dimension. Fractal just means �fraction� so a fractal dimension is really just a fractional dimension and would look something like �2.34784� or �3.48723� instead of exactly 2-D or 3-D. Each person has a unique fractal dimension and will respond to different fractal images in different ways.
Fractal healing.
Fractals can be seen in trees, rivers, clouds and all other natural objects. Your own person (body and mind) is a fractal with patterns that repeat on every level of your being. Fractal patterns can be used to restore yourself to the healthy beautiful creature that you really are. A healthy body has all it�s fractal patterns intact on every level, mind, body and soul. An unhealthy body is a body that has had its fractal pattern disrupted in some way causing things to change unexpectedly. You can easily tell a diseased tree from a healthy one because it begins to lose its fractal pattern.
Because of the self-similar nature of fractals, it is possible to work on many levels simultaneously. It has been proven that looking at pictures of nature can increase the healing process dramatically. This is because nature is a fractal. Looking at pictures of fractals has the same effect in generating healing energy and installing the healthy fractal pattern back into the body and soul. The trick is to find the fractal that suits you best. Meditating on a fractal image can be both relaxing and stimulating. The images can trigger the natural fractal properties of our minds and bodies. It is important to choose a fractal that is pleasing to you since this is more likely to be compatible with your fractal dimension.
What is a Buddhaverse?
A buddhaverse is a universe that is perfected by enlightened beings who realize that they are an itegral part of the universe they live in and that their actions effect how the universe will turn out. They create the buddhaverse by transforming the ordinary world of desire and suffering into a world of happiness and self-sufficiency through right thought, right action and right livelyhood. In the pure land of the Buddha field, or buddhaverse, beings are in a state of perpetual bliss, where each individual represents a different aspect of Buddha, the embodiment of truth. This pure land or realm of peace and bliss is not a physical place but a living dimension of formless reality intimately resonating with our being. The buddhaverse experience is timeless, perpetually existing, eternally evolving, with no beginning or end.
In Chaos Theory, forms emerge, dissolve, and reform through the creative process known as self-organization. These forms resonate with our spirit and foster a cultivate of connection. Healing originates in the chaotic consciousness and emerges as the process of creative self-organization. Chaos is ubiquitous in nature. It reveals in its own depths, hidden degrees of order and structure that resonate with the soul and reveal to us the basic forms and structures repeated throughout nature and throughout our nature. It underlies the fundamental laws of nature and natural processes, and implies a universe of evolution and constant re-creation. Structure arises in the moment and is in resonance with its environment. But in a universe of constant evolution each form eventually becomes dissonant as the evolving environment surrounding it changes.
Chaos Theory restores the balance to the entropic forces which lead to the decay and death of outworn systems. It is the means of creative self-organization which arises from the undifferentiated disorder that lurks within the processes of creation. Chaos holds infinite possibilities of new form, and these forms are eventually revealed and emerge from chaos as new structure. This also applies to consciousness dynamics, such as thought, spontaneous behavior and creativity. Reality is neither structure nor chaos, but a process in which structure and chaos dance between form and formlessness. This is the eternal cycle of death and renewal, the dance of Shiva.
Emergence is a process by which order appears �spontaneously� within a system. When the mind lets go of its rational order and enters into unstructured chaos, it emerges later with a new structure or form. Since order arises spontaneously from disorder through self-organization, healing is an ever-present potential.
1 comment:
I just read about it here:
http://en.wikipedia.org/wiki/Udo_of_Aachen
It is a funny story although it is made up :)
"The hoax was lent an air of credibility because often medieval monks did discover scientific and mathematical theories, only to have them hidden or shelved due to persecution or simply ignored because publication prior to the invention of the printing press was difficult at best. Mr. Girvan adds to this suggestion by associating Udo with several other more legitimate discoveries where an author was considered ahead of his time in terms of a scientific theory of some sort that is now established as a mainstream theory but was considered fringe science at the time.
Another aspect of the deception was that it was very common for pre-20th century mathematicians to spend incredible amounts of time on hand calculations such as a logarithm table or trigonometric functions. Calculating all of the points for a Mandelbrot set is a comparable activity that would seem tedious today but would be routine for people of the time."
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