The Fractal Revolution

by Peter Bearse (1)

Introduction: Fractal Patterns

It's too bad. With the popularity of fractal calendars and popular writers who know nothing of mathematics and little of science playing on "chaos", the new concept has become a cliche'. It's too bad, because even those who understand chaos and the mathematics underlying fractal art have missed the revolutionary implications of the new concept for understanding and improving human society.

Human life is inherently chaotic. People have felt it to be so since the beginning of recorded time. They have sometimes sensed, but mostly prayed, that the chaos may have an underlying structure. Only recently, however, has this hope been expressed in scientific/mathematical terms, as the tracings of an underlying reality rather than merely the subject of deep human yearning. Until the French Revolution, the structure of human existence was an article of transcendental faith rather than human knowledge. The basic "structure" was millennial -- the apocryphal City of God, reified by vain men in the form of monuments and causes.

The discovery that the "geometry of nature" is fractal has radical implications for human beings' understanding of their society and of their role in things social and political. What does this mean? It means that there is now, for the first time in human history, a firm mathematical and scientific basis for a continuing (r)evolution of society worldwide in ways that focus upon the fulfillment of individual potential as the fundamental aim of human development. Individuals and their actions, however small and localized they may be, can finally be recognized as influences on historical patterns. The "big picture" is a construct of many tiny, interactive patterns. The "Organization Man" is dead or dying in any of his "top down" variations.

Some would say that such a basis was provided over two hundred years ago by the Enlightenment based upon Newtonian physics. Yet, even at the time, the great English artist and poet, William Blake, recognized that this was not so. Subsequent history up to the present time was to prove him right. The Enlightenment was grounded in scientific values, especially relentless testing of theories in light of experimental facts. Mathematical theories provided the basis of Newtonian physics, however, were also employed to rationalize hierarchical systems of power -- the dominance of the "little" by the "big," of the "lower" by the "higher," etc. Now, such inversions can revert to turn the right side up. The true math-ematical-scientific basis for a continuing American revolution has only recently come into view. And such a view it is! -- the potential empowerment of the individual in all spheres depicted by color graphics, creative advertising and fascinating geometrical figures as well as mathematical formulae and scientific studies; enabled by "decentralization," "devolution," "flattening of hierarchies," "reinvention" of selves and organizations, "learning organizations," "grass roots" individual or community-based initiatives; "think globally; act locally," and many other ways.

The basis of the fractal revolution is the principle underlying chaos and other natural patterns, that of "self-similarity." This means that the basic patterns are the same at any scale. They are the same at large "macro" scales as at small "micro" scales. The large is revealed by, and grows out from, the small. Wholes mimic parts (and vice versa); the bigger is revealed in the smaller.

We can see this, for example, in the branching patterns of plants, trees, rivers organizations or even the lines in an old man's face. In your kitchen, repeat a simple demonstration.employed by Benoit Mandelbrot, the "father of fractals."(2) Pick up a cauliflower. Look at it, then pick it apart. Note how the pieces, however small, replicate the structure of the whole. That is self-similarity. Look at the veins of the leaf of a tree in summer and then at the bare dendritic structure of its branches in winter. The patterns are self-similar. Notice, from an airplane, the branching patterns of a great river. Then, close to the ground, notice the intricate, tiny pattern made by rain flowing from a puddle. The patterns are self-similar. See below.

Next, examine the organization charts of organizations and their branch units. The patterns are sometimes self-similar. To the extent that the latter are not so, they are likely to become more so as organizations, following the advice of advanced management "gurus" like Peters (1987) and Handy (1994), introduce concepts borrowed from chaos and evolutionary biology into organizational designs. Germany's Fraunhofer Institute, for example, an advanced industrial R&D organization, advertises the fact that its structure is "fractal." So does Renault, the giant French auto company. Renault uses a Serpinski diagram, a famous fractal figure, to illustrate the essentials of its "total quality management" structure, as shown below. Both software developers and entrepreneurs growing technology-based enterprises think and talk in terms of the "scalability" of designs and companies.

Thus, the mathematics and science of fractals have finally revealed what some philosophers have only hoped or surmised over the past two millennia; namely that:

The implications of these insights are so revolutionary that some of those who have been so foolish or courageous to speak or write them in the past have been burned at the stake. We like to think of ourselves as being more civilized or "modern," but the brutal, inhuman, bloody history of the century now ending but still continuing belies any such naivete.

A fractal figure is a snapshot of a dynamic system at a stage of development. The snapshot is a clue to a dynamic process - a pattern of development. One basic pattern is that of a flowing stream. In response to a force (gravity), the water's potential energy becomes actual. The stream will branch in various ways as it runs into various barriers and constraints. As the stream speed increases, the flow pattern may become chaotic, as with rapids for example.

Another pattern is that of a growing tree. In response to another source of energy, the sun, a tree grows and branches in various ways depending on a variety of resources, opportunities, barriers and constraints. Compare the growth patterns of a fan palm in a tropical garden with that of a scrub pine in a northern rocky crag. The fractal patterns are basically similar but also quite different, one graceful; the other, gnarled.

Fractal patterns are all around us, exhibited by human social organization, works of art, music and poetry as well as by nature. There are computer graphics and the fact that fractal mathematics provides compression algorithms through which complex graphical patterns can be stored. There is poetry with fractal metaphors; e.g. -

Researchers have detected fractal patterns in widely disparate areas of scientific observation and practical concern. The most recent examples are only a small subset of a great many that could be cited as showing the broad and varied applicability of fractal concepts; e.g. --

Thus, it seems that no matter where one looks, fractal patterns can be seen now that we have been enabled to see them. What Blaise Pascal, French mathematician and philosopher, observed over three centuries ago can now be truly noted:

"Nature imitates herself. A grain thrown onto good ground brings forth fruit; a principle thrown into a good mind brings forth fruit. Everything is created and conducted by the same Master - the root, the branch, the fruits, the principles, the consequences."

Dynamic Systems

Fractal patterns are not simply revealed by geometries of things as they are or have become; they are also revealed by processes of change from one state of development to another. Self-similar fractal patterns characterize diagrams depicting the dynamics of change; e.g., phase diagrams of rates of change in weather patterns, heartbeats and water flows. The discovery of such a pattern behind the behavior of a model of weather changes by Edward Lorenz, for example, led to his coining the now famous descriptor "butterfly effect." The model's behavior indicated (hypothetically) that the flap of a butterfly's wings in Brazil could lead to a tornado in Texas. The validity of such inferences has since been confirmed many times over in systems of all types. The implication? -- that small moves by micro-actors can lead to big ("macro") changes.

A key feature linking fractals and chaos in the dynamics of change is "starting positions." Even miniscule differences in these lead to increasing divergence over time. The results are unpredictable; that is, one cannot say that, if the starting positions are 'x' and 'y', then the ending positions will be 'x+m' and 'y+n' with any acceptable degree of confidence -- even if we understand precisely how the 'xy' "system" behaves.

How is the "fractal revolution" in the understanding of dynamic systems relevant to human lives? -- in more ways than a brief article can recount. Consider, for example, the current debate over "inequality." In an increasingly open and competitive market economy which, since the mid '70's, has been increasingly dependent upon the initiatives of "micro-actors" (entrepreneurs), it should be no surprise to see increasing inequality of income and wealth. This would happen even if each of us entered the economic race with equal endowments (of wealth, intelligence, education, et. al.) but, of course, we don't. Each of our "starting positions" is different. Any attempts to improve the life chances of those whose starting positions are, in some sense, "disadvantaged" are doomed to be counterproductive if the nature of the dynamics of our "new economy" is not understood.

The manifold human implications of fractal patterns, however they may be revealed, will take some time to be worked out in detail; yet, even now we can begin to see some of their potentials of the fractal perspective in several spheres of human concern. After all, even in this so-called "'modern" or "postmodern" age, major features of human society still rest on the following pre-modern; indeed, ancient assumptions (3) -- that:

Fractal structures and dynamics contradict these assumptions. That is why the adjective "revolution" is used in the title of this article.

Preliminary, Suggestive Observations Reflecting Some Applications and Implications of Fractal Perspectives

Consider the following dichotomies, in which "--" denotes the type of relationship commonly assumed; that is, a super-ordinate -- sub-ordinate ("superior to") relationship between the two sides of each divide:

We can begin to see whether there is anything revolutionary in the fractal perspective through preliminary, trial applications to see whether its features offer any new insights into traditional patterns and interrelations. First, recognize essential self-similarities between the levels marked by each of the above divides. Then acknowledge their implications.

The major implication is the need to shift responsibility between the levels and to reduce barriers and constraints that the (presumptively) "higher" levels place on the development of the (presumptively) "lower." Where this is not acknowledged and overcome, we see misalignment of responsibilities and dominance/sub- dominance relationships that limit and distort development at "lower" levels by presumably less significant, less powerful or smaller entities. The (--) divide in each of the above dichotomies signifies such potential problems.

What are the more specific "revolutionary" implications of the fractal structure of features that are human as well as natural? They are often paradoxical. Consider each of the several major divides in turn.

Man - Nature

Since the environmental movement has spread worldwide, it may seem surprisingly backward to posit a divide between "Man" and "Nature." Yet, we still lack a solid foundation for an ethic of ecology. The lack of such underlies many conflicts; for example, those which surfaced at the Global Warming Conference in Kyoto. We still need to reaffirm man in nature and nature in man and resolve the paradox of Man as animal and not-animal.

The fractal branching pattern of development in both Man and Nature provides at least part of a foundation. Any adult reflecting on his past history has come to appreciate, with Robert Frost, the meaning of "The Path Not Taken." Similarly, we can look back on how paths taken have traced patterns as "one thing led to another" through odd combinations of chance and design. If we fail to see this dynamic pattern in human life, starting with the development of the fetus and, over far greater spans of time, through the course of evolution, then we will continue to believe the destructive myth of Man -- Nature -- Man apart from or over Nature, as if the whole of creation lies at our behest. As a result of this arrogant, unnatural presumption, we can now see the contraction rather than the radiation of species. If reason rather than presumption were to govern, the inversion would be more justified, Nature -- Man, The truth revealed by the fractal structure of nature reinforces that from evolution, that nature is in man and man in nature.

The synthesis of science and art revealed by computer art generated through self-similar fractal geometry provides one of the finest representations of the convergence of Man and Nature. For centuries, the gap between art and science supplied the aesthetic symbols and intellectual substance of the presumed divide between Man and Nature.

No one can reasonably question that the environmental movement has been a revolutionary force in our time. The force, however, is far from spent. It can be renewed by insights derived from the fractal structures of Man and Nature interacting, each the ecological framework for the other.

National Government -- Local Government

The divide here is deceptive since the two levels appear to be quite self-similar in a decentralized democratic system. The national President and Congress, state governors and legislatures, and local Mayors and Councils are self-similar structures of government at each level. An older, more basic and democratic unit of government, however, is the Town Meeting, a form that is receiving increasing attention as a means to reduce the growing, nationwide alienation of people's lives from politics.

National government is not a community, family or village, nor can it be made so. To the extent national government has any essential meaning except as an instrument of national security, it is as a servant of community, family or village -- the basic, "micro" units of democracy. National issues are local and non-local. Local concerns are national and not national. The paradox is resolved if we view national government, not as a body with a life of its own, but as a federation of city-villages. This view has been set forth by Charles Handy in The Age of Paradox.

That this view has any but revolutionary implications for the structure of government and the nature of our political life would be hard to deny. Among other things, it implies a revised national Constitution and significant diminution of the powers, plus revisions of the roles of, higher levels of government. Some of this has already been seen to a limited extent via the kinds of decentralization sometimes referred to as "devolution" -- efforts to shift power and money from the national government to the states in the United States.

God - Man

God is dead and alive; alive if alive in us, if not, dead. Like many features of human life, if God did not exist we would have to invent Him (or Her). We have invented Him. Whoever or whatever our God may be, that God is the construct of our highest hopes, yearnings, creativity, values and imagination. As the famous evolutionary biologist, Gaylord Simpson, wrote, "Man Makes Himself." This includes the God(s) we worship unless, unlike most religions, "God" is equated to "Universe." If the universe is, as physicist Freeman Dyson indicates, "Infinite in all Directions," then man and man's world can once again be seen at the center.

The alternative, conventional view, that God exists apart from Man, somehow higher, apart from and independent of humanity, has been the source of incredible inhumanity, brutality and grief throughout recorded history. The end of such history is not yet in sight. This non-self-similar view has allowed Man to project himself on God and to project God upon states, wars and alliances. Thus, Kings and other rulers, their rationales for oppression and their wars are made holy, through unholy alliances between rulers and organized religion. As Ernest Becker (1975) has shown (and as William Blake realized 200 years ago), such projections are the essence of evil.

Adults -Children

The quality of our lives and, indeed, the humanity of our communities, would be greatly improved if the essential self-similarity between adults and children were recognized. Sayings attest to this over hundreds of years, such as "the child in the man" and "childhood is lost on children." Adults are children and not-children. Children strive to imitate and become adults. The great English artist and poet William Blake resolved the paradox about 200 years ago in his Songs of Innocence and Songs of Experience.

Conscious realization and application of the concept of self-similarity would help us to make significant, desirable changes and to understand some that are already underway. One such change, for example, is the growing acceptance of "lifelong learning" as a goal, rather than seeing "learning" as something limited to what young people do in schools.

Other artificial, limiting and counterproductive divides and distinctions between childhood and adulthood could also be overcome. For at least 200 years and even now, schools have served to drive the best qualities of children out of them, thinking that such purgation is required for children to become adults. These qualities include creativity, playfulness, openness, critical thinking, questioning and challenging behavior. Fortunately, the new economy values the entrepreneur more than "The Organization Man", so some of the creative qualities of children are now valued in adults.

Self-similarly, on the side of childhood, our growing up would be helped greatly through recognition that much of the adulthood>>childhood divide in American society undermines both childhood and adulthood. We take pride in extending our children's feckless, irresponsible childhoods rather than insisting that they take on responsibilities in the household and community that give them tests and tastes of adult life. It is no surprise that American society contains so many childish adults.

Would the changes implied here be "revolutionary"? -- not in the sense of 1789 or 1917 but in the more basic sense of real, desirable change -- towards a more open, creative, democratic and adaptable society. Such changes would be all the better for working their way through our lives gradually rather than suddenly or violently. It is not inconceivable even to imagine a new, 21st Century Renaissance based on the best qualities of adults who are more like children and children who are more like adults -- the American character triumphant.

Politics -- Life

Politics and government in even the most self consciously democratic societies have evolved to the point where they are specialized, separate -- activities which appear to influence our lives without being part of them. In many countries, including the United States, this has gotten to the point of "us" vs. "them." "Them" is the small minority of people, including career politicians and political "spinmeisters," consultants, pundits and junkies whose lives are consumed by politics while, for the great majority, politics are not a part of their lives at all.

An important part of an answer to the growing political alienation featured above is the realization that politics and life are self-similar. Politics is like life; life is like politics. Politics is not "out here" or "up there." Life without politics is not complete; politics disconnected from the rest of life is tyranny.

The self-similarity is most apparent to those who run for office or manage a political campaign. Then, politics is like life only more so -- life when the tape is run at four times normal speed, life at its most intense. Decisions are made, large and small. The path(s) branch and divide. Paths not taken become the source of later speculations of the "what went wrong" variety. The inner circle, the confidants of a candidate, become like family. One learns who one's friends truly are. There are disappointments and betrayals. There are triumphs and victories. Most of all, there is the growing realization of adult life -- that most of what one does has consequences that affect others and that the politics of our most intimate, "private" interactions in family, business and other settings, are essentially similar to those in larger, "public" world(s).

Society - Family

A family is a society in miniature. Many concepts of community or society are predicated on a model of the nuclear family. But society is more than just an aggregation of families. It is, as is every higher level entity -- a projection and a construct of meaning by more integral and behaviorally basic entities, the fundamental "micro" units of being, becoming and concern -- families and individuals. The latter "invest" in their higher level constructs in many ways, especially to invest them with meaning(s). Such investment is most meaningful to the extent that the higher level structures are self-similar to the lower level -- in being:

A closed system may be impervious to input. A society which is not open to innovation or diversification lacks even the tolerance for diversity which family members need to deal with differences among themselves. Both a family and a society have some integrity to the extent they honor traditions based on their own history (ies). Both a family and a society need to have ways of learning to adapt to changing circumstances. And if there is not a set of shared values; indeed, values-in-practice that the society is able to represent, then none of the above may count for much.

Are these similarities "revolutionary"? -- only to the extent that they are missing, and, unfortunately, there is much that seems to be missing. At this point, without searching the literature for studies that might shed some light, one can only speculate. One could claim that "society" has it has been known in the past no longer exists. At least, "society" as it is now viewed by most through the media differs substantially from that of the past and appears to differ substantially from "family" as it is known to the majority of families. Perhaps this has been one factor leading to the establishment of mini-societies -- small social worlds of relatives, friends, affections or belief for which people have affinity and affection. The larger society, especially as depicted by the media, may be seen as foreign, unethical or distasteful. The implications of tendencies to devolve to small worlds as an antithesis or escape from the perceived shortcomings of the large are quite serious but, for now, they are left for others to imagine.


In the past, major scientific discoveries, theories or revolutions have led to paradigm shifts in the social sciences. The latter have been known for having inferiority complexes relative to the physical and mathematical sciences. Thus, too often, social scientists have almost slavishly imitated, adapted or applied physical and mathematical science frameworks to social worlds. We must be wary of this danger as we recognize how fractal geometries enable us to see many things in new ways and extend interpretations of these into the human domain.

There is one aspect of the above "danger" that is readily avoidable within the new fractal paradigm even though it was not avoided within earlier scientific/ mathematical frames. This is the careless use of the latest scientific theory to justify undemocratic and inhumane forms of social organization or government policies. Toulmin (1994) documents what has also been noted by many others -- the use of Newtonian science to rationalize and uphold hierarchical systems of domination and governance over the past 300 years. These implications of Newtonian science were reinforced by the use of Darwin's theory of natural selection to justify "Social Darwinism." These developments were possible because Newtonian science provided strictly quantitative, closed, hierarchical and predictable models of the cosmos.

By contrast, the new fractal paradigm presented here, supported by other major 20th century developments in physics and biology, provides a qualitative, open, non-hierarchical, unpredictable set of patterns which support decentralized, humane and democratic lines of human development. What we see is not a closed set of equations that suggest that human development "will" or "should" follow a predetermined or predicted trajectory dictated from "above," but an open, variegated set of patterns that suggests possibilities for their realization via human choices at the most basic, individual level.

For what is truly revolutionary in the preceding is that the presumably "micro" or "lower level" units have become the most important points of reference. The lack of self-similarity of the "higher" level units becomes, then, a normative prescription -- they should be self-similar to the structure and developmental behavior of the lower level units in certain essential ways.

It is such involution -- overturning presumed authorities and their assumptions -- that is tantamount to a revolution, just as it seemed to be in the late 18th century with the eruption of the American and French Revolutions. Since then, beginning with the Civil War, we have raised up a national government as if it had some larger meaning and higher level authority above that of the constituent micro- units that make up American society, whereas the self-similar truth is the other way around. "America", unlike most other countries, has authority, meaning and integrity only to the extent that its government recognizes that authority, meaning and integrity reside in its constituent micro-units, individuals most of all but also families and local governments.

Even now, in all sorts of ways, we attach authority and meanings to higher level entities. Many such attributions or imputations are unjustified. Many are justified only to the extent that their structure and behavior are self-similar to that of the micro-units they supposedly "govern" -- so that they support micro-units and nurture rather than constrain their development potentials.

It may seem odd to hold out as "revolutionary" a vision that is over two hundred years old, yet we need to return to our revolutionary roots in order to construct a vision that is up to the challenge of a new century in a time of rapid change. We also need to recognize that the "Enlightenment" that informed the revolutions of two hundred years ago was deficient even then. (4) It was based upon the science and mathematics; that is, the "natural philosophy" of its time. Even then, one man recognized the deficiency, the English poet, William Blake. The fractal concept of self-similarity based upon the science and mathematics of our own time is consistent with a Blakean vision expressed in terms of myth and allegory, partly because of the bridge between art and science that has been built of fractal geometries.

As indicated by the discussion of "divides," a new/old vision starts with a recognition that it is the micro-behavioral units that are fundamental -- most of all, individuals, but also including other "micro" units such as families, small businesses and small sub-units of larger organizations. This is also the basic assumption underlying early 21st century management practices. It is the realization that, if you want to change things for the better, you have to be able to change behaviors of independent, self-organizing behavioral units. If you can't do that, then no desirable changes are likely to emerge. After all the bloody, inhuman, so-called "revolutions" of this century and the two preceding, this is a peaceful approach to change. We may be doing something wrong, dangerous or inhumane to the extent that we project higher level attributes or meanings on aggregations of people that have not arisen through organic, self-organizing, constitutionally liberal, democratic and competitive processes of development. (5)

Does the fractal concept provide the basis for "empowering" the disempowered? Yes, but in a way that goes beyond the usual "from the top down" passage of "empowerment," as Charles Handy in his Age of Paradox has already suggested. The words of the Magi in the movie animation Fern Gully: The Last Rainforest are apropos of an empowering fractal vision: "There are worlds within worlds. Everything is connected by the delicate strands of life. Anybody can call upon the cosmic forces of life. You have to find the power within yourself."

Eventually, this vision may provide a valid intellectual basis for integration of the themes of the French Revolution and the realization of Blake's dream -- that all three, "liberty, equality, fraternity," might be fulfilled, together. This possibility may yet ignite the hopes and engage the energies of a new generation. Back to the future?

1 Peter Bearse, Ph.D., is President and Economist, Development Strategies Corporation, Gloucester, MA. Feedback from readers would be gratefully received, via telephone (508-281- 6992, fax (508-281-6588), e-mail ( or snail-mail (47 Pleasant St., Gloucester, MA 01930-5944).

2 As referred to by Manfred Schroeder (1991) (and many others in similar terms). The phrase "geometry of nature" is Mandlebrot's, from the title of the second edition of his seminal book (Mandlebrot, 1977).

3 Bruno Latour, in his book WHY WE ARE NOT MODERN, challenges the "modern"/"pre- modern" distinction and shows just how artificial it is in many ways.

4 This point and its larger implications have been elaborated by Stephen Toulmin in his book COSMOPOLIS (Toulmin, 1994).

5 The distinction between a "constitutionally liberal" democracy, arising from "the American system," and "plebiscitary democracy," arising from the "the French (Revolution) model," is an important one which has been well made recently by Zakaria (1997). The former is based upon "an avowedly pessimistic conception of human nature;" the latter, upon "faith in the goodness of human beings." The distinction was made earlier by Lipow (1996), who shows how the American model, especially if shed of its "Progressive" era overlays, is the basis for a truly democratic system while the French model is fundamentally undemocratic.

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