Order In Chaos: How The Mandelbrot Set
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The magnification of the last image relative to the first one is about 1010 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border shows an astronomical number of different fractal structures.
Physicists like to order. Their job is to look at intricate systems and find ways to represent them as simply as possible with equations. Small deviations from the large-scale order are dismissed as noise. Only the large-scale regularities are given attention. There is even a joke about a farmer who hires a physicist to design a chicken coop: the physicist begins by assuming spherical chickens.
Reality seems more real to us when it is modeled on chaos. Science seeks order in nature and finds it, but that order is so complex that it seems to give rise to disorder. But in examining the disorder of chaotic systems closely, we find a new order.
It is an approximation of the locus of connectedness for the Julia sets of the family of functions f(z) = z^2 + lambda/(z^2) (rotated by pi/2). This is analogous to the standard Mandelbrot set (which applies to the family f(z) = z^2 + c), but holds additional fascination because for lambda values which are in the interior of one of the subdomains of the connectedness locus, the Julia set is a Universal Curve. To me this represents the structure unifying chaos (since Julia sets are chaotic) and order (since Universal Curves act as a sort of catalog of all planar curves).--Aaron
The science of chaos seems to answer some of the most fundamental questions humanity struggles to answer: How does life begin? What is turbulence? Above all, in a universe governed by entropy, which creates greater disorder, how does order arise? A science that addresses the age-old question: how does the microcosm weave itself into the macrocosm? When studied in isolation, one atom or neuron behaves in one way, but billions of atoms and neurons behave completely differently. A science that unravels the link between aperiodicity and unpredictability.
Fig- Lorenz SystemThe map formed a sense of infinite complexity that embodied chaos and order. It always stayed within certain bounds, but at the same time, it never repeated itself. The generated chaotic system moved predictably toward its attractor in phase space, but strange attractors appeared instead of points or simple loops. Strange attractors represent a chaotic system in a specific phase space, but attractors are also found in many nonchaotic dynamical systems.The shape looked like a double spiral in three dimensions that looked like a butterfly. And thus came to be known as the butterfly effect.
The lungs are an excellent example of a natural fractal organ. The volume of a pair of human lungs is only about 4-6 liters, but the surface area of the same pair of lungs is between 50-100 square meters. The lung surface area to volume ratio is very high and useful. This is possible only because the structure of the lungs is fractal. There are 11 orders of branching, from the trachea to the alveoli at the tips of the branches. Fractal branching geometry provides an incredibly useful way to make a very large surface area extremely compact.
A fascinating feature of quantum chaos is that it reveals a significant amount of universality in the behavior of extraordinarily different physical systems. For example, acoustic wave intensities found in problems with strong multiple scattering that lead to a probability density known as the Rayleigh distribution, Ericson fluctuations in the cross-sections of neutrons scattering from medium to heavy nuclei, and conductance fluctuations found in chaotic or disordered quantum dots can be seen to possess a common underlying statistical structure. One is thus able to see essential parallels between systems that would normally otherwise be left uncovered. Universality implies a lack of sensitivity to many aspects of a system in its statistical properties, i.e., an absence of certain information. Furthermore, quantum chaos brings together many disparate, seemingly unrelated concepts, i.e., classical chaos, semiclassical physics and asymptotic methods, random matrix ensembles, path integrals, quantum field theories, and Anderson localization, and ties together in unexpected ways.
It is not surprising then to see that quantum chaos has found application in many domains. A partial list includes: i) low energy proton and neutron resonances in medium and heavy nuclei; ii) ballistic quantum dots; iii) mesoscopic disordered electronic conductors; iv) the Dirac spectrum in non-Abelian gauge field backgrounds; v) atomic and molecular spectra; vi) Rydberg atoms and molecules; vii) microwave-driven atoms; viii) ultra-cold atoms and optical lattices; ix) optical resonators; x) acoustics in crystals and over long ranges of propagation in the ocean; xi) quantum computation and information studies; xii) the Riemann zeta function and generalized L-functions; and xiii) decoherence and fidelity studies. There are many other examples.
Chaos theory has a bad name. It conjures up images of unpredictable weather, economic crashes, and science gone wrong. But there is a fascinating and hidden side to chaos, one that scientists are only now beginning to understand. It turns out that chaos theory answers a question humankind has asked for millennia... How did we get here? In this breathtaking program, Professor Jim Al-Khalili reveals the science behind much of the beauty and structure in the natural world and discovers that far from it being magic or an art of God, it is in fact an intrinsic part of the laws of physics. Amazingly, it turns out that the mathematics of chaos can explain how and why the universe creates exquisite order and pattern.
The concern in this Annex to Sustainability through the Dynamics of Strategic Dilemmas is to indicate features associated with the Mandelbrot set (hereafter the M-set) in order to point to their significance in configuring complex experience -- rather than in describing natural phenomena, as is normally the case. The assumption is that the features offer templates for innovative thinking in response to highly divisive strategic and value dilemmas. An assumption is also made that the mind is uniquely capable of undertaking operations that explore the features of complex spaces such as those with which the M-set is associated. Note that contextual arguments and references are provided in the main paper.
Insights distinguishing levels of abstraction, of the same subtle order as the M-set, may have been deliberately (or inadvertently) anchored in cultural artefacts such as myths, legends, leys and the like. Whether they are termed "archetypes" or not, their fundamental nature and importance is nourished by the societies by which they are valued for the order they offer.
The much-cited Chinese work discusses the "circulation of the light" of awareness through various conditions during meditation [diagram] reminiscent (if only in the metaphors used to describe them) of stages of the Carnot heat cycle discussed above. The Taoist practitioners of "internal alchemy" (nei-tan) refer to a continual circulation of the ch'i (vitality principle) up the primary yang (positive), back or Governor Channel (tu mai or du mo) and down the primary yin (negative), front or Functional Channel (jen mai or ren mo) of the body. (see Lu K'uan Yü, Taoist Yoga, Alchemy and Immortality, 1970) [more]. It is possible that the representation of the M-set is to be usefully related to the traditional concept of an alchemical "vessel" in which all "base matter" can be dissolved -- the container for the "universal solvent". Alchemy postulates the existence of such a universal solvent as being capable of transforming base metals into gold and bestowing eternal youth and therefore immortality on human beings. The universal solvent -- counterpart to the "philosopher's stone" -- is not ordinary water, but "philosophical" water, the water of life, aqua permanens, aqua mercurialis. For Robert Grinnell (Alchemy in Modern Woman, 1973) it is basic to the transformative alchemical process of solutio which facilitates the fluid, mobile basis of consciousness: For aqua permanens is a mode of the arcane substance; its symbol is water or sea-water, an all-pervading essence of anima mundi, the innermost and secret numinosum in man and the universe, that part of God which formed the quintessence and real substance of Physis, at once the highest supercelestial waters of wisdom and the spirit of life pervading inorganic matter. For Iona Miller (Chaos as the Universal Solvent, 1993): In reducing all to pure water, the prima materia and the ultima materia become synonymous. That primal consciousness state, that creative and chaotic consciousness is the beginning of the operation of "water", and its ultimate realization. It becomes easy to see why the operation of water is the "root of alchemy." Through consciousness journeys which liquify our rigid notions of self and world, we re-create the adventures of the hero or heroine. The theme is the loss and recovery of identity. Chakra system: In Hinduism, its spiritual systems of yoga, in some related eastern cultures, as well as in some segments of the New Age movement, a chakra (from the Sanskrit word for "wheel, circle") is considered to be an energy node in the human body [more]. The seven main chakras are described as being aligned in an ascending column from the base of the spine to the top of the head (see fractal representations). Each is associated with a certain color, multiple specific functions, an aspect of consciousness, a classical element, and other distinguishing characteristics [more]. Given the hypothesized organizing function of the M-set, it is interesting to consider where these chakras are each located in relation to the geometry of its graphical representation (in the "seated-Buddha" orientation): Muladhara (Root chakra) : Understood to be at the base of the human spine. On the M-set representation it is most closely associated with the junction of the inward-turned portion of the cardioid. Svadhisthana (Sacral / Hara chakra): Understood to be below the human navel. On the M-set representation it is most closely associated with the axial origin from which the cardioid is generated. Zazen as the study of the self, is a particular kind of meditation, unique to Zen Buddhism, that is at the "very heart" of the practice. Fundamental to that practice is the centering of attention in the hara as the physical and spiritual center of the body. In Hinduism, this is recognized as governing the function of the gonads and the reproductive system. It is associated with the emotional body, willingness to feel emotions and accept change. As the centre of gravity of the body, this awareness is also fundamental to some Eastern martial arts. Manipura (Solar Plexus chakra) : Understood to be at the level of the human solar plexus. On the M-set representation it is most closely associated with the intersection of the vertical axis with the horizontal line between the symmetrically positioned 3-period primary bulbs attached directly to the cardioid. Anahata (Heart/Lung chakra) : Understood to be at the level of the human heart. On the M-set representation it is most closely associated with the intersection of the vertical axis with the horizontal line between the symmetrically positioned 5-period primary bulbs attached dfirectly to the cardioid. Visuddha (Throat chakra) : Understood to be at human throat. On the M-set representation it is most closely associated with the junction of the cardioid with the circular period-2 primary bulb above the cardioid. Ajña (Third Eye chakra) : Understood to be between the human eyes. On the M-set representation it is most closely associated with the centre of the circular period-2 primary bulb above the cardioid. Sahasrara (Crown chakra) : Understood to be at the crown of the human head. On the M-set representation it is most closely associated with the crown-like features about the circular period-2 primary bulb above the cardioid. There is a case for investigating the extent to which the position of these chakras emerges from more complex mathematical features of the cardioid, its associated curves, and especially from its generation. These might be understood as significant associative pathways for those who embody this system. Types of concentration/meditation : In the terms, for example, of the Yoga Sutras of Patanjali (1.17-1.18): All objects are in one of four stages: Virtually all types, styles, methods, or objects of meditation are included in one of these four stages, levels, or categories (1.17): Savitarka/Gross: relates to concentration on any gross object while still accompanied with other activities of the mind, including meditation on sensory awareness, visualized objects, the gross level of breath, attitudes, syllables of mantra, or streams of conscious thought. Savichara/Subtle: relates to subtle objects, after the gross have been left behind; the subtleties of matter, energy, senses, and the mind are, themselves, the objects of meditation, inquiry, and non-attachment. Sananda/Bliss: emphasizes the still subtler state of bliss in meditation. In this state, the concentration is free from the gross and subtle impressions that were at the previous levels. Sasmita/I-ness: focuses on I-ness, which is even subtler, as it relates to the I that is behind, or witness to all of the other experiences. Objectless concentration: The four stages (above) all have an object to which attention is directed (samprajnata). Beyond these four is objectless concentration (1.18), where all four categories of objects have been released from attention (asamprajnata). Emptiness of mind: As with many spiritual disciplines (for which there are extensive web references [more | more]), Buddhism in its various forms (notably Mahayana Buddhism) places much emphasis on the emptiness of mind and the ways in which that understanding is obscured. Offering an intriguing association to the M-set cardioid, in The Heart Sutra the Buddha reportedly states, for example: ...form is emptiness and the very emptiness is form; emptiness does not differ from form, form does not differ from emptiness; whatever is form, that is emptiness, whatever is emptiness, that is form, the same is true of feelings, perceptions, impulses and consciousness. In another example, the Tibetan Kalu Rinpoche reportedly stated: You live in illusion and the appearance of things. There is a reality, and you are that reality, but you don't know it. If you should ever wake up to that reality you would realize that you are nothing, and being nothing, you are everything. That is all. [more] This could be understood in terms of the the human tendency to focus on, and identify with, the experience of the sensible dynamics which the M-set "embodies" -- but from a higher order of abstraction with which the individual may alternatively choose to identify. It is therefore interesting to note the Buddhist comparison between such understandings of emptiness and the origin of numbers from emptiness. For Buddhists, emptiness (sunyata) implies a sense of potential rather than nihilism. In Mathematics, mind, ontology and the origins of number the recognition by Geshe Kelsang Gyatso (1995) is noted that there are three levels of dependent relationship: Gross dependent relationship - causality - the dependence of phenomena on their causes. Subtle dependent relationship - structure - the dependence of phenomena on their perceived parts (including aspects, divisions and directions). Very subtle dependent relationship - the dependence of phenomena on imputation by mind. For Buddhists, these ideas are noted [more] as being remarkably similar to the theory of the origins of mathematics, as proposed by the mathematician John von Neumann (1923) in the light of the theory of sets. He was the originator of the architecture used in most non-parallel-processing computers. Von Neumann suggested that all numbers could be bootstrapped out of the empty set by the operations of the mind. A set is a collection of things. An empty set is a collection of nothing at all. An empty set can be thought of as nothing with the potential to become something (that is to be become a set with at least one member). Again, the three orders of abstraction might be fruitfully related to the structure of the M-set. From a perspective of hermetic philosophy (About Spiritual Emptiness or the Void): Emptiness designates a state of mind, an inner level of consciousness based upon the renunciation of what one believes to be real, beyond all comprehension or lack of comprehension. Emptiness is, therefore, a higher level of the mind, an attunement with "nothing," i.e., with Pure Being that has no reflection at all. This is the revelation that the abstract or higher mind of a spiritual man receives. He is called "spiritual" because his mind is open and in harmony with the whole of creation, or the Unknown God. Hence, such a man is empty of a personal "interpretation" and is in touch with the world of spirit. As he manifests the world of spirit within him, a new one opens up before him, giving him a new understanding of life. Such references point to many possibilities of using the M-set as a means of holding the distinctions between various fundamental states of consciousness associated with "spiritual" insight. For example, given the various distinct modes of long term behaviour characteristic of the dynamical systems mapped by the M-set: 2b1af7f3a8