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November 10th, 2014

Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.[7]

Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesco broccoli are two examples from nature." He points out an unexpected conclusion:

One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot

Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphic computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body."...


A Mandelbrot set
Mandelbrot, however, never felt he was inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke:

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.

According to Clarke, "the Mandelbrot set is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity?" Clarke also notes an "odd coincidence:" "the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas."

Fractals and the "theory of roughness"
Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature. He began by asking himself various kinds of questions related to nature:

Can geometry deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?

Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured." Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in The Fractal Geometry of Nature had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new:

The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer ... bringing an element of unity to the worlds of knowing and feeling ... and, unwittingly, as a bonus, for the purpose of creating beauty.


Fractals are also found in human pursuits, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
  —Mandelbrot, in his introduction to The Fractal Geometry of Nature

Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.

Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.]


Benoit Mandelbrot - Wikipedia, the free encyclopedia
"....Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.

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, and is one of the best-known examples of mathematical visualization....."

Mandelbrot set - Wikipedia, the free encyclopedia

The Holographic Universe - Crystalinks

The Holographic Universe - Crystalinks

Saved for future reading.  Who knows what wonders i will discover/imagine/invent?.
Who knows how much more ignorant i will feel  than i do now?
Will this light or impede my path?  Trash or treasure?
"...Bekenstein asks "Could we, as William Blake memorably penned, 'see a world in a grain of sand,' or is that idea no more than 'poetic license,'" referring to the holographic principle."...

Holographic principle - Wikipedia, the free encyclopedia

saved for future reading.
He followed Aristotle in asserting that the subject matter of metaphysics is "being qua being" (ens inquantum ens). Being in general (ens in communi), as a univocal notion, was for him the first object of the intellect. Metaphysics includes the study of the transcendentals, so called because they transcend the division of being into finite and infinite and the further division of finite being into the ten Aristotelian categories. Being itself is a transcendental, and so are the "attributes" of being—"one," "true," and "good"—which are coextensive with being, but which each add something to it.[original research?]

The doctrine of the univocity of being implies the denial of any real distinction between essence and existence. Aquinas had argued that in all finite being (i.e. all except God), the essence of a thing is distinct from its existence. Scotus rejected the distinction. Scotus argued that we cannot conceive of what it is to be something, without conceiving it as existing. We should not make any distinction between whether a thing exists (si est) and what it is (quid est), for we never know whether something exists, unless we have some concept of what we know to exist.[20]

Individuation[edit]
Scotus elaborates a distinct view on hylomorphism, with three important strong theses that differentiate him. He held: 1) that there exists matter that has no form whatsoever, or prime matter, as the stuff underlying all change, against Aquinas (cf. his Quaestiones in Metaphysicam 7, q. 5; Lectura 2, d. 12, q. un.), 2) that not all created substances are composites of form and matter (cf. Lectura 2, d. 12, q. un., n. 55), that is, that purely spiritual substances do exist, and 3) that one and the same substance can have more than one substantial form—for instance, humans have at least two substantial forms, the soul and the form of the body (forma corporeitas) (cf. Ordinatio 4, d. 11, q. 3, n. 54). He argued for an original principle of individuation (cf. Ordinatio 2, d. 3, pars 1, qq. 1–6), the "haecceity" as the ultimate unity of a unique individual (haecceitas, an entity's 'thisness'), as opposed to the common nature (natura communis), feature existing in any number of individuals. For Scotus, the axiom stating that only the individual exists is a dominating principle of the understanding of reality. For the apprehension of individuals, an intuitive cognition is required, which gives us the present existence or the non-existence of an individual, as opposed to abstract cognition. Thus the human soul, in its separated state from the body, will be capable of knowing the spiritual intuitively.

Formal distinction[edit]
Like other realist philosophers of the period (such as Aquinas and Henry of Ghent) Scotus recognised the need for an intermediate distinction that was not merely conceptual, but not fully real or mind-dependent either. Scotus argued for a formal distinction (distinctio formalis a parte rei), which holds between entities which are inseparable and indistinct in reality, but whose definitions are not identical. For example, the personal properties of the Trinity are formally distinct from the Divine essence. Similarly, the distinction between the 'thisness' or haecceity of a thing is intermediate between a real and a conceptual distinction.[21] There is also a formal distinction between the divine attributes and the powers of the soul.


Duns Scotus - Wikipedia, the free encyclopedia

For my crude and simplistic understanding ...Read more...Collapse )

The Holographic Universe (3)

University of London physicist David Bohm, for example, believes Aspect's findings imply that objective reality does not exist, that despite its apparent solidity the universe is at heart a phantasm, a gigantic and splendidly detailed hologram


The Holographic Universe

This article is written by Michael Talbot.
It is seven pages of well written, simple prose.
You will be told what a holograph is
You will see how the notion of a holographic universe suggests itself to some physicists.
You will get a sense of the connection between quantum mechanics and the ancient sense that the world is maya,

The Holographic Universe (3) Fragment

"....the apparent faster-than-light connection between subatomic particles is really telling us that there is a deeper level of reality we are not privy to, a more complex dimension beyond our own .... And... we view objects such as subatomic particles as separate from one another because we are seeing only a portion of their reality.,," Michael Talbot.
The Holographic Universe

The Holographic Universe (3) Fragment 2

"...In addition to its phantomlike nature, such a universe would possess other rather startling features. If the apparent separateness of subatomic particles is illusory, it means that at a deeper level of reality all things in the universe are infinitely interconnected.

The electrons in a carbon atom in the human brain are connected to the subatomic particles that comprise every salmon that swims, every heart that beats, and every star that shimmers in the sky.

Everything interpenetrates everything, and although human nature may seek to categorize and pigeonhole and subdivide, the various phenomena of the universe, all apportionments are of necessity artificial and all of nature is ultimately a seamless web.

In a holographic universe, even time and space could no longer be viewed as fundamentals. Because concepts such as location break down in a universe in which nothing is truly separate from anything else, time and three-dimensional space, like the images of the fish on the TV monitors, would also have to be viewed as projections of this deeper order.

At its deeper level reality is a sort of superhologram in which the past, present, and future all exist simultaneously....".  Michael Talbot

The Holographic Universe

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