Polynomial

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 z_n+1 = z_n² + 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.

i. Coastlines:

http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension

ii. Stock Market Prices:

http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street

iii. Lichen

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:

http://wmi.math.u-szeged.hu/xaos/doku.php

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.

2. Diversity

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?

3. Divergence

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:

http://blog.wired.com/wiredscience/2008/09/biologists-on-t.html

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

Part 1: http://www.youtube.com/watch?v=e1tk-NLrIKY

Part 2: http://www.youtube.com/watch?v=B3Z05zf17k0

Part 3: http://www.youtube.com/watch?v=QfVMLeBPOQk

Part 4: http://www.youtube.com/watch?v=VQLVJ8Rjkvc

Part 5: http://www.youtube.com/watch?v=Vbl91VKxThA

Part 6: http://www.youtube.com/watch?v=86GJIcRZBWI

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?

4. Evolution

But perhaps it is What Darwin Didn’t Know that really solidifies the ideas of Diversity and Self Similarity…

What Darwin Didn’t Know:

Part 1: http://www.youtube.com/watch?v=D7bL-jr2rh4

Part 2: http://www.youtube.com/watch?v=P25u_1RSmxg

Part 3: http://www.youtube.com/watch?v=O2bydu3qAj0

Part 4: http://www.youtube.com/watch?v=6Ba4ibVhy0U

Part 5: http://www.youtube.com/watch?v=jAsuMGe3gjM

Part 6: http://www.youtube.com/watch?v=Fr2PnrVwZ6c

Part 7: http://www.youtube.com/watch?v=1MXW0-DBUGs

Part 8: http://www.youtube.com/watch?v=oXsHOITZ0nM

Part 9: http://www.youtube.com/watch?v=Qcpb5zlSJ5Q

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…

Postscript

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.