For All Fractal Purposes ... an introduction

by William J. Jackson

Gateway One:Encountering Fractals in Nature

Gateway Two: An Architectural Fractal - St. Patrick's Cathedral

Gateway Three: Kandariya, a Fractal Pattern Temple, Khajuraho, North India

Gateway Four: Fractal Images of Personhood

Gateway Five: A Fractal Vision of Hindu Traditions

Gateway Six: The Soul, Earth's Life, and the Cosmos

Gateway Seven: The Ganglia of Five

 

Gateway One: Encountering Fractals in Nature

Strolling along a shady forest path among the tree trunks like pillars rising to the canopy of branches in the sunshine you glance down. At your feet a fern lies across the winding path -- its structure is very fractal. The featherlike frond of the fern is composed of rows of smaller fronds, and the smaller fronds have their rows of even smaller fronds, too.

Trees branches, rivers meander, cloud structures, lightning, the patterns of insects and birds in their environments, the flames of fire, the ways of wind, soil particles and aggregates, the breaks of metal and earthquakes, all involve fractal processes. In lungs, bile duct system, bowel and kidney, in the neural networks of the brain, in the placenta and the heart, in the way we walk and talk are full of fractals. Fractals of vision show us order.

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Gateway Two: An Architectural Fractal - St. Patrick's Cathedral

You are travelling through the grid-like streets of a busy city -- block after block of rectangular buildings with gridlike walls of windows almost as regular as checkerboard squares surround you. Occasional rollerbladers glide between the gridlocked lanes of frozen traffic.

Suddenly you look up and see something of a different order, an immense fractal. The two spires of a cathedral point up, each tall thin peak narrowing to a high point with a small cross at the top. And at various points along the way up there are smaller spiresññ the same sharp angle appears again and again, but at a reduced scale. There are "self-similar" arched windows, tall and narrow, and around the rest of the building there are many smaller "self-similar" spires and other details which reflect the whole, giving the building a kind of many-spiked, multi-splintered rhythm -- a celebration of aspiring, making an impression markedly different from the boxy rectangular shapes of the surrounding buildings. The cathedral is simple, as a snowflake or a great fir tree is simple, yet complex, ornate, even cosmic in the way it suggests infinity. While the grids seem very manmade, neatly modern and rational, the fractal has natural structure too, like a forested mountain or burst of hoarfrost
shards. The grids seem mechanically linear, like prison walls and barred windows, next to the dynamical vitality of the fractal cathedral. The fractal cathedral embodies rhythmic laws of nature that seem older than grids, crystalline as frost, organic as thistle, free as life to express deep impulses with rhythm.

As Benoit Mandelbrot noted, " Fractal geometry is not just a chapter of mathematics" -- if it were it would be relegated to the plethora of abstract technical obscurity. It is important because it “ helps Everyman see the same old world differently,” to notice patterns with principles underlying complicated forms, to observe subtle order. It suggests the infinite in the dynamic cosmos, and access to transcedence. In beauty humans find an intimation of the holy, a way to the blissful, a taste of the resonance of consciousness.

Such a sign of creativity is a fitting structure for the idea behind the cathedral, “ a magnificence that breaks the circle of rational satisfaction completely apart.” Such a sign of creativity enacts an aesthetic linking vast systems and the human soul. It is an eruption of exuberance, lifting us up, impelling our vision beyond the inward workings of logic, to awe.

You are explaoring a fractal; flexibily you expand by degrees, magnifying the pattern to the vastest whole imaginable; just as fluidly you telescope down, rhythmically arabesquing around to munute versions of the whole.

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Gateway Three: Kandariya, a Fractal Pattern Temple, Khajuraho, North India

Looking through a book on architecture in India you notice the rising spires of a temple silhouette are like mountain peaks. Light seems to be playing in the spaces between the peaks, like streams of life-waters flowing from the heights, down through the space of empty valleys. The Kandariya temple contours in silhouette make a fractal impression. The supreme shikhara or spire of utmost aspiration is made up of smaller scale self-similar subsidiary shikharas, rising from yet smaller ones. The eye is drawn up higher to the utmost peak. The main point of the whole which is reflected in the parts. This is a vision of the cosmic mountain, center of the world. The ideal form gracefully artificed suggests the infinite rising levels of existence and consciousness, expanding sizes rising toward transcendence above, and and at the same time housing the sacred deep within. The gated enclosures-within-enclosures enshrine an inner sanctum at the inmost depths of divine mystery.

The mountain peaks are idealized, describing a fancy edge between our time and space and the perfection of eternity and infinity. Perhaps, it is as if ordinary people are privileged to see the glorious mountain reflected in the consciousness of the sage. The pilgrim arrives and looks up at the temple against the bright sky. Light is flooding down along the peaks and in between, like streams of life-waters flowing from Himalayan heights. The temple, like all Hindu cosmic images, is a reflection of the play and patterns of consciousness in the pool of time, a context in which pilgrims find themselves and make their way on their inner journey to inmost depths to glimpse spiritual truth, the beyond which is within. Again, imagine yourself within a fractal. Flying to different scales you observe tiny details, each of which is related through some self-similarity or organic participation in the over-all configuration. You can envision coordinated growth, nested relationships, branches splitting smaller, cycles of shrinking and expanding... The fractal provides structures which are not static or linear but fluid with life-like creative possibilities. The Taj Mahal has fractal self-similarities in different sized domes and arched windows with their empty spaces.

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Gateway Four: Fractal Images of Personhood

You talk with friends about fractals. Though you think that maybe fractal beauty speaks for itself. You say "Fractals usefully extend our vocabulary, helping us recognize patterns in natural processes." One friend says it seems more than just another term. "Is it a totalizing concept? How do people fit into it? " You haven't thought it out, and say you don't intend to go around imposing the pattern everywhere but think it's helpful to explore wherever that kind of pattern seems to appear. Another friend says DNA code, miniaturized traits and family resemblences would be a possible application. Of course a DNA molecule is not a little man, a homunculus -- a misconception that midieval Europeans had about sperm. But the idea of seeds is suggestive: the whole coded in densely packed form, the traits condensed in toto, the whole in the parts and the potentially infinite generationññ all humanity from "Adam and Eve," endless forests from an acorn, the same theme played out in time in many selfñsimilar adaptations. A self similarity on different scales is evident if you consider DNA, with its sequences which show ,“ long-range fractal correlations and prominent short range periodicities;” DNA code reflects the physical organism as a whole, and the sharing of selfñsimilarities in genetic lines of inheritance, seen in family resemblences. And there are culturally shared human behavior values. Someone asks "Is there a danger of regimentation into groups here -- anti individualism? Does fractal structure undercut autonomy and freedom?" You recall that Francois Lyotard in The PostñModern Condition said just the opposit -- since each part is whole, the structure of fractals ensures free will, decentralization. You have to ask yourself, "What is a person?" This issue brings up archetypes, psychological images, patterns which attract attention and call for certain kinds of behavior. For example, think of a person made up of people, and in turn they are made up of smaller personlike entitiesññ and you are thinking of the cosmic person archetype found in ancient cultures.

The Cosmic Person containing living beings organically is an image found in Vedic Purusa, Chinese P,’ anku, Germanic Ymir, Mother Earth Goddesses, including Gaia and Sedna. Leonardo envisioned earth as person in his note book. The Mystical Body of Christ encompassing members of the faith, and Buddha Mind shared by all, symbolize corporate sharing of sacred nature.

Mankind is sometimes pictured altogether as a cosmic person, a universal being, as in Rabindranath Tagore,’ s philosophy. A quality of consciousness is to exist without necessarily drawing attention to itself or revealing its secrets. Collective , “ Unconscious,” means the knowledge you don,’ t know you know, the whole you aren,’ t necessarily conscious you,’ re a part of. You speak a language without being aware that it,’ s part of a vast pool of languages, maybe without even thinking it,’ s a common symbol system--it's reality, the way you think where you are.

Carl Jung wrote: ,“ If it were possible to personify the unconscious, we might think of it as a collective human being combining the characteristics of both sexes, transcending youth and age, birth and death, and, from having at its command a human experience of [more than] two million years, practically immortal. If such a being existed, it would be exalted above all temporal change; the present would mean neither more nor less to it than any year in the hundredth millenium before Christ; it would be a dreamer of ageñold dreams and, owing to its immeasurable experience an incomparable prognosticator. It would have lived countless times over again the life of the individual, the family, the tribe, and the nation, and it would possess a living sense of the rhythm of growth, flowering and decay.,” Freud spoke of the recapitulation involved in one person's maturation: "The psychic development of the individual is a short repetition of the course of development of the race." The Cosmic Person myth is a vision of a kind of fractal intuition: organisms are organically related, share common fates. Each reflects the whole, with selfñsimilarities, selfñaffinities, etc.

The person in JudeoñChristian outlook is said to be made ,“ in God s image,” and each society is a body politic made up of people playing different roles. Some cultures envision ,“ persons within each individual, in subtle ancestors,” from previous generations, and in fact a reflection of the story of life in the universe. Moderns call the remnants DNA memories, tribal people speak of spirits of the ancestors. Hindus speak of Indriyas -- inner rulers or powers governing different abilities, to see, to hear, etc. Thus each individual is an ensemble, just as the body is made up of "members" and "organs." The dynamic fractal of humanity, sub specie aeternitatis , with each person belonging to the full spectrum, shows the human story,’ s long continuum; the players are wholeñlike parts of the total drama, recapitulating earlier and presaging subsequent lives, in greater and lesser degrees. They are microcosms of the macrocosm, souls seeking wholeness. ,“ Unless psyche was there at the fireball origin of all this, there is no psyche possible,,” as Brian Swimme puts it.

Childhood, as a phase of the life cycle, presents certain experiences like cameos, making a fractal on various scales of one,’ s personality and destiny, one,’s unique calling, a prediction of what the full portrait will be like. This is metaphorically speaking; since life is a moving picture. The conceptual images of fractals help us picture this dynamic vitality.¨

Fractals help visualize the quality of nestednessññ withing within within like Russian dolls. They depict the interrelatedness, and show alternatives to static linear separation images (which, due to Euclidian presuppositions about what is correct and possible, and due to materialism and individualism, may be prevalent and out of touch with what is known by both traditional and modern scientific outlooks). Images of a person made up of people, a being of beings, a composite personified, incorporating or incarnating a congeries of powers and impulses together, are found in the arts and literatures of the world. Whitman: ,“ I contain multitudes. ” R.W. Emerson: “ In the depths of each man,’ s biography lies the story of all men.,” Sufis say all the prophets are subtly inside a person.

On the level of religious imagination expressed in mythic images, Proclus pointed out a fractalñlike configuration of divine powers in mythic figures. There is an innumerable multitude of daemons who have the same appellation as the leading deity: Venus and her cupids, Mars and all his warriors who are smaller versions of the archetype. In Hindu traditions Shiva is the Lord of all the yogis great and smallññ the greatest yogi, who transcends and encompasses all. The vengeful paranoid jealous God of extreme fundamentalists in prophetic religions is reminiscent of those extremists -- they seem to incarnate his terrorizing wrath and vengence on varying scales. French artist Gustave Dore, illustrating Dante,’ Divine Comedy , engraved a vast angel made of angels, and a vast serpent made up of demons and damned souls.

Thus there are many examples of selfñsimilarity on varying scales in conceptions and perceptions of the person. Entities within persons, persons in societies, societies in the human race, and the collective unconscious as reservoir of shared images. Whole within whole within whole within whole: many art works and images in world literature depict this reproductive, selfñregenerative, selfñreferential quality of consciousness. It is possible to imagine the essence of the human race as a person; the essence of the person as DNA; the essence of DNA as intelligence; the essence of intelligence as light.

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Gateway Five: A Fractal Vision of Hindu Traditions

In specific traditions we can see fractalñlike patterns in society and images of the nature of the divine. For example, Indologist Madeleine Biardeau shows the atemporal structure of classical Hinduism’ s world: “ each territorial ‘ unit'- which can be as small as a village with its chief - one finds, theoretically at least, the totality of the Hindu socio-religious structure and its values”5 - very much a fractal-like image: all in each.

The different castes are associated with different parts of the cosmic Person or Purusa (Rig Vedav X.90) and the Laws of Manu assert that therefore if one acts contrary or oversteps boundaries of one’ s role in society, one disturbs the overall order. The logic (as in European feudalism) gives an order in which the members are participants, a chain of being.

Thus, there are fractalñlike patterns of part/whole correspondence and transcendence on the part of persons in the Hindu worldview: “at the level of secular society, both theoretical and concrete, there was no place for the human individual as such; the individual is only a unit within his group, caste or lineage, which determines his place in the hierarchy... from his group he receives a certain idea of his duty... The day he feels too strongly the relativity, or indeed the vanity, of what is expected of him he breaks away from the group and attends to what is most essential and permanent within himself -- the ... which still derives from the social group the revelation of its existence... it attains liberation, through knowledge or yoga, only by shedding all differentiation, by being absorbed into the Absolute where it finally loses all that remains of empirical individuality.” In this view the ego is dissolved and there is a merging in a more encompassing state of being.

And there is a fractal-like pattern of multiple forms of God in Hinduism: "Each Hindu deity is part of a complex. If the Absolute is located anywhere, it is this complex and not one or another of its members. It is in this sense that ‘ all the gods are merely forms of Brahman .' Let us say that together they all constitute Brahman." This concept is a vision of cosmic totality, selfñsimilar organically related parts of a larger composite on different scales: fractal-like.

Fractal patterns show natural processes and principles of organization.

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Gateway Six: The Soul, Earth's Life, and the Cosmos

There is a pressing need to understand both the universe around us and ourselves in a manner commensurate with the complexity found in the dynamics which characterize existence. Oversimplification distorts. But there is a deep need as well to realize the functional simplicity of which creative minds are capable. Splinteringly complex derangements may amuse but they cannot provide enduring answers which satisfy and support the needs of life. “ The simpler the solution the more powerful it architect I.M. Pei has said. The simpler the solution (if truly enduring and satisfying) the more complexity it skillfully holds. Poets, artists, scientists prove that functional elegance is beautiful simplicity.

Let'ss start small and begin at home, considering an image used to depict the situation of the soul, a simple image capable of holding complexity. In its central image, language, and style, Interior Castle , like so many works of genius, is extremely simple. St. Teresa envisioned the soul as ‘ castle made of a single diamond... In which there are many rooms, just as in Heaven there are many mansions.’

Teresa narrates the progress of the soul being purified and refined as it moves through the mansions toward the inmost chamber, toward the promise of spiritual communion with God. The small perfect diamond image holds the whole unfolding of the complex soul, reflects the structure of heaven, conjuring in the reader’ s mind, fractal selfñsimilarities. Next let’ s consider the order of a place on earth considered to be most sacred to a particular community. Here too the concept must be able to hold much complexity with unifying simplicity. “ Palestine, Jerusalem, and the Temple severally and concurrently represent the image of the universe and the Center of the World. This multiplicity of centers and this reiteration of the image of the world on smaller and smaller scales constitute one of the specific characteristics of traditional societies,”

Finally we’ ll consider a system which holds the planets of the cosmos, living beings on earth, and spiritual knowledge in humans. “ The Vedic system of knowledge is based on the equivalences between stars adhidaivika ) living beings ( adhibhautika ), and man’ s cognitive structure adhyatmika “ Corresponding patterns, symbols which work on several levels, a vision of order held in a system of dynamic relations. Without unity things fly apart, incoherently; complexity is elusive unity.

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Gateway Seven: The Ganglia of Five

Consider the fiveness of the person. Take for example yourself. Note how basically your physique is formed in five limbs branching from your torso: two legs, two arms and your head. This branching (“ poor bare forked animal as Shakespeare’ s King Lear says) is fractally replicated at the end of each limb --hands branch with four fingers and thumb, feet with five toes, head with senses. There are five ññ sight, smell, hearing, taste and touch, with cordinating powers in central nervous system and brain.

In some traditional worldviews (e.g., Hindu) the five senses correspond with the five elements which make up the universe: wind, fire water, earth. Hindu philosophy as found in the Prasnopanisad holds that the body is a complex produced by the combination of five: Akasa (space), Vayu (wind), Agni (fire), Jala (water), Bhumi (earth). Each of the elements composing the body has a presiding deity.

In Upanisadic thought a person is made of five sheaths: 1. Annamayakosa gross physical body nurtured by food, 2. Pranamayakosa , sheath of vital airs, soul of the gross body, 3. Manomayakosa , sheath of mind, able to weigh good and bad, 4. Vijnanamayakosa , purposive will, decisive intelligence of commitment, 5. Anandamayakosa
, the joy of achievement, sheath of bliss. The pranamayakosa , vitalizes the whole body with motivation, and has five varieties: prana, apana, vyana, udana, samana. The Chinese classical text Tao Te Ching warns "The five colors blind a perso s eyes. The five musical notes deafen a person’ s ears. The five flavors ruin a person’ s taste buds.” Asian music systems make use of five basic notes. The five tastes are salt, sweet, bitter, sour and savory. .

The branching quality of the body is replicated at smaller and smaller levels, in lungs, nerve paths and blood vessels in the brain and eyes. The bile duct system, as well as the urinary collecting tubes in the kidney, and the structure of the lining of the bowel, the neural networks, the placenta, and the heart all display fractal patterns with intriguing (and very useful) selfñsimilarity and scaling. And when one dies, in the language of some Sanskrit texts one’ s body panchates returns to the five” elements. Thus, while the bodies are lively, and when they die, they do both fively. The starfish has thrived a long time, much longer than humans; it presages us and waves at us with its five-rayed design. We wave back and flow on.

"I think, therefore I am;" I fractal, therefore I am part of more than me.

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