ABSTRACT Two types of stable systems can be found in the physical universe: the death state of perfect equilibrium and the infinitely fertile condition of self-organized non-equilibrium. Self-organization provides useful models for many complex features of the natural world, which are characterized by fractal geometries, cooperative behavior, self-similarity of structures, and power law distributions. Pilgrimage systems may be viewed as natural and self-organizing structures wherein complexity develops through the spontaneous and coherent movement of people. The dynamical nature of the pilgrimage process may be analogous to that of other non-equilibrium systems that are being actively investigated today in the physical and biological sciences such as self-organized criticality and stochastic resonance. Examples of such systems in the natural world range from earthquakes, solar flares, forest fires, to biological evolution. Openness to the environment and coherent behavior are necessary conditions for self-organization and the growth of complexity. Biological, physical, and spiritual forces that disrupt equilibrium may drive cultural systems and individuals into a state of metastability or self-organized criticality lying between frozen inactivity of equilibrium and disordered hyperactivity. As systems depart more from equilibrium the amplification of small fluctuations by non-linear processes may result in stress-releasing avalanche-like movement of people to pilgrimage centers. The resulting cultural systems may be highly non-adaptive as judged by prior paradigms. Self-organization lacks an overall goal or teleology, although the emergence of new purpose (teleonomy) provides new meaning. Because of a common conceptual framework or microphysics, self-organizing systems are characterized by self-similarity and fractal geometries, in which similar patterns are repeated with different sizes or time scales without changing their essential meaning. The body, temple, and universe, as well as movement throughout them, may be self-similar. The pilgrim may follow a pathway that represents the universe from center to periphery or mimics the evolution of the universe from creation to death.

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To be presented at the conference "Pilgrimage and Complexity", Indira Gandhi National Centre for the Arts, New Delhi, Jan 5-9, 1999.

I. THE NON-EQUILIBRIUM UNIVERSE

When two different liquids are placed in the same vessel they will diffuse into a homogeneous mixture, thereby achieving an equilibrium state with the greates disorder and maximum entropy. Such behavior is the prediction of the Second Law of Thermodynamics. At some time far into the future all configurations of matter and energy in our universe will be similarly homogenized and will approach the condition of thermodynamic equilibrium. Stars will cool and eventually temperature differences between them and their surroundings will vanish. Depending upon whether or not the universe collapses, even black holes and neutron stars may dissipate their concentrated mass and fade into a homogeneous background. In its most perfect form, thermodynamic equilibrium involves a detailed balancing of all processes, such that each process is precisely balanced by its inverse. In such a balance no information, inhomogeneity, or complexity can persist. The state of thermodynamic equilibrium is frozen in time, remembering no history, containing no information, and incapable of sustained change.
Today portions of our universe depart significantly from equilibrium. The universe of stars, galaxies, and clusters of galaxies is highly inhomogeneous: hot, bright, dense stars are surrounded by cold dark vacuous space. The expansion of the boundaries of the universe prevents stagnation, insures the universe's continuing departure from equilibrium, and allows the formation of complex low entropy structures. Within the non-equilibrium biosphere of planet earth there are dense concentrations of information combined with complex and unlikely behavior.

II. DISSIPATIVE STRUCTURES

The physical chemist Illya Prigogine (1980) was one of the first to demonstrate the thermodynamic and chemical benefits of non-equilibrium by showing that non-equilibrium configurations of low entropy and great complexity can be stabilized if they are open to the inward and outward flow of energy and matter. Prigogine and his colleagues (Prigogine 1980; Nicolis and Prigogine 1989; Jantsch 1980) call these stable systems dissipative structures and argue that they may spontaneously develop great complexity, coherent behavior, and become reservoirs of considerable information.
The pathway to the formation of stable dissipative structures involves a condition of metastability in which non-linear processes amplify small fluctuations so that they can have influences far beyond themselves. As a result of such non-linearity, the system can suddenly transform into something radically different, similar to rapid phase transitions from a supersaturated liquid to ice. In the metastable state just prior to transformation, the future system may be entirely unknowable. Order and complexity in the dissipative structure are created from disorder precisely because the system is out of equilibrium and dominated by non-linear processes. We humans should feel comfortable by the ideas of self-organized complexity and dissipative systems for we are the ultimate improbable structures in our cosmic neighborhood.
In order to achieve a decrease in entropy, the system needs to be open and dissipate "low quality" energy into its surroundings. A chicken egg is able to develop internal complexity by dissipating heat to the outside. Preventing dissipation by surrounding the egg with an insulating layer would kill its contents. Likewise enclosing the inner part of the solar system by an insulating shell and thereby preventing dissipation of "low quality" energy into inter-planetary space would eventually destroy life on our planet. The de-coupling of an internal combustion engine from its radiator will similarly kill the engine.

III. SELF-ORGANIZED CRITICALITY

An alternate approach to self-organizing systems that emphasizes discreet events rather than a continuous flow of matter and energy is that of self-organized criticality (SOC). The theory was first developed by Bak (Bak, Tang, and Wiesenfeld 1988; Bak and Chen 1991; Bak 1994) modeling avalanches of sand piles that achieve critical slopes. When sand grains are first added to a pile, they function as entirely separate and isolated objects, having little if any effect upon neighboring grains. Once he pile has achieved a critical slope (34o for dry sand) individual grains may acquire large ranges of influence and, acting in cooperation with other grains, initiate avalanches; the sand pile acquires thereby a cooperative and coherent mode of behavior. The system is open, requiring a constant throughput of energy and matter: sand must be constantly added and must be eliminated at the edges. Building a wall around the base of the pile would destroy the coherent behavior of the system. Because it is an improbable low entropy structure, any randomizing process such as a strong gust of wind would also eliminate its coherency. The equilibrium condition of maximum entropy would be one in which sand is scattered randomly across the floor.
Once criticality has been achieved in the sand pile model, avalanches occur with a power law distribution in sizes. [Avalanches of real piles of rice grains when measured in the laboratory actually show fractal geometries and power law distribution better than real piles of sand (Frette et al 1995).] Occurring at all scales of time and space, these avalanches have no special size or frequency. A power law distribution is used as a crucial test for SOC and is consistent with scale invariance and self-similarity. The initial conditions of the pile or the rate as which sand [or rice] are added does not influence the distribution of avalanche sizes. An important feature of the SOC state is that its details are not determined by fine tuning or initial conditions, but it is a "robust" process that often may be a necessary outcome of a wide variety of situations, and as a result is very attractive for modeling a variety of intermittent processes in the natural world such as earth quakes, forest fires, solar flares, and biological extinction events. In contrast to the sudden transformation of the chemical systems studied by Prigogine, the sand pile model approaches the state of criticality slowly.
Flares on the sun appear to be other examples of metastability and avalanches in a self-organized system (Lu 1993). In the early phase of its 11 year sunspot cycle, the magnetic field of the sun gradually builds up into a critical state such that there are many metastable sites (sunspots and active regions) each containing twisted magnetic fields that are just at the edge of instability. Very slight perturbations on those magnetic fields will cause sudden release of stress and avalanches of flares of all sizes, which have a power law distribution. When it reaches a state of SOC the correlation length of the sun's magnetic field has expanded from the 10,000 km of an individual sunspot group to the 4 million km of the sun's surface.

IV. RESONANCE

A very powerful process that can transform dynamical systems is resonance, whereby small fluctuations may slowly accumulate, if they are synchronized with other cycles, and may produce major changes. Resonant phenomena are examples of non-linear behavior where small fluctuations may be amplified and have influences far exceeding their initial sizes. The natural world is filled with examples of resonant behavior that can produce system-wide changes, such as the destruction of the Tacoma Narrows bridge due to the resonant amplification of small gusts of wind. Elsewhere in the solar system gaps in the asteroid belt and in the rings of Saturn are established by non-linear resonant amplification. Some features of human behavior and most pilgrimage systems are influenced by resonant phenomena involving the heavens, such as solar days, the 27.3 day (sidereal) and 29.5 day (synodic-new moon to new moon) cycles of the moon or the 365.25 day cycle of the sun. A major resonance involving three solar-lunar cycles occurs at Varanasi on the morning of the first full moon after makarasankranti (when the sun enters the constellation of Makara-Capricorn). Resonance between the synodic lunar cycle and human social rhythms may have influenced the length of the human menstral cycle and other features of early human culture (Knight 1898) The 11.86 year cycle of Jupiter resonates with the dates of the Kumbha Mela (Dubey 1987). The 18.6 year standstill cycle of the moon establishes periods of high and low ocean tides at middle and high latitudes and consequently has great influence on coastal fishing cultures. Attention to the standstill cycle is evident in the archaeological record of the ancestral Pueblos of the American Southwest (Malville and Putnam 1993).
An important example of resonance occurs in noisy systems such that a background of random fluctuations can enhance the influence of a particular periodic event in the process known as stochastic resonance (Wiesenfeld and Jaramillo 1998). For each situation there is an optimum level of noise that will have an influence but will not overpower the signal. This phenomenon of stochastic resonance can enhance the ability to detect a particular signal such as weak electrical signals that sharks can use to detect prey fish in the noisy ocean (Adair, Astumain, and Weaver 1998). One of the first natural systems in which evidence for stochastic resonance was detected is the changing climate of the earth, particularly that of the ice ages. The period of advance and retreat of ice is close to 100,000 years, which synchronizes with a very weak periodic variation of solar energy due to the changing shape of the earth's orbit, first pointed out by Milankovitch in the 1920s. In the presence of random fluctuations of the earth's atmosphere, the small periodic signal from the sun may have been amplified to produce the large climate fluctuations of the ice ages. The earth's climatic system and biosphere were self-organized by the process as judged by the complex set of phenomena associated with the ice ages.
Large periodic aggregations of people at the Kumbha Mela or other great pilgrimage events are further examples of large consequences of small periodic signals that are amplified in our noisy world. In the natural world with a rich spectrum of background noise, such as a rural Indian village, certain relatively weak influences due to the cycles of the sun and moon may be amplified by stochastic resonance to result in periodic pilgrimage or other social aggregations. By contrast, people living in noise free "sterile" modern environments or alternately in ones that are dominated by noise may not become aware of periodic signals emerging from their terrestrial and celestial surroundings.

V. SELF-ORGANIZATION IN THE BIOSPHERE AND IN PILGRIMAGE

All three types of self-organization of non-linear systems are present in the complex biosphere of the earth. We acquire high quality energy from the sun (T= 5800 K) and export lower quality energy (T = 300 K) into interstellar space. In his Gaia hypothesis, Lovelock (1979) has proposed that the total biosphere functions as a single organism that is integrated by the movement of atmospheric oxygen and carbon dioxide. Lovelock's ideas have provoked certain controversy, but the self-organization of the biosphere is beyond dispute. Our biosphere is a highly non-equilibrium system with concentrations of O2 and CO2 that greatly depart from equilibrium. The non-equilibrium levels of these gasses are maintained by a complex network of self-interacting feedback loops, which were initiated in atmosphere and oceans of our planet by the appearance of life. Like blood flowing through the human body, atmopheric oxygen and carbon dioxide provide the medium by which separate features interactwith each other. The biosphere functions as a single organism in the inter-dependency of its parts: a loss of function of a part influences the whole. Described in another fashion, the correlation length of the biosphere is the circumference of the earth. When an comet or asteroid impact kills part of the system, such as the K-T event 65 million years ago or the great Permian extinction event some 230 million years ago, it can imperil most of the biosphere.
Similar to the biosphere, pilgrimage systems contain characteristics of Prigogine's chemical systems, Bak's discrete sand-rice piles, and stochastic resonance. The movement of people and ideas through a pilgrimage landscape stabilizes the system and is analogous to the movement of O2 and CO2 through the biosphere and blood through the body. The mega-system of pilgrimage throughout the whole of Hindu India is a self-organizing, spontaneous, and natural system similar to the biosphere in which it is embedded. The fully developed pilgrimage system may start suddenly, like a phase transition, through non-linear amplification of small human impulses, and it includes avalanches of people moving toward pilgrimage centers. The complexity of many pilgrimage systems, especially in their initial phases, may develop spontaneously through the free and autonomous actions of ordinary people in search of an ideal. These systems are not in equilibrium or balance with the socio-economic needs of the participants, as they often involve temporary abandonment of domestic responsibilities as well as arduous and sometimes expensive travel. Openness and departure from equilibrium are necessary conditions for self-organization of pilgrimage, which may evoke strange and often unconventional responses from individuals, new meaning, and entirely new modes of behavior. In its beginning phase, a pilgrimage may lack an overall goal or teleology, although eventually emergence of new purpose, teleonomy, may provide new meaning to the experience. When functioning as a natural system, pilgrimage may draw upon the whole panoply of human emotion, reason, and history and accordingly develop far greater complexity than could have been pre-planned by political or ecclesiastical authority. Non-verbal communication among people and between people and the landscape may be an important artery for transferring information and energy within the pilgrimage system.

VI. SELF-SIMILARITY

Self-organizing systems are characterized by self-similarity and scale invariance. Similar geometric patterns are repeated at different sizes and are expressive of a fundamental unity of the system such as braiding patterns ranging from streambeds to root systems and the human lung. The "physics" is continuous across multiple layers of scale; the same processes of nature reoccur at different levels. With no fundamental scale or characteristic size, self-similar structures are unchanged after increasing or shrinking their size.
The parallelism of macrocosm and microcosm is a deep and ancient human insight, which led to the first naming of nighttime constellations as well as to astrological speculation. Wheatley (1971) alludes to the self-similarity of macrocosm and microcosm when he speaks of "an intimate parallelism between the regular and mathematically expressible regimes of the heavens and the biologically determined rhythms of life on earth" Pilgrimage systems are replete with self-similar structures. The pathway walked by the pilgrim may represent the universe from center to periphery and may mimic the entire age of the universe from beginning to end. The cosmic cycles, which entrain the pilgrim, are similar to the cycles of birth, maturation, and death of terrestrial life. Since they can be enlarged or decreased without changing their essence, the geometries of pilgrimage, whether held in the mind of a pilgrim, drawn on the page of a manuscript, or embedded in a pilgrimage landscape retain their universal meaning.
The carinal dhamas, the four "abodes" of the gods of the subcontinent parallel the cardinality of the Hindu temple. Varanasi-Kashi is a paradigmatic example of self-similarity. It contains those four dhamas within itself as well as most if not all of the other pilgrimage destinations that span the sub-continent (Eck 1983; Singh 1993; Saraswati 1985). Kashi is one of the Seven Cities, saptapuri, and it achieves another form of self-similarity by including the other six cities within itself. The Panchakroshi route encloses the universe, but it is also contained within the body of the individual pilgrim.
The cosmic mountain that the pilgrim circumambulates may be both the axis of the universe and the spire of a temple; a change in scale does not change the significance or efficacy of the movement. The garbhagriha of the temple is both the dark and watery womb of life and the point of creation from which the universe expanded outward from chaos. Circulation about the temple may be revolution about the cosmic axis, the motion of the sun in the sky, and a spiral backward in to the beginning of time. The labyrinthine patterns in the floors of certain Gothic cathedrals through which penitents would crawl were meant as miniaturized pilgrimage journeys to the center. The 660 ft circumambulation path at Chartres cathedral symbolically recreates the pilgrimage to Jerusalem (Barrie 1996). Self-similarity is the key to the story of Shiva and Parvati asking their two sons to demonstrate their love by racing each other around the universe. Ganesh circumambulates his parents, who are also the entire universe, while his brother begins the immensely long journey along its perimeter. Both pathways are equally valuable and meaningful for the pilgrim, each providing different benefit.

VII. FRACTALS

Fractals are composed of self-similar structures and have overall geometries with fractional dimensions. The well-known fractal named after Mandelbrot is a striking example of the endless inclusion of patterns within patterns. No matter how much its tiniest element is magnified it still contains the essence of the entire structure. Fractals are the free spirits of the world of mathematics and geometry and range in the natural world from clouds, cauliflowers, leaves, braided streambeds, and coastlines (Porter and Gleick 1990; McGuire 1991). It should come as no surprise that pilgrimage landscapes have fractal geometries when the natural movement of people generates their geometric patterns.
Meyer (1994) suggests that "psychic maps" and cosmograms are created by most cultures and are "absorbed and held unconsciously by members" of those cultures. Mentally-held cosmograms may often be elements of the pilgrimage experience as pilgrims move along fractal pathways of their own. The complex geometry of the Hindu yantra is an example of such a mental map, which holds great power and significance for the Hindu pilgrim and which may play a role in organizing the ideal landscape through which pilgrims move (Khanna 1979). In its expression of unity across all scales of the universe, the Sri Yantra is fractal both in geometry and in fundamental meaning.

VIII. POWER LAW DISTRIBUTIONS

Self-similarity of the large and small is expressed in power laws in which different sizes and lengths are united. In fractals, for example, there are many more small structures than large ones. Their respective numbers are represented by a power law distribution. Such a distribution is a litmus test for self-organization and and fractal geometries. The natural world is full of power law distributions between the large and small: earthquakes, words of the English language, interplanetary debris, and coastlines of continents. Each of these power law distributions results from a commonality of laws and processes at all scales.
The separations of the 108 shrines of the Panchakroshi encircling Varanasi have a power law distribution with an exponent of -1.5. This kind of structure of the shrines means that pilgrims create a fractal time series as they move along the Panchakkroshi. The implications of such a mathematical structure are rather fascinating and potentially full of meaning, suggesting a subtle embedded structure in the geometry of pilgrimage circuit. The shrines are not regularly spaced or randomly scattered, but their sequential placement may obey a law that is hidden and profoundly natural.

IX. ORIGINS OF PILGRIMAGE

The catchment basin of a pilgrimage center contains villages which themselves may be self-organized complex systems, as has been noted by Saraswati (1995). In a manner like slowly increasing the slope of a rice pile or the build-up of magnetic stress on the sun, a village may be brought to a critical state through what Saraswati identifies as bio-spiritual stress. The village, itself, is another example of a complex Gaia-like ecosystem that is stabilized by a myriad of interconnecting feedback loops. Release of stress in a metastable village may result in periodic departures of people on pilgrimages and other forms of creative activity.
The critical event leading to the onset of a particular pilgrimage tradition may have been a charismatic leader, a powerful individualistic vision, a miraculous cure, a novel idea, or simply a wealthy patron. A pilgrimage tradition may start with surprising suddenness when the cultural system departs sufficiently from equilibrium that a very slight fluctuation or bio-spiritual impulse can produce a transforming avalanche of spontaneous change. The circulation of pilgrims from village to pilgrim center creates an open system that in some cases transcends regional diversity and social stratification and opens up possible pathways for innovation and individual social mobility (Bhardwaj 1973). Like the stirring of the cosmic ocean to create amrita, pilgrimage stirs the cultural landscape, produces new life-giving options, and prevents the stagnation of equilibrium.
Preston (1992) alludes to the "spiritual magnetism" that draws people into a pilgrimage center, but, it is important to note that such attraction is not toward a simple point in space and time. The condition of equilibrium creates an attractive center that is different from pilgrimage. Equilibrium systems attract matter like a sink drain attracts water, the ocean attracts rivers, and black holes attract matter, but in the case of pilgrimage there is a substantially different kind of attraction for there is no single, persistent goal to be pursued. The goal changes as the pilgrim changes and the journey may be endless and ever different.
Changing dynamical systems can be described mathematically in terms of fixed point, limit cycle, or "strange" attractors (Cambel 1993). Closed systems moving toward equilibrium are drawn by a fixed point attractor toward a predictable end such as in the cases of the diffusion of a drop of ink in a glass of water or the gradual slowing of a swinging pendulum. The strange attractor is an important element in the growth of complexity in many self-organizing systems, as matter is pulled into self-transformation by unpredictable, changing, and ambiguous forces.
There clearly are powerful attractors at work to cause pilgrims to journey far and undergo sometimes extreme physical hardships in search of an ideal. In its ambiguity and unpredictability the attractor of pilgrimage shares features with the strange attractor of mathematical chaos theory. In his description of complex physical systems Cambel (1993: 4) comes remarkably close to a description of pilgrimage: "Complex systems are dynamic and not in equilibrium; they are like a journey, not a destination, and they may pursue a moving target."

X. BIFIRCATION AND INDETERMINANCY

As a system is driven further from equilibrium, it may encounter a bifurcation point where two possible futures are available. Exactly at that point there is nearly complete indeterminacy and chaos, since it is practically impossible to predict which one it will follow. The point is metastable in that non-linear processes at that point will amplify small random (stochastic) fluctuations and drive the system along one of the alternate pathways. Bifurcation results in a breaking of symmetry as the system follows only one pathway. The onset of many pilgrimage traditions may take place in conditions of metastability and indeterminacy.

XI. COMPLEX LANDSCAPES

Interactions between humans and the heavens, between microcosm and macrocosm, probably have initiated and maintained many pilgrimage traditions. Such appears to have been the case in the Nubian Desert of southern Egypt, when repetitive visits by nomadic cattle herders during the summer monsoons between 10,000 and 7000 years before the present established a ceremonial complex at the western shore of Nabta Playa. An incipient pilgrimage tradition appears to have developed involving cattle worship and megalithic structures aligned to the sun and stars (Malville et al 1998). The rising of the sun at its summer solstice point on the horizon coupled with the onset of the monsoon to make the desert habitable again for a few months. Since Nabta Playa is close to the Tropic of Cancer, the summer solstice sun reached the zenith at noon, when vertical megaliths would cast no shadows. Multiple process of human life, the landscape, and the heavens were brought into resonant coupling at the time.
A similar resonance between humans, the terrestrial landscape, and the cycles of the heavens may have played a role in the development of the Chaco regional system among the ancestral Pueblos of the American Southwest. The region may have been pushed into self-organized criticality by environmental forces and population growth. The result was a rapid onset of periodic movement of people into Chaco and construction of great houses, great kivas, roads and other possible accoutrements of pilgrimage (Malville and Malville 1995). Near the middle of the 11th century there appears to have sudden transformation of the culture as its correlation length, Lc, of meaning, ritual, and movement increased by a factor of more than 100 from the kilometers of a villages to the hundreds of kilometers of the Chaco regional system.
The first cities of the world, which appeared during the third millennium B.C. were self-organizing, non-equilibrium systems, which often underwent rapid self-transformation. The first major examples of urban order appeared suddenly in the Indus valley between 2600 and 2550 B. C., giving the appearance of "a paroxysm of change" (Possehl 1990; Jansen 1991; Parpola 1994). A variety of stimuli may have triggered the vast modifications of socio-cultural behavior that led to the Mature Harappan culture: the onset of maritime trade, a powerful new ideology, and a writing system. The Harappan culture area covered an area of more than 1,250,000 km2 and consisted of approximately 1000 settlements, many with orthogonal systems of streets, ritual baths, standardized construction bricks standardized, and sophisticated sanitation and water systems. Parapola (1994) has suggested that the sudden transformation of the Harappan landscape was paralleled by the development of a complex astronomy, which may be manifest in part by the Harappan script containing a self-similarity stars, planets, and fish, each swimming in their respective oceans.
The tirthas of India provide further examples of self-amplifying interactions between people and their landscapes. Tirthas are "crossing-over" places with many levels of meaning. Some were initially plaes to pause in the process of fording a river, such as the Ganga at Varanasi, at which place a symbolic resonance was evident because the Ganga flows northward out of the realm of death toward the place of birth. Repetition of simple acts and the confluence of countless mytho-historical events have led to the vast significance of pilgrimage to Kashi. In that location multiple realms can be crossed and many of the features of self-organizing systems can be experienced such as self-similarity, resonance, and fractal geometries (Saraswati 1985; Singh, 1993; Malville and Singh 1995).

XII. CONCLUDING REMARKS

In its nascent stage, pilgrimage may be a vigorously open self-organizing system, which creates new meaning, myths, and rituals. In its more mature state as a well-developed process opportunities for spontaneity and innovation may be reduced, although not entirely eliminated. Openness always invites spontaneity, surprises, and unconventional behavior.
When a system has reached the critical state, it may sustain ideational avalanches as new concepts, theories, perspectives, and cosmologies are developed and then tested by experience. Within the "catchment basin" of a pilgrimage system, these avalanches involve people as well as ideas. In the case of Hindu India, the scale of interaction extends across the entire country, involving the coherent behavior of hundreds of millions of people who are not organized by any central authority, but who, by their movement, provide social continuity and cohesion for the entire land (McNamara 1995). The physical scale of coherent behavior proceeds from the four cardinal abodes, through a hierarchy of self-similar structures that are characterized by circumambulation, cardinality, and centrality to the scale of the temple, and home. Self-similarity continues to married couple, in the rite of the saptapada, in which the bride takes seven steps, symbolic at the macrocosmic level of the seven realms of the heavens, around the groom who is the axis of the universe.
Often pilgrimage acquires cosmogonic and cosmological meaning as pilgrims reenact the origin myths of the cosmos, mimic great events of the past, and move in parallel with the cycles of the heavens. As self-organizing systems, the complexity of the pilgrimage landscape may be natural and internal, and we do not need to look for the evidence of a structure imposed by imperial decree or by a city planner, pundit, or priest when encountering complex geometries and complex ritual.
There is great depth to the pilgrimage landscape, which is generated by many levels of meaning, many sources of energy, and many centers of attraction. The entry of the mathematics, physics, and chemistry of self-organization into the discussion does not diminish in any way the deep mystery and grandeur of the pilgrimage experience. There is breath taking mystery and spaciousness to the fractal of Mandelbrot, for example, which in its scale invariance continues inward to the heart of the tiniest piece of matter and outward to its expanding boundaries of the universe. In the actuality of dawn on the Ganga, there is mystery to the rising of the sun greeted every morning by thousands of pilgrims. The deepest mystery of all is that these complex pilgrimage structures were created by free individuals who walked the landscape in search of meaning and in pursuit of an ideal.

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REFERENCES

Adair, R. K., R. D. Astumain and J. C. Weater. 1998. Detection of weak electric fields by sharks, rays, and skates. Chaos: An Interdisciplinary Journal of Nonlinear Science. 83: 576-587.
Bak P. 1994 Self-organized criticality: A holistic view of nature. In Complexity: Metaphor, Models, and Reality, ed. D. Pines and D. Meltzer. Reading, MA: Addison-Wesley
_____. 1996. How Nature Works: The Science of Self-Organized Criticality. New York: Copernicus, Springer Verlag.
Bak, P. and K. Chen. 1991. Self-organized criticality. Scientific American 246: 46-53.
Bak, P. and K. Sneppen. 1993. Punctuated Equilibrium and criticality in a simple model of evolution. Physica Review Letters, 71, 4083-4086.
Bak, P., C. Tang, and K. Wiesenfeld. 1988. Self-organized criticality. Physical Review A, 38, 364-374.
Barrie, T. 1996. Spiritual Path, Sacred Space: Myth, Ritual and Meaning in Architecture. Boston: Shambhala.
Bhardwaj, S. M. 1973. Hindu Places of Pilgrimage in India: A Study in Cultural Geography. Berkeley: University of California Press.
Cambel, A. B. 1993. Applied Chaos Theory: A Paradigm for Complexity. Boston: Academic Press.
Coleman, S. and J. Elsner. 1995. Pilgrimage: Past and Present in the World Religions. Cambridge: Harvard University Press.
Dubey, D. P. 1987. Kumbha Mela: Origin and historicity of India's greatest pilgrimage fair. The National Geographical Journal of India 33: 469-492.
Eck, D. 1983. Banaras, City of Light. London: Routledge & Kegan Paul Frette, V., et al. 1992. Avalanche dynamics in a pile of rice. Nature 379, 49.
Gluckman, B. J., P. So, T. I. Netoff, M. L. Spano, and S. J. Schiff. 1998. Stochastic resonance in mammalian neuronal networks. Chaos: An Interdisciplinary Journal of Nonlinear Science. 8, 588-598.
Jantsch, E. 1980. The Self-organizing Universe: Scientific and Human Implications of the Emerging Paradigm of Evolution. Oxford: Pergamon Press.
Knight, C. 1989 The lunar analogue and the origins of culture. Cosmos 3: 72-106; also, Menstruation and the Origins of Culture, Ph.D. thesis, University College, London (1987).
Lu, E. T., R. J. Hamilton, J. M. McTiernan, and K. R. Bromund. 1993. Solar flares and avalanches in driven dissipative systems. Astrophysical Journal 412:841-852.
Lovelock, J. 1979. Gaia, A New Look at Life on Earth. Oxford: Oxford University Press.
Malville, J. M. 1995. Emergent geometries of self-organizing systems: from chaos to pilgrimage. Paper presented at the seminar on Cosmic Order and Chaos, Indira Gandhi National Centre for the Arts, 18-22 December 1995.
Malville, J. M. and N. Malville. Pilgrimage and astronomy at Chaco Canyon, New Mexico. Paper presented at the National Seminar on Pilgrimage, Society of Pilgrimage Studies, Ardha Kumbha Mela, January 21-23, 1995.
Malville, J. M. and C. Putnam. 1993 Prehistoric Astronomy in the Southwest. Second Edition. Boulder: Johnson Books.
Malville, J. M. and Rana P. B. Singh. 1995. Visual astronomy in the mythology and ritual of India: The sun temples of Varanasi. Vistas in Astronomy 39:431-449.
Malville, J. M., F. Wendorf, A. A. Mazar, and R. Schild. 1998. Megaliths and Neolithic astronomy in southern Egypt. Nature 392: 488-491.
McGuire, M. 1991. An Eye for Fractals. Redwood City: Addison-Wesley.
McNamara, J. D. 1995. Hindu pilgrimage as a force of Indian spatial cohesion and temporal continuity. In Pilgrimage Studies: Sacred Places, Sacred Traditions, ed. D. P. Dubey, pp. 31-44. Allahabad: The Society of Pilgrimage Studies.
Nicolis, G. and I. Prigogine. 1989. Exploring Complexity. New York: W. H. Freeman.
Parpola, A. 1994. Deciphering the Indus Script. Cambridge: Cambridge University Press.
Porter, E. and J. Gleick. 1990. Nature's Chaos. New York: Viking.
Possehl, G. L. 1990. Revolution in the urban revolution: the emergence of Indus urbanization. Annual Reviews of Anthropology 19: 216-282. Paolo Alto: Annual Reviews.
Preston, J. J. 1992. Spiritual magnetism: an organizing principle for the study of pilgrimage. In Sacred Journeys: The Anthropology of Pilgrimage, ed. A. Morinis. Wesport: Greenwood Press.
Prigogine, I. 1980. From Being to Becoming. New York: W.H. Freeman.
Saraswati, B. N. 1985. Kashi pilgrimage: the end of an endless journey. In Dimensions of Pilgrimage: An Anthropological Appraisal, ed M. Jha, pp. 91-104. New Delhi.
_____1995. Lifestyles in traditional cultures: a conceptual framework. In Computerizing Cultures, ed. K. Vatsyayan and B. N. Saraswati. New Delhi: Indira Gandhi National Centre for the Arts.
Singh, Rana P. B. 1991. Pancakroshi Yatra, Varanasi: Sacred Journey, Ecology of Place, and Faithscape. Varanasi: Tara Book Agency.
_____1993. The sacred geometry of India's holy city, Varanasi: Kashi as cosmogram. In The Spirit and Power o Place, ed. Rana. P. B. Singh, pp.189-216. Varanasi: National Geographical Society of India.
Schroeder, M. 1991. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freemen and Co.
Turner, V. 1979. Process, Performance, and Pilgrimage: A Study in Comparative Symbology. New Delhi: Concept Publishing Company.
Wheatley, P. 1991, The Pivot of the Four Quarters. Chicago: Aldine.
Wiesenfeld, K. and F. Jaramillo. 1998. Minireview of stochastic resonance. Chaos: An Interdisciplinary Journal of Nonlinear Science. 8:539-548.

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