Complexity and Self-organization in Pilgrimage Systems
John McKim Malville
Department of Astrophysical, Planetary, and Atmospheric Sciences
University of Colorado at Boulder
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.
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
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
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
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
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
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.
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
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.
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
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
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
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
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.
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,
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
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
Lovelock, J. 1979. Gaia, A New Look at Life on Earth. Oxford: Oxford
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
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
Nicolis, G. and I. Prigogine. 1989. Exploring Complexity. New York: W. H.
Parpola, A. 1994. Deciphering the Indus Script. Cambridge: Cambridge
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.