March 30, 2009
I read this article back in 2005… And as I feel it brings credence to some of the ideas I have discussed thus far, I am now posting it here for all to see:
March 30, 2009
Having introduced the idea of infinity, I would like to now introduce and develop four other important ideas. Namely ‘Self Similarity’, ‘Diversity’, ‘Divergence’ and ‘Evolution’. These ideas will become the back bone to most of the future writings found on this blog.
1. Self Similarity
In mathematics, a self-similar object is one that exactly or approximately looks the same on any scale i.e. the whole has the same shape as one or more of the parts. Many objects in the real world, such as coastlines and fluctuations in stock market prices, are statistically self-similar: parts of them show the same statistical properties at many scales.
It is also known that self-similarity is a typical property of fractals. A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, as stated in Mandelbrot’s The Fractal Geometry of Nature (1982 W. H. Freeman and Company). The term “fractal” was coined by Benoît B. Mandelbrot in 1975 and was derived from the Latin word fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation (in the case of the Mandelbrot set, the complex quadratic polynomial zn+1 = zn2 + c ) that undergoes iteration, a form of feedback based on “recursion” i.e. an expression, such as a polynomial, each term of which is determined by application of a formula to preceding terms.
When a fractal equation is iterated ad infinitum OR infinitely, they are then considered to be infinitely complex. Through out this infinitely complex structure self similar patterns abound through out the whole on all scales.
Perhaps now would be a good time to develop the idea of fractals further. Below I have provided a link to an hour documentary narrated by the late Sir Arthur C. Clarke:
Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, lichen and snow flakes. Some man made systems are fractal too i.e. stock market prices have been shown to be fractal in nature.
ii. Stock Market Prices:
Here is a picture of a lichen pattern:
Below is a picture that I generated using Corel Draw’s ‘Fract Flame’, where I’ve used a similar color scheme to that found on the lichen above:
Notice anything similar between the the two of them?
iv. Bacterial Growth
While the above eddies and flows of growth might not be quite as stable a near perfect mathematical models (possibly due to imperfections in the surface structure of the agar), there is none the less a similarity shared with the Mandelbrot set.
v. Obviously if one was to zoom into one of these real life objects, such as a snow flake for example, one would eventually loose sight of the self similar cascade of iced tips, as molecules of atoms, then atoms themselves, then electrons, protons and neutrons, and eventually quarks became visible. It’s almost as if there are sudden disparities between two structural scales of observation; a boundary where self similarity flips from one larger field of view into another smaller field.
However, even then, self similar structures that are seemingly unrelated to one another can jump out at an observer from the most unlikely of places:
vi. Cosmological Phenomenon
The Cat’s Eye Nebula (NGC 6543) is a planetary nebula in the constellation of Draco. Structurally, it is one of the most complex nebulae known, with high-resolution Hubble Space Telescope observations revealing remarkable structures such as knots, jets and sinewy arc-like features.
Above is an outline of the Julia set, another very complex mathematical structure exuding infinite complexity, which structurally resembles the Cat’s Eye nebula.
vii. Generating Fractals
If you would like to view a fractal on your home computer, you can download a freeware program called GNU Xaos (which has both a PC and Mac version available) from the following link:
Note: obviously the above pictures are only meant as suggestions to coax one into keeping an eye out for new ideas and patterns observed in nature’s mesmerizing flow.
In the English dictionary, the word ‘diversity’ makes a point about difference. It is defined as:
a. the state or fact of being diverse; different; unlikeness
b. variety; multiformity
c. a point of difference
It denotes separateness, division, discrepancy, fluctuation, heterogeneity, incongruity, inconsistency, even mishmash. For example, when we talk about biodiversity, we often look as the variation of life forms within a given ecosystem, biome, or for the entire Earth, at given time. The biodiversity found on Earth today consists of many millions of distinct biological species, which is the product of around and about 3.5 billion years of evolution.
But where did all these distinct biological species come from? Were they always here? Was difference always present? Or did they evolve, as Darwin’s theory of evolution states, from one ‘point’ i.e. one common ancestor? Did they transmutate over ages, slowly being reinvented as iterated nuances of the original form, redefining their habits into separate niches that gave them greater domain to sustain their lively needs and modes of being, modified ad infinitum into a distillation of harmonious countenance with the surrounding and ever changing environment, so as to separate inordinately competitive struggles into a slightly more relaxed interplay?
In “The Vestiges Of The Natural History Of Creation”, written by the Scottish journalist Robert Chambers, though anonymously published in October of 1844 for fear of ridicule, the idea that natural phenomena arise and evolve through natural laws of development. It also “boldly” postulated that there could therefore be some sort of rational explanation as to how everything in the universe came into being… “The whole train of animated beings, from the simplest and oldest, up to the highest and most recent, are then to be regarded as a series of the principles of development. It has pleased providence to arrange that one species should give birth to another, until the second highest gave birth to man.” However, due to the orthodox views of the time, it suffered greatly at the hands of many critics.
Darwin, having been discouraged by the cruel obloquy that the “The Vestiges Of The Natural History Of Creation” had received, decided to postpone publishing his ideas on evolution until he had bolstered the theory with near irrefutable evidence. One thing in particular troubled him about his concept… And in hind sight he wrote, “At that time I overlooked one problem of great importance. The problem is the tendency in organic beings descended from the same stock to diverge in character as they become modified.” Darwin further noted in his quest for refinement of the basis of evolution, the principle for divergence, which turned out to be the missing piece of his great puzzle. In November 1854 he wrote, “And I can remember the very spot in the road whilst in my carriage, when to my joy, the solution occurred to me. The solution, as I do believe, is that the modified offspring of all dominant and increasing forms tend to become adapted to the many and highly diversified places in the economy of nature.”
So, in short, Darwin realized that the more individual species differed from each other, the better able they would be to take advantage of the particular environment in which they all shared. Just as importantly, species would adapt even more as they adjusted to each other. And it is this interdependence, this balancing out of supply and demand, that has a parallel in what would eventually become the Victorian factory system. As can be seen from Adam Smith’s ideas about industry, a wonderful resemblance between the notion that one can produce more wealth if one has people who are specialists i.e. instead of everyone being farmers, if some people became tailors, while others become leather workers, you can produce more wealth of better quality, than if everyone was to do everything themselves i.e. be a jack of all trades and a master of none.
After all, using a notion of Self Similarity, a branch of a tree comes from the main tree trunk, which stemmed up from seed and nut… And no two branches ever precisely overlap. For what would be the efficiency of leaf upon leaf upon leaf, stacked one a top another with only the upper most being exposed to the Sun’s light?
So perhaps the ‘point’ of life’s diversity (the main trunk) could have ‘seeded’ from this simple idea of non living matter transmuting into a living form:
But perhaps I should let a “grand” documentary describe how basic tenets of these ideas gave rise to the intermingling of Self Similarity and Diversity. After all, man’s folly for almost time immemorial was to separate himself from nature’s base of animal like bewitchery. How could ‘He’, descendant of Adam, be linked to the surrounding unenlightened, earthy, distinctly corporeal and depraved natural world? Darwin, having been slammed by his colleges from Cambridge for his heretical ideas about man’s ‘obvious’ links to the natural world, found new angles of commonality within the real world which provided a solid and sure footing for his theory of evolution to remain in the fore front of scientific review, regardless of theistic doctrine…
Darwin’s Struggle – The Origin Of Species
Diversity, seen in pigeons… Natural selection, seen in corporate pressures of the industrial world… What more does one need to see that Self Similarity flows unifyingly through the Diversity of life and all of creation, man-made or natural? A Diversity that originated through Self Similar patterns branching out and away from the ‘seed’ of life’s origin, chaoticly evolving through time’s eternal languid flow, using fixed laws of physics, chemistry and biology, iterating subtilties into ever more refined complexities of balanced spread, till entwined ecosystems abound throughout the world!
But what perhaps brings me closest to Darwin was his compassionate and humanistic direction through the comparison of his children to the orangutan babies he had seen in London. We know from his diaries that he loved his children with all his heart, and so this comparison is not a debasement of their being. Rather he is ‘painfully’ aware of the similarities between both human and orangutan, as well as mankind’s own denial of the obvious truth.
Clearly it can be seen just by zooming into the Mandelbrot set that Diversity and Self Similarity abound in beatific balance. And, as M. C. Escher duly noted in the endeavors of his life’s work, these patterns also abound in the natural world, and flow from one another’s essence…
Darwin noticed that even the hexagons created by honey bees in their hives emanated from instinct rather than divine providence. So what he realized was that, while divine origin gave credence to the existence of these sturdy and structured forms which Euclid had disclosed and discussed in his Elements, it was innate in their being to do so. Much in the same way, as the previous photographs of a brain cell and the universal arrangement of galaxies demonstrates, the idea of the whole is used to know itself.
This notion that we are part of universe knowing itself is perhaps not such a divine myth as one might initially presume. Rather when empirical knowledge is in place, it becomes evident. After all, we are all comprised of atoms, which nearly everything in the universe is made from. So would we surely not use the same forces and methods that the universe uses to simply be?
But perhaps it is What Darwin Didn’t Know that really solidifies the ideas of Diversity and Self Similarity…
What Darwin Didn’t Know:
As one can probably see, the exquisite diversity found here on Earth stems back from time immemorial. Having been a molecular geneticist myself, I too have seen good cause for the ‘evidence’ that DNA has illuminated. Namely that the beautiful unfolding of species throughout the eons of time slowly distill back into one basic precursor of life, that central ‘point’ if you will, where we all came from.
Obviously not every branch will survive all the way through to the present day. Resources are limited and space is finite here on our spherical world. The “Terrible Lizards” were the dominant vertebrate animals of terrestrial ecosystems for over 160 million years, from the late Triassic period (about 230 million years ago) until the end of the Cretaceous period (65 million years ago). However, Dinosaurs eventually became extinct. What happened to them, no one exactly knows. But the fossilized remains of their bones are shinning evidence that these creatures once had the upper hand, as humans now do.
When I zoom into the Mandelbrot set, it’s almost as if similar patterns to what we vaguely know about our past, can be found writhing and rippling through it’s sublime topography. Subtle changes that allow greater efficiency and diversity, so that life settles into a new nest of chaotic equilibrium. As the first law of thermodynamics mandates the conservation of energy… So how better would life conserve energy, other than to diversify and specialize, rather than engage in the draining aspects of unnecessary competition?
But what astounds me most is that this rather obvious (but much over looked) beauty of fractals also gives credence to the idea that self similarity has been repeating throughout the ever evolving structures of organic life here on earth… Over the course of billions of years, basic structures have been reused again, and again… Eyes, teeth, brains, stomachs, bones, limbs, etc… All these anatomical textures of interwoven molecular weaves have ‘worked’ for life in some way or another, and over the years have simply been reworked/refined into better ideals, so that better ‘results’ occur. Some of these organs occur in very successful combinations, while others in fleetingly strange and unviable anomalies that only serve to demonstrate nature’s ‘bugged out’ chaotic approach to this complex chemical reaction called life.
Obviously it can be seen that universal modes of being have simply been reused, recombined and refined over and over again to give rise to what we now know and see around us today. I would hazard a guess that the eyes of Dinosaurs and man do not differ that much from one another. And I’d bet that they too would have had very similar internal organs to our own. And yet the outer bodies only vaguely (if at all) resemble one another’s. Just in this way, as I delve into the M set, I see islands of similarity in an eternal sea of change. Could there be some credence to my mode of thinking?
That I will leave for you to decide…
Again, I would like to bring to the reader’s attention my own intentions by writing this blog… It is not my aim to disclose a hidden meaning to life’s eternal flow. Neither is it to procure new scientific or religious standings. Nor is it my aim to put into disrepute current world views OR Religious ideals. Rather it is to ‘suggest’, using analogies recently disclosed through science that appeal deeply through intuition to my reasoning, new modes of possible understanding about ‘what’ We are and ‘why’ We came about in this Garden Of Eden.
March 29, 2009
The sage whose words are ambiguous you call great.
Those who advocate discipline you shun.
With one, you treat the words the way you want.
With the other, you resent having no quarter.
It is unfortunate that we need the words of the wise. Though they are essentially to out beginnings on a spiritual path, they can cause problems because they must be interpreted to be understood. Because words are imperfect, every generation rewrites itself.
People love ambiguity, especially when it comes to religion. They can interpret things any way they want. If they are unhappy with the cast given to a particular teaching, they invent ways to circumvent it, which is why we have so many authorities, schools, and sects.
It is no accident that the most revered sages are dead. They aren’t around to correct our misguided notions, to change their teachings, or even to make mistakes that might mitigate our reverence. Christ, Mohammed, Buddha, Lao Tzu – how many of us are actually devoted to the wisdom that they embodied? Or have we made them mere screens upon which we project our own ideas?
It is important to spend time with a living teacher, one who can correct mistakes and discipline you. But the object of such study should not be the creation of a new orthodoxy. Rather, your goal should be to bring yourself to a state of independence. All teachings are mere references. The true experience is living your own life. Then, even the holiest of words are only words.
March 19, 2009
March 17, 2009
“If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all.” Bertrand Russell
In this brief essay on the ‘infinite,’ I do not want to ramble on about uncertainties or truths. My aim here is not to lecture. Rather it is to encourage… So instead of joining the dots together in an obvious proclamation of basis, I am happy to quote certain others’ works that have more pertinently and eloquently touched aspects of the ‘infinite’ over the years, with a hope that the reader’s mind will naturally settle on the splendor lying behind the complex and distracting facades of catechism.
1. William Blake
William Blake (28 November 1757 – 12 August 1827) was an English poet, painter, and printmaker. Largely unrecognized during his lifetime, Blake is now considered a seminal figure in the history of both poetry and the visual arts of the Romantic Age. His prophetic poetry has been said to form “what is in proportion to its merits the least read body of poetry in the English language”. His visual artistry has led one modern critic to proclaim him “far and away the greatest artist Britain has ever produced”. Although he only once journeyed farther than a day’s walk outside London during his lifetime, he produced a diverse and symbolically rich corpus, which embraced ‘imagination’ as “the body of God”, or “Human existence itself”.
In one of his most insightful poems “The Auguries of Innocence”, he states:
“To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.”
For the full poem, please visit: http://www.artofeurope.com/blake/bla3.htm
What could such majestic tapestry mean? Well… Perhaps to the bovine logician, or the unenlightened literary reader, this idea might fall short of the lofty missive prescribed by Blake’s godly eye. But thankfully, with today’s scientific awareness of all things great and small, this notion may be better ‘understood’ with only a lax dedication towards ‘knowing’ the world around oneself better.
2. Niels Fabian Helge von Koch
Niels Fabian Helge von Koch (January 25, 1870 – March 11, 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.
He was born into a family of Swedish nobility. His grandfather, Nils Samuel von Koch (1801–1881), was the Attorney-General of Sweden. His father, Richert Vogt von Koch (1838–1913) was a Lietenant-Colonel in the Royal Horse Guards of Sweden.
Von Koch wrote several papers on number theory . One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem.
He described the Koch curve in a 1904 paper entitled “On a continuous curve without tangents constructible from elementary geometry” (original French title: “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire”).
The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. (Actually Koch described what is now known as the Koch curve, which is the same as the now popular snowflake, except it starts with a line segment instead of an equilateral triangle. Three Koch curves form the snowflake.)
The Koch curve is a special case of the Cesaro curve where:
which is in turn a special case of the de Rham curve.
One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:
- divide the line segment into three segments of equal length.
- draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
- remove the line segment that is the base of the triangle from step 2.
The Koch snowflake/star is generated using the same recursive process but starting with an equilateral triangle rather than a line segment. After doing this once for the Koch snowflake, the result is a shape similar to the Star of David.
The Koch curve is the limit approached as the above steps are followed over and over again.
The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. Hence the total length increases by one third and thus the length at step n will be (4/3)n: the fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line (dimension 1) but less than Peano’s space-filling curve.
Ever smaller and smaller… As one zooms into the Koch curve, steady self-similarity is exuded infinitly:
3. Karl Menger
In mathematics, the Menger sponge is a fractal curve. It is the universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge. It is a three-dimensional extension of the Cantor Set and Sierpinski Carpet. It was first described by Austrian mathematician Karl Menger in 1926 while exploring the concept of topological dimension.
Each face of the Menger sponge is a Sierpinski cerpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M0 is a Cantor set.
The Menger sponge is a closed set i.e. it contains its own boundary (unlike the Mandelbrot set); since it is also bounded, the Heine-Borel theorem implies that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesque measure 0.
The topological dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge, where here a curve means any compact metric shape of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.
In a similar way, the Sierpinski cerpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions. Thus any geometry of quantum loop gravity can be embedded in a Menger sponge.
Interestingly, the Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. This idea of the infinite held within the finite is perhaps not such a revelation as it might initially seem…
A ‘simpler’ more visual way to understand the complexity of Menger’s idea can be seen in the follow animation:
4. Benoît B. Mandelbrot
Benoît B. Mandelbrot (born 20 November 1924) is a French mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Cente; and Battelle Fellow at the Pacific Northwest National Laboratory. He was born in Poland. His family moved to France when he was a child, and he was educated in France.
In mathematics, the Mandelbrot set, named after Mandelbrot himself, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z0=0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets.
In other words… Part of the charm of the set is that it springs from such a simple equation: z2 + c. The terms z and c are complex numbers, which consist of an imaginary number (a multiple of the square root of –1) combined with a real number. One begins by assigning a fixed value to c, letting z = 0 and calculating the output. One then repeatedly recalculates, or iterates, the equation, substituting each new output for z. Some values of c, when plugged into this iterative function, produce outputs that swiftly soar toward infinity. Other values of c produce outputs that eternally skitter about within a certain boundary. This latter group of c‘s, or complex numbers, constitutes the Mandelbrot set.
When plotted on a graph consisting of all complex numbers, the members of the set cluster into a distinctive shape. From afar, it is not much to look at: it has been likened to a tumor-ridden heart, a beetle, a badly burned chicken and a warty figure eight on its side.
A closer look reveals that the borders of the set do not form crisp lines but seem to shimmer like flames. Repeated magnification of the borders plunges one into a bottomless phantasmagoria of baroque imagery. Some forms, such as the basic heartlike shape, keep recurring but always with subtle differences.
The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called “zooming in”. The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures, and explains some of their typical rules.
The magnification of the last image relative to the first one is about 10,000,000,000 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometres. Its border would show an inconceivable number of different fractal structures…
And here I will leave you with a quotation…
“Pure mathematics is, in its way, the poetry of logical ideas.” Albert Einstein
March 16, 2009
Lake shadows color of cold,
Willow branches weep ice.
Swans rises dazzling in the sunlight.
After long self-cultivation, one’s accumulated energy reaches a threshold and then bursts out full, breathing, and vibrant. Without the careful building of momentum, this moment of release would never have been possible. With long years of preparation and experience, the freeing of the soul will not be mere dissipation but will be so strongly focused that it lifts one into a higher state of awareness. When one’s spiritual energy emerges, it feels like a swan rising from the water.
Once you have reached this level of stored energy, you will be a different person. On one hand, you may take genuine comfort in the point of attainment that you have made. On the other hand, you now see all the other possibilities that remain for you to explore.
With emergence of great possibilities comes the need for responsibility. If you diverge from your life’s path in order to explore new vistas, remember how far you are flying, and remember to return at the proper times. Only you can decide how to arrange your life. Once you are a strong flier, you must still use wisdom to direct your flight.
March 14, 2009
Stand at the precipice,
That existential darkness,
And call into the void:
It will surely answer.
The precipice represents our dilemma as human beings, the sense that this existence is all too random, all too absurd. Is there order? Is there a force directing things? These are the important issues, so important that we cannot rely on scripture, but must instead explore on our own.
The followers of Tao compare the void to a valley. A valley is void, yet it is productive and positive. The emptiness of the valley permits water to accumulate for plants. It allows life-giving sunlight to flood its surface. Its’ openness gives comfort to people and animals alike. The void should not be frightening. Rather, it contains all possibilities. Peer into it, call out, not just with your voice, but with your whole being. If your cry is deep and sincere, an echo is sure to return. This is the affirmation of our existence, the affirmation that we are on the right path. With that encouragement, we can continue our lives and our explorations. Then the void is not frightening, but a constant companion.
March 12, 2009
Heaven and hell:
Our meditation opens seldom glimpsed areas of our subconscious. When that happens, extraordinary thoughts and awareness come to us with seeming spontaneity. We realize truths that were opaque to us before; we perceive events that were previously too distant. But no one ever became superhuman because of meditation. They only opened their own latent potential. Everything is locked inside of us and need only be opened. That is why it is said that heaven is within us.
In the same way, the pains and the struggles of the past sometimes haunt us with astounding vehemence. Problems and conflicts are difficult to exorcise. Although we may practice spirituality and move on to new endeavors and relationships, past hurst still come back to in our memories and dreams. These are not demons from another world, nor are they karmic manifestations of previous lives; they are scars in our subconscious. No matter how diligently we try to make progress, there still are pains that curse us day after day. This is why it is said that hell is within us.
We ourselves are the battleground for good and evil. There is no need to look beyond our world. Everything to be understood is within us. All that must be transcended – the pains and scars of the past – is within us. All the power of transcendence is also within us. Tap into it and you tap into the divine itself.
March 11, 2009
A great Taoist master once dreamt that he was a butterfly fluttering here and there. In the dream he had no awareness of his individuality as a person. He was only a butterfly. Suddenly, he awoke and found himself laying there, a person once again. But then he thought to himself, “Was I before a man who dreamt about being a butterfly, or am I now a butterfly who dreams about being a man?”