Polynomial

The following exerts I have taken from James Gleick‘s veritable summation of one of science’s most enigmatic subjects, entitled “Chaos – Making A New Science.”

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Nature forms many patterns. Some are orderly in space but disorderly in time, others orderly in time but disorderly in space. Some patterns are fractal, exhibiting structures self-similar in scale. Others give rise to steady states or oscillating ones. Pattern formation has become a branch of physics and material science, allowing scientists to model the aggregation of particles into clusters, the fractured spread of electric discharges, and the growth of crystals in ice and metal alloys. The dynamics seem so basic – shapes changing in space and time – yet only now are the tools (i.e. computers and programs to run lengthy iterative ideas with) available to better understand them.

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So perhaps…

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It is a fair question now to ask a physicist, ‘Why are all snowflakes different?‘

Ice crystals form in the turbulent air with a famous blending of symmetry and chance, tapping into the special beauty of six-fold indeterminacy (due to di-hydrogen monoxide’s physical properties). As water freezes, crystals send out tips, the tips grow, their boundaries become unstable, and so new tips shoot out from the sides. Snowflakes obey mathematical laws of surprising subtlety, and it was impossible to predict precisely how fast a tip would grow, how narrow it would be, or how often it would branch. Generations of scientists sketched and catalogued the variegated patterns: plates and columns, crystals and polycrystals, needles and dendrites. The treatises treated crystal formation as a classification matter, for lack of a better approach.

Growth of such tips, dendrites, is now known as a highly nonlinear unstable free boundary problem, meaning that models need to track a complex, wiggly boundary as it changes dynamically. When solidification proceeds from outside to inside, as in an ice tray, the boundary generally remains stable and smooth, its speed controlled by the ability of walls to draw away the heat. But when a crystal solidifies outwards from an initial seed – as a snowflake does, grabbing water molecules while it falls through the moisture-laden air – the process becomes unstable. Any bit of boundary that gets out ahead of its neighbors gains an advantage in picking up new water molecules and therefore grows that much faster – the “lightening-rod effect”. New branches form, and then subbranches.

One difficulty was in deciding which of the many physical forces involved are important and which can be safely ignored. Most important, as scientists have long known, is the diffusion of the heat released when water freezes. But the physics of heat diffusion cannot completely explain the patterns researchers observe when they look at snowflakes under microscopes while growing them in the laboratory. The heart of the new snowflake model is the essence of chaos: a delicate balance between forces of stability and forces of instability; a powerful interplay of forces on atomic scales and forces on everyday scales.

Where heat diffusion tends to create instability, surface tension creates stability. The pull of surface tension makes a substance prefer smooth boundaries like the wall of a soap bubble. It costs energy to make surfaces that are rough. The balancing of these tendencies depends on the size of the crystal. While diffusion is mainly a large-scale, macrosopic process, surface tension is strongest at the microscopic scales.

Traditionally, because the surface tension effects are so small, researchers assumed that for practical purposes they could disregard them. Not so. The tiniest scales proved crucial; there the surface effects proved infinitely sensitive to the molecular structure of a solidifying substance. In the case of ice, a natural molecular symmetry gives a built-in preference for six directions of growth. To their surprise, scientists found that the mixture of stability and instability manages to amplify this microscopic preference, creating the near-fractal lacework that make each snowflake different and original. The mathematics came not from atmospheric scientists but from theoretical physicists, along with metallurgists, who had their own interest. In metals the molecular symmetry is different, and so are the characteristic crystals, which help determine an alloy’s strength. But the mathematics are the same: the laws of pattern formation are universal.

Sensitive dependence on initial conditions serves not to destroy but to create. As a growing snowflake falls to earth, typically floating in the wind for an hour or more, the choices made by the branching tips at any instant depend sensitively on such things as temperature, the humidity, and the presence of impurities in the atmosphere. The six tips of a single snowflake, spreading within a millimeter space, feel the same temperatures, and because the laws of growth are purely deterministic, they maintain near perfect symmetry. But the nature of turbulent air is such that any pair of snowflakes will experience very different paths. The final flake records the history of all the changing weather conditions it has experienced, and the combinations may well be infinite.

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For if we start to look at infinities, as Gregor Cantor did (see a BBC documentary called “Dangerous Knowledge“) i.e. the flea, on the back of the flea, on the back of the flea, ever smaller and smaller and smaller… Then we could start dividing up a single degree’s change in temperature into infinities… And if one then combines this ‘exacting’ variable with all the possible other infinite variables of humidity, impurity density, wind speeds, etc… Then perhaps we can begin to see the endless possibilities that allow each flake to be truly unique.

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It is here that I feel a powerful metaphor exists for the way our conscious/unconscious minds and attitudes are shaped along our journey through everyday life. When we are born into this world, we enter with a small body and a (nearly) blank and highly sensitive mind. As we move through the currents of care and love that we receive from those who tend to our daily needs, so our minds too begin to accumulate these patterns of love, caring and joy, as well as the hurt and even neglect, that we will use to express ourselves later on in life. Just as the tiniest scales of moisture, temperature, wind speed, etc… affect the infinitely sensitive molecular structure of the solidifying water that proves crucial to the geometrical shape that each snowflake will become; so too we respond just as sensitively to our early experiences… Experiences which prove crucial to our future choices and directions that we will take in life. Love bolsters the stability we seek to grow from, that we learn and create with; while neglect forges an instability and needy yearning. Both the stability and instability within our mind sets are gradually amplified from their microscopic beginnings, creating a near-fractal lacework of memories and emotions, that oscillate like a ‘strange attractor’ does, making each person so individual and so unique.

I cannot stress enough how sensitive we all are as human beings to our initial conditions. We absorb all the subconscious expressions of those around us. Expressions that emanate from body language, subtle intonations in the way we speak to one another, as well as what we think. These expressions stretch back the course of one’s life. Just as the final “flake” records the history of all the changing “weather” conditions it has experienced in its brief voyage here on Earth, so too the character of a person relays the history of experiences that has “weathered” upon their consciousness and physical form. Like a growing flake that falls to earth, floating in the winds for an hour or more, the choices made by the branching tips (that surround the seeding crystal) at any instant, depend deeply on such things as temperature, the humidity, and the presence of impurities in the atmosphere… So the six senses of any human being, living, feeling and growing within the social and environmental spaces it inhabits, are shaped by the same emotions and energies as those that are being projected into the world around them. Just as the laws of growth for a snow flake are purely deterministic i.e. every event is causally determined by an unbroken chain of prior occurrences, so we too will maintain a near perfect symmetry to ideas and attitudes projected into our surrounding environment. The nature of life’s turbulent flow is such that any pair of humans will experience very different attitudes to one another within life’s flow.

According to the Buddhist theory of interdependent origination, body and mind are dependent upon each other. In order to keep the body pure and normal, one must abstain from killing and harming others, stealing another’s property, committing adultery, drinking and using addictive drugs. Speech must also be kept pure by abstaining from lying, verbal abuse, deceit, and the avoidance of idle talk. The suitability of words can be assessed by examining five pairs of antonyms: words that are suitable to their occasions and those not so suitable; words that fit the facts and those that don’t; words that sound pleasant and those that sound rude; words that are beneficial and those that are harmful; and words that are sympathetic and that those are hateful.

So something we should all remember is, these experiences that we provide to others always serve to create… They can create both positive and negative aspects of mind. So be mindful of your current situations and your attitudes towards life. Whatever words we utter should be chosen with care, for people will hear them and be influenced by them for good or ill. When our minds are filled with sympathy and compassion, they will be resistant to the negative cycles we experience. We should not really let callus, unruly, indulgent words pass from our lips, especially when these words grow out of feelings of anger and hatred. The words we speak should always be words of sympathy and wisdom.

Let us all be mindful of the initial sensitivity of our life’s force and direction. And let us respect the chaos within and use it wisely to inspire the ever changing and wondrous patterns of life to abound. Because, while all the patterns of nature and mind are equally as beautiful as the rest, we must understand that negative actions can just as easily propagate as positive ones. We don’t want to shorten life, as wars and misunderstanding seem to so often do. All life really is so special and unique here on Earth. We have all surely won the lottery of existence! And so it should be treasured and adorned with encouragement to grow and encourage growth.

I’ll leave you with a quote from Wilson “Snowflake” Bentley:

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“Under the microscope, I found that snowflakes were miracles of beauty; and it seemed a shame that this beauty should not be seen and appreciated by others. Every crystal was a masterpiece of design and no one design was ever repeated. When a snowflake melted, that design was forever lost. And just to think, that that much beauty was gone, without leaving any record behind…”

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P.S. Many thanks to Ms. C. J. Kingsley for pointing out the lack of acknowledgement to James Gleick’s work, which has now been rectified.