 / 
Introduction
Relationships between "incommensurables"
Thesis
Methodological approach
Challenging aspects of this exploration
Dissipative systems and their illusory continuity
Structure of the visual representation of the Mandelbrot set
Interpreting features of the Mset
Potential implications: orders of abstraction and "explanation"
Meshing mathematical and experiential understanding
Psychosocial significance of the Mset (in Annex
2)
Imagination, Resolution, Emergence, Realization and Embodiment: iterative comprehension (in Annex 3)
Potential implications for interdisciplinary and intersectoral
initiatives
Managing intractable differences: relevance to particular polarities
"Real" vs "Imaginary"
Relevance to strategic dilemmas
Enhancing insight through audiovisual techniques
References
Annex 4: Features of Mandelbrot and Julia Sets (not completed)
This exploration endeavours to frame the concerns of two earlier associated papers in terms of the insights of dissipative systems and the Mandelbrot set (hereafter referred to as the Mset). The first paper (Being Positive Avoiding Negativity: management challenge  positive vs negative, 2005) was concerned with the appropriate handling of "positive" and "negative" from a strategic perspective and as a judgement on the relevance of feedback. The second paper (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005) sought to demonstrate the importance of a set of 8 patterns of interaction in defining a coherent pattern within any system of relationships  highlighting the role of the cardioid in that pattern, following the work of Edward Haskell (Generalization of the structure of Mendeleev's periodic table, 1972).
Given the prime importance of the cardioid in representation of the Mset, the argument that follows is initially descriptive in clarifying an explanation of dissipative systems in terms relevant to the strategic challenge of interpersonal and intergroup relationships of those papers. There is an extensive body of literature of varying levels of technicality that explains the Mset and related issues. The concern here is the potential relevance of those insights to contexts which have not as yet been a prime concern. Reference is therefore only made to the technicalities where they suggest insights of relevance to the strategic challenge that might otherwise go unrecognized.
The purpose here is to explore imaginative leads and framings  possibly primarily metaphorical  that may be a guide to more concrete interpretations. In that respect the isomorphism with Haskell's cardioid may bear a less than rigorous relationship to that discussed here. [This question is currently under investigation by Kent Palmer]
This approach is consistent with that advocated by Ralph H. Abraham (Human Fractals: the arabesques in our mind. [text]
To many pure mathematicians, especially those to whom fractal geometry itself is not mathematics but heresy, these applications of new mathematical ideas to anthropology will seem anathema, vulgarization, fractal evil itself. In my perspective, however, they are the first steps of a major paradigm shift, a critical renewal arriving in timely fashion, of an entire area of cultural studies. Let us encourage this trend, which could be advanced spectacularly by a new generation of students welltrained in mathematics as well as in a social or human science.
The stimulus for this discussion came from the dynamic between "positive" and "negative" and the developing widespread movement in favour of "positive" thinking and in opposition to "negativity" (see J K Norem and E C Chang. The positive psychology of negative thinking, 2002). Negativity may even be condemned as "bad", even sinful. The earlier paper (Being Positive Avoiding Negativity: management challenge  positive vs negative, 2005) endeavoured to show that in both strategic and practical contexts there was clearly a need for both modes  if only in the cybernetic terms of positive and negative feedback required for systems control.
This polarity can however be seen as merely a rather "pure" and obvious example of many other forms of "incommensurables" between which an operational relationship has somehow to be ensured in practice and in daily life. Other examples range from economy vs environment, through peace vs conflict, female vs male, and including abstract vs concrete. They also include the ordering of the many interpersonal and strategic dilemmas faced in society (cf Value polarities).
The (re)discoverer of the Mset, Benoit Mandelbrot, recognized repeating patterns on all scales in numerous phenomena  cotton prices, clouds, and coastlines. Whilst his research showed that the changes were unpredictable  namely random  the sequence of the changes was independent of scale. This means that the variation in each case is no more of a period of centuries, than over decades or years  socalled scale invariance. This applied equally to shapes such as clouds, trees or earthquakes and resulted in the formulation of the concept of fractals as a measurement of roughness or irregularity that demonstrated selfsimilarity on all scales. In natural systems, the structure of the whole system is often reflected in every part of it  especially when similar forces act at many levels of scale. Natural forms tend to reveal transformed copies of the whole in every part. Fractals are therefore widely found in nature. So in many ways fractal structures are potentially more relevant than more conventional idealized scientific concepts.
It was later established that chaos is a feature of many nonlinear dynamical systems. Their deeply cyclic structure does not however imply that the cycles repeat exactly. Whilst the amount of the variation within such cycles is constant, the variations with variations makes them inherently unpredictable at every level of scale. These nested cycles may be simulated by iterative procedures.
It is appropriate that relating the apparently incommensurable should be achieved through "fractal" techniques in contrast with the techniques of algebra" that have proved so appropriate to relating the commensurable. Algebra derives from the notion of "binding together", in contrast with Fractal that derives from "breaking into irregular fragments".
The term "Mandelbrot set" is used to refer both to a general class of fractal sets and to the particular instance of such a set derived from the quadratic recurrence equation z_{n+1}= z_{n}^{2} + c. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set (Jset) is connected and not computable. It is the particular instance that is discussed in what follows (and referred to as the Mset). [more]
Although the Mset is indeed perhaps the best known fractal, there are many other types. In its more general form, the power in the quadratic recurrence equation may be increased from the standard "squared" form (with one symmetry axis) to the cubic form (with two such axes), the quadric (with three)  with any number of "poles" by suitable choice of exponent. These raise fruitful questions which are not however addressed in what follows. It is the particular instance that is the focus here because it is the simplest that gives rise to an object of such great complexity.
The Mset fractal corresponds to the simplest nonlinear function  but is also as complicated as a fractal can get. It distinguishes the simplest boundary between chaos and order. It is recognized as the simplest nontrivial example of a holomorphic parameter space. Given the significance of pi in defining a circle as a simpler object, the generation of the Mset by iteration may be compared to the iterative calculation of pi (cf Alex LopezOrtiz. How to compute digits of pi ?; Dave Boll. Pi and the Mandelbrot set) [more]. One method, with a striking formal resemblance to that required for the Mset, is: z_{n+1}= z_{n} + sin(z_{n}), especially if initialized to z_{n} = 3 [more].
In the search for solutions to complex equations, experiments with iterations by computer have highlighted intricate global properties related to nonconvergence and the stability of convergence. The behavior of quadratic functions, as the simplest of all nonlinear mappings, combines ease of calculation with sufficient generality to illustrate most of the abstract properties of iterations. Just as using complex variables often clarifies the properties of functions of a real variable, studying complex iterations can be expected to generalize and illuminate real nonlinear mappings as well.
The complex space in which the conceptual and value concerns are significant has been usefully described by Vladimir Dimitrov (Glimpses at Mathematics and Physics of Social Complexity)
We use the concept of strange attractor to describe emergence of meanings in the mental space of an individual or a group (organization). The phase space where meaning emerges is the 'multidimensional' mental space of an individual or a 'swarm' of individuals  a nonmaterial ("transcendental" in Kantian term) space energized by continuously generated thoughts and feelings. Meaning has fractal structure  once a certain dynamical sign makes sense to somebody, s/he can 'zoom' deeper and deeper into the meaning of this sign. Although each level ('scale') of meaningexploration may differ from any other level, there is similarity between the levels, as they all relate to the dynamics of one and the same sign interpreted by one and the same individual. The strange attractor of meanings can exhort human actions. Although, the actions may appear randomly skipping around, they relate to the attractors of meanings, which propel them. If there is no attractors of meanings behind one's actions, the actions are simply meaningless; they are running at physical level only. The lack of intelligent support, be it mental, emotional or spiritual, is incompatible with one's growth as a holistic individuality.
Dissipative systems, and the Mset, offer a language through which to explore and identify viable patterns of sustainable relationship between essentially incompatible modes of behaviour or antithetical modes of thinking. It is these which are typically fundamental to the strategic dilemmas in pyschosocial systems  whether intrapsychic, interpersonal or intergroup. It is the continuing search for the resolution of these dilemmas that characterizes the dynamic of such systems. Typically however the resolution is of four types:
This approach offers a pattern language to explore the complexities of the periodic resolution to strategic dilemmas  the space of notthis, notthat (the neti neti of Sanskrit). The emergent patterns there are those which characterize a multitude of dynamically stable experiential resolutions of strategic dilemmas. These dynamic resolutions can be depicted (through the Mset) as characteristic patterns of great variety. The set of all such patterns (the Mset as a whole) is of a coherent form that is reflected in many ways (isomorphically) in their detail ("when two or three are gathered together in my name, there am I").
The pattern language is of significance because it enables agonizing psychosocial dilemmas, such as employment vs unemployment (environment vs employment, "affairs of the heart", etc) to be addressed in new ways  unconstrained by the conventional binary logic and the logically excluded middle. In effect it is a language for exploring the viable patterns of the "middle way". It gives form, space and locus to particular dynamic resolutions of strategic dilemmas. The viability of these patterns, and the challenge to their comprehension, arises, however, from their characteristic dynamic  in contrast with the stability normally sought in nondynamic resolutions to such dilemmas.
The characteristic form taken by the set of patterns as a whole is also of particular significance because of the way in which its aesthetic potential can be used to mnemonic advantage. As with delightful melodies, it offers memorable features that reinforce the coherence of the pattern in practice. In addition, as depicted, these emergent patterns are in many respects intuitively recognizable and familiar rather than being alien to the human psyche. It is in this respect that they may echo  and be echoed by  cultural symbols of great archetypal significance. In these senses, "Mset" might more usefully be understood as the "Memorable set" or the "Mnemonic set". But the challenge to comprehension  through "iterative "remembering" of it as a gestalt  might then be understood in the light of Antonio de Nicolas' poetic title (Remembering the God to Come: a book of poems, 1988).
The particular concern here is with how the geometry of the dynamic pattern is sensed experientially  how the "geometry is felt" (using the "computing" and "graphics" capability of the brain)  rather than with the technicalities that are important to its rigorous description. The challenge is to ensure that the latter serves in improving the quality, richness and viability of experience in engaging with strategic dilemmas. As a mathematician, Ron Atkin (Multidimensional Man; can man live in 3dimensional space?, 1981) has addressed how geometry may be "felt" in a communication space (see Social organization determined by incommunicability of insights). .
The Mandelbrot set is not an invention of the human mind; it was a discovery. Roger Penrose 
The following points endeavour to provide a rationale for the approach taken:
In support of this approach, for example, Chris C. King (Fractal and Chaotic Dynamics in Nervous Systems, 1991) presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel.
This exploration offers an intriguing challenge in attempting to render comprehensible some rather subtle insights. For mathematicians the Mset is recognized as one of the most complex objects  whilst at the same time claiming that its intricacies are basically accessible to those with a background in high school mathematics. For many, like this author, exposure to mathematics at that level may no longer be meaningfully remembered  effectively grouping them with those who have not had that exposure. On the other hand, as visualized through dramatic fractal displays, the object lends itself to easy exploration and has aroused much enthusiasm  supposedly avoiding the need for any mathematics.
However the purpose of this exploration is to benefit to a higher degree from the mathematics, without getting lost in its technicalities, and to focus on its implications for offering an ordering for psychosocial insights that may have been acquired or intuited through other disciplines. That said, there remains the problem of how to structure this exploration so as to offer a link to the mathematics for those who may have some willingness to benefit from it (and be reassured by its formal features)  without disturbing the flow of the argument and distracting from its integrative commitment.
Clearly the argument is primarily speculative  a rightbrain exercise. The mathematics may offer a leftbrain framework for some. It must also be said that, for the author, endeavouring to make the technical arguments of mathematicians meaningful proved to be an extremely valuable exercise in triggering such intuitive rightbrain insights.
This paper therefore carries the speculative argument. Extensive links to introductory explanations elsewhere are provided in the table below.
It should be noted that with respect to any "nonmainstream discipline", any reference to it here is not to be considered as an endorsement of that perspective. Its significance may however lie in the size of the constituency holding that view  namely in the political and cultural implications of the dynamic arising from such alternative views in a global system. The purpose here is to raise issues for imaginative exploration, not to seek premature closure.
A very useful articulation of the challenge is in terms of dissipative systems about which the remarks of Kent Palmer (Steps to the Threshold of the Social: the mathematical analogies to dissipative, autopoietic, and reflexive systems, 1997) seem the clearest and most relevant for the above purpose. For him (pp 587588):
Dissipative systems hold two strands of illusory continuity together. They concern the situation where there are two orders that are in imbalance so that one order is displacing the other. Notice that if there is only one order there cannot be a dissipative system. Also if the two orders are in balance or stasis there cannot be a dissipative system. A dissipative system is when there are two different orders or ordering mechanisms that are out of balance with each other so that one ordering mechanism is disordering the other and creating a boundary between the two ordering mechanisms where one is dominant and the other is being dominated.
Such language would seem to be a helpful way of handling the many fundamental strategic dilemmas that affect both the coherence of global debate and the experience of interpersonal relationships. The challenge is indeed one of two different "ordering" mechanisms, whether these are culturally defined (Huntington's "Clash of Civilizations", Snow's "Two Cultures", political cultures ( "right vs left", "mainstream vs alternative"), gender defined ("Men are from Mars and Women are from Venus"), or in terms of epistemological mindsets (Systems of Categories Distinguishing Cultural Biases, 1993).
As Palmer argues, this situation can be approached using the "imaginary" qualities of complex numbers, stressing the nature of the "illusion" involved:
This case has the basic form of vector arithmetic or the complex number system that holds the order of the real numbers together with the ordering of the imaginary numbers. The complex number system includes both real and imaginary numbers. The differentiation between the two is indeed imaginary because either number could be designated as real and the asymmetry between imaginary and real numbers is an illusion which comes directly from their conjunction not from the numbers themselves. In the case of the complex number system the reals are dominant and the complex numbers are subservient.
In the other previous paper (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005), the challenge of "positive" and "negative" was handled through a coordinate system developed by Edward Haskell to map pairs of interacting biological species in terms of the nature of their transaction or "game". This gave rise to a "coaction cardioid". But as Palmer indicates in endeavouring to map out such relationships:
We only actually see the relation between the two if we place the complex axis at right angles to the real axis. When we look at the field of these numbers what becomes apparent to us is the form of the mandelbrot set. The Mandelbrot set is the most complex mathematical object known to man. This set is composed by iteratively taking each point and multiplying it by itself and measuring the rate at which it escapes toward infinity. All real numbers escape toward infinity at the same rate. The numbers that represent the intersection between real and complex have different rates of escape toward infinity. We will... call each of those escape velocity weights the line of flight of a particular point. Dissipative systems have an interface between their two orderings (that of the system and that of the environment) which is very complicated. It involves myriad lines of flight that produce an infinitely complicated pattern which is still determinate.
It was suggested that the cardioid intrinsic to Haskell's approach could possibly be understood as that feature of a Mset. It is indeed the case that the systems to which Haskell's coaction cardioid was applied could be understood as dissipative systems  even though he did not use the axial representation conventionally used for complex numbers (as described by Palmer).
In order to offer a framework for any more detailed discussion of some of the technicalities of how the Mset emerges as a coherent pattern  and its significance for the above purpose  it is useful to provide a focus through the features of a visual image to which reference can be made.
Figure 1: Mandelbrot set 
Generated by Xaos: realtime fractal zoomer 
As a representation of the Mset, Figure 1 is rotated 90^{o} from that used conventionally. This is an orientation similar to that used in the earlier paper on the cardioid as an attractor. It is also that favoured by the many who compare it to the seated Buddha, especially when coloured to highlight the concentric "auras" (as in Figure 1). It could be rotated a further 180^{o} to give prominence to the cardioid effect, for those who associate more strongly with representations of the heart.
Selected Web Resources on Mandelbrot and Julia Sets There are extensive resources available (especially on fractals in general)  of different quality and duration, since many are developed as student projects 

Static images / Animations  Xaos  . 
Mandelbrot Explorer Gallery  .  
Interactive applets  Mandelbrot Fractals  . 
Julia and Mandelbrot Sets 
David E. Joyce 

Mandelbrot Applet  .  
The Mandelbrot/Julia Set Applet  James Denvir  
Mandelbrot and Julia sets  Alexander Bogomolny  
Fractal Microscope  http://www.shodor.org/master/fractal/software/mandy/  
Java Julia Set Generator  Thomas Boutell (Boutell.Com)  
Mandelbrot Set  Eckhard Roessel  
Mandelbrot Explorer  Panagiotis Christias  
Mandelbrot Fractals Generator  Salvatore Sanfilippo  
The Fractal Microscope  Alton Patrick.  
The Mandelbrot Set Iterator  James Denvir  
Julia and Mandelbrot Explorer  Julia Thrower (Texas Tech University)  
Overview / Features / Tutorials  The Mandelbrot Set Explorer  Systematic overview by Robert L. Devaney 
The Mandelbrot and Julia sets Anatomy  Evgeny Demidov  
Xaos  Separate tutorial (associated with a demonstration)  
Studying Mandelbrot Fractals 
Suzanne Alejandre: 

Overview  SUNY (Binghamton)  
The Mandelbrot Set and Julia Sets  Yale University  
The Mandelbrot Set  University of Utah  
Introduction to the Mandelbrot Set  David Dewey (A guide for people with little math experience)  
Chaos Hypertextbook  Glenn Elert  
Chaos and Fractals  Dale Winter (introductory course, notably Julia sets and Mandelbrot set)  
Fractal Explorer: Mandelbrot and Julia sets  Fabio Cesari  
Downloadable interactive demos  Xaos: realtime fractal zoomer  Excellent open source demonstration software (free), with many interactive facilities and an excellent tutorial  focused on Mandelbrot and Julia sets 
Winfeed fractal exploration program  Allows user to explore functional iteration, including Mandelbrot and Julia sets (Exeter University)  
Fractal Explorer  Free. See also tutorial  
Glossaries  MuEncy  Mandelbrot Set Glossary and Encyclopedia  Robert Munafo 
Mandelbrot Set Explorer  . 
As noted by Len Warne (A Meditation on the Mandelbrot Set, via Walt Whitman):
The Mandelbrot Set emerges from the behavior of a famously simple mathematical function. The Set itself is like a black hole in the abstract space it inhabits. Most of that space is a vast, featureless void. But the points near the boundary of the Set are torn between the temptation to join the Set and the lure of infinity. When their behavior is coded in colors, the result is a beautiful filigree of infinite depth and complexity.
For detailed descriptions of features of the Mset, see the web resources in the above table.
The Mset can be divided into an infinite set of figures (typically represented as black, as in Figure 1) with the largest figure (in the center) being a cardioid. An (infinite) number of circles are in direct (tangential) contact with the cardioid  but they vary in size, tending asymptotically to zero. Each of these circles has in turn its own infinite set of smaller circles in contact with it, and these surrounding circles also tend asymptotically in size to zero. Repeatedly indefinitely, this branching produces a fractal. In addition the Mset is characterized by filaments or tendrils within which some new cardioids appear, not attached to lower level "circles". [more]
The shape of the main Mandelbrot cardioid is given by the formula z = e^{it}/2  e^{2it}/4 . The largest bud, the west bud, appears at t=2*pi/2. The next largest, the north bud, is at t=2*pi/3, and so on, each at t=2*pi/n.... In fact, the importance of Fibonacci Numbers and Farey Numbers for the description of the Mandelbrot set is well known. (See, for example, R.L. Devaney How to Count).
Figure 3: Periods of Mset "bulbs" and cardioid 
You first plow in the dynamical plane and then harvest in the parameter plane. Adrien Douady 
The emergence of the Mset as a focus of disciplined reflection enables extremely useful distinctions to be made between levels of abstraction and the nature and credibility of any corresponding explanations. Any assumption that the Mset offers a simplistic, singlefactor explanation should be carefully examined  perhaps in the light of the relation between multiperiod attractors and multifactor explanations. The potentials of the Mset relate in fact to the concerns with metasystemic perspectives, notably those developed by Kent Palmer (Autopoietic MetaTheory, 1998; Deep Mathematics and MetaSystems Theory, 1997; Vajra Logics and Mathematical Metamodels for Metasystems Engineering, 2001).
Distinctions might be made in terms of deductive ("topdown") and inductive ("bottomup") approaches:
It is the first of these which is associated with invariance in its most
fundamental sense. But as such it is at a level of abstraction that is largely
beyond explanation and comprehension. It points to (or "maps")
the variety of models of dynamic behaviour, effectively providing an explanatory
context for them. We then encounter the level of systems analysis that abstracts
from sensible phenomena particular behavioural functions by which the behaviour
is modelled. Only finally do we encounter the level of the actual sensible
phenomena that are experienced in the confusing variety that is organized
by any form of systematic analysis at the preceding level.
There are a number of sources distinguishing levels of abstraction:
G. Experiential rebirth (operacy, flow, embodiment of mind, speaking with God, bornagain, possession, psychedelic experience, embodiment in song, spiritual rebirth)  F. Cognitive perspective (metacognition, critical thinking, philosophy, aesthetic sensibility, orders of thinking, systematics, orders of abstraction, disciplines of action) 
E. Therapeutical rebirth (release from trauma, mentors, selfhelp, discipleship)  D. Developmental rebirth (education, perspective, initiation, cultural creativity, individuation) 
C. Psychobehavioural rebirth (sintovirtue, changing patterns of consumption, conversion,  B. Socioreligious rebirth (birthright, destiny, reincarnation, social status, ceremony, ritual, group affiliation, games, sports) 
A. Cultural rebirth (renaissance, aesthetic birth, mythopoesis) 
Levels of abstraction, however clustered, focus two intertwined questions:
These questions might be discussed in terms of metalevels and how communication about them can be ensured with minimal confusion (** Palmer). For example, David Wright (Talking 'about': Languaging, emotioning and communicating knowings, 2002) asks:
How do art and anecdote meet analysis in the pursuit of understanding? This paper proposes that we are in a constant process of negotiating genres in recognition of the structural limits to our knowing, further that our knowing is itself in negotiation with those structural limits and that emotion and language are key mediators and moderators in this process.
Whilst mathematics may offer formal distinctions, the mind may be much challenged to comprehend these distinctions. Much use can be made of metaphors in making the distinctions, as is typical of spiritual discourse. These do not necessarily offer an experiential sense of the distinction. It is here that the proposal of Kathleen Forsythe to use the term "isophor" to depict a feeling response to one thing in terms of another is relevant. Forsythe describes an isophor as something that is "experienced emotionally and, (that) as such, define(s) the experience of understanding." (1987)
Kathleen Forsythe (Cathedrals in the Mind, 1986) in a paper to a meeting of cyberneticians argues: "Analogy and its poetic expression, metaphor, may be the "metaforms" necessary to understanding those aspects of our mind that make connections, often in nonverbal and implicit fashion, that allow us to understand the world in a whole way."
Forsythe uses the term isophors for isomorphisms experienced in the use of language. Isophors are distinct from metaphors in that they are experienced directly. With the isophor there is no separation between thought and action, between feeling and experience. The experience itself is evoked through the relation. She suggests that the experience of one thing in terms of another, the isophor, is the means by which domain is mapped to domain and that consciousness of this metaaction, when recognized, lies at the heart of cognition. Forsythe has postulated the development of an epistemology of newness in which learning is the perception of newness and cognition depends on a disposition for wonder leading to this domain of conceptionperception interactions. She argues that the notion of metaphor is commonly understood to mean the description of one thing in terms of another  presupposing an objective reality. This objectivity may be questioned and if, as suggested by Maturana, (objectivity) is placed in parentheses:
"we can begin to appreciate clearly the role we play in the construction of our own perception of reality. for this reason, the notion of the experience of one thing in terms of another, the isophor, suggests that it is this dynamic constructing ability that involves conception and perception  unfolding and enfolding, that this gives rise to the coordination of actions in recursion which we know as language." [more]
The challenge in what follows is to determine whether felt experience and insight resonate with formal representation. This is the challenge of aesthetic proportion in general and of sacred geometry in particular. The concern is not to force such an association but rather to provoke an imaginative exploration of possibilities. This exploration is not aimed at closure or reductionistic grasping of a subtle integrative pattern. The concern is more to offer a form (the Mset) that ensures an interplay between suggestive possibilities. It is however important to recognize that the very complexity of the Mset, however well understood intuitively in some way, poses a real challenge to explanation and comprehension of the Mset as a 4dimensional gestalt. The challenge might in some ways be compared to explaining movement up a spiral staircase, without pictures, to someone who has never seen one. The length of this paper is perhaps a measure of the lack of full understanding of that gestalt by its author!
It is important to be as clear as possible concerning the challenge of relating what mathematics can discover, the possible beauty of its graphical expression, with reality as known and experienced. Each constitutes a different focus and their relationship is not necessarily evident or rigorously established.
It is perhaps appropriate to recall the early recognition of the paradoxical challenge of explaining chaos as articulated by Heraclitus (540475 B.C.E):
Dissatisfied with earlier efforts to comprehend the world, Heraclitus of Ephesus earned his reputation as "the Riddler" by delivering his pronouncements in deliberately contradictory (or at least paradoxical) form. The structure of puzzling statements, he believed, mirrors the chaotic structure of thought, which in turn is parallel to the complex, dynamic character of the world itself.
Rejecting the Pythagorean ideal of harmony as peaceful coexistence, Heraclitus saw the natural world as an environment of perpetual struggle and strife. "All is flux," he supposed; everything is changing all the time. As Heraclitus is often reported to have said, "Upon those who step into the same river, different waters flow." The tension and conflict which govern everything in our experience are moderated only by the operation of a universal principle of proportionality in all things. (Garth Kemerling. Origins of Western Thought, 2001)
In contrast with the uniqueness of a particular experienced phenomenon, a fractal models the essence of a species or type, not the appearance of a particular individual. Fractals can be found to fit any set of data, including inherently unpredictable and chaotic systems  where linear equations fail entirely (as representations of the rate of change of a system at any given point). The question is how to understand the relation to the Jsets and Mset discussed here. Fractals emerge at the border between harmony and dissonance  when rhythms fall into or out of sync. The human heart and the brain are dynamical systems in motion. In this respect, psychiatrist Arnold Mandell is quoted (by James Gleick. Chaos: making a new science, 1986) as follows:
Is it possible that mathematical pathology, ie chaos, is health? And that mathematical health, which is the unpredictability and differentiability of this kind of structure, is disease? ... When you reach an equilibrium in biology, you're dead.
There is now an interest in "dynamical diseases"  when fractal rhythms fall out of sync. On this point, Dick Oliver (Fractal Vision, 1992) notes:
Life is rhythm. But it's a special kind of rhythm, a rhythm where resistive friction is always dragging it toward rest and almostcoordinated pushes are always pumping it back into sync. Graphs of heartbeats and brainwaves are not smooth pulsations, but fractals with a dimension nearer to two than one. They thrive on chaos, and sicken with smoothness. when this measure of dimensional roughness falls closer to linearity, a heart attack or seizure is probably on its way.
In a personal communication in 2005 on the questions posed by this paper, Chris Lucas (CalResCo: The Complexity and Artificial Life Research Concept for SelfOrganizing Systems) makes the points:
The problem would be that mathematicians who study fractals concentrate on low dimensional (analysable) systems and do not think their work is applicable to the messy "real world". Those who recognise real world fractals like lungs, trees, fern and coastlines have rarely the mathematical knowledge to create a formula (which probably would be beyond the mathematicians anyway). We can estimate fractal dimension for simple examples like trees, but in itself this says little about the generating function. We can though find a sort of "formula" for real fractals by using Michael Barnsley technique of fractal compression, which finds a "seed" which when iterated recreates the image, some examples of the results of this can be seen in Fractal Vision
What we must bear in mind is that nature does not generate fractals by computer iterations, and certainly not in the rather strange way in which we create the Mset (where we iterate disjoint points thousands of times (for a 640x480 screen we do it 307,200 times in total)  the picture we then see is an artefact of this and not a single ongoing iteration.... This is again the "map is not the territory" distinction, the painting is not the landscape. So even if we ask the questions that you pose, it is not at all clear that they could even have a meaningful answer.
The Julia set for example, although related to the Mset in definite ways is yet another concatenated set of disjoint iterations (which are just adjacent) and has no physical reality as such. In other words, for a tree we go along a trunk and this "branches" fractally, but in neither the Mset or Jset is any part of the picture mathematically connected to the adjoining points, we "construct" the picture analogously to making a mosaic, i.e. the resultant is more like an abstract "phase portrait"  a "map" of state space than anything natural. All connections we make to real world "fractals" are just analogies (as far as I understand it, but bear in mind I'm not a mathematician, some may see it all very differently  especially those who think the world *is* a vast computer !). It is interesting to speculate *why* we created the Mset in this strange way  is our love of fractal structure such that we made it in our own image subconsciously ? What other ways of "assembling" such things may be possible... !
The theoretical disconnect from experienced reality has been expressed differently by Dick Oliver (Fractal Vision, 1992):
You obviously can use fractal templates based on affine transformations to model nature. But all this Julia set business seems about as far from nature as you can get. chaotic, nonlinear transformations such as z squared plus c can produce pretty pictures, but they don't seem to have any connection with the physical world at all.... No one is certain how the spirals and branches in the Mandelbrot and Julia sets arise from nonlinear equations, let alone why they follow the archetypal patterns of nature so closely. these topics are at the forefront of current mathematical and scientific research.
It is curious the extent to which so much hangs on the understanding of "iteration". Clearly there is a sense in which the cyclic phenomena studied by Mandelbrot can be usefully understood through iterative procedures. It is also clear that recurrent daily experience  such as starting the day with the state of one's office as left the previous day  can be understood as an iterative process. Experience itself may be understood as having been built up through iteration  repeatedly taking past experience and using it as the configuring seed for experience in the present moment.
An iteration takes all the past  the past as a whole  and feeds it into the function in the moment. From any "seed thought" (or intuition), one is then always recomputing the whole  so the maths of iteration are extremely close to moment by moment thinking (life as "constantly computing z from c"), and our various cognitive habits (see Antonio de Nicolas. Habits of Mind: An Introduction to Philosophy of Education, 2000).
It is the pattern recognition (or imposition) ability of the human mind that bridges the mathematical disjointedness to which Chris Lucas refers. The iterative process of recomputing points to constitute an image has similarities to the pointillist, postimpressionist painting style founded by artist Georges Seurat. The semiotic challenge in relation to artificial intelligence has been addressed by Swinton Roof (MandelBug: A New Project, 2003).
There has to be some kind of balance and homeostasis amidst interesting dynamics. Semiotic unity has a similar problem. If symbolic unity is achieved in a brain, how does it do it? A brain is a fantastically complex nonlinear system. What enables it to avoid fractally treacherous and chaotic event horizons that rip up any possibility of a unified coalescence of the state vectors. And how does it do it yet still remain flexible and novel? Some sort of critical edge would be good for precise decision making, but conversely, we can't have every minute change affect a system with hairtrigger sensitivity. Symbols have to retain a certain fixed identity yet still connect adaptively to novel changes.
The foundations of mathematics has a similar problem. Recursive statements lead to inconsistency and lack of completeness. Attempts at unified foundations for symbolic systems thus becomes fractured by holes and forked tongued truths. Mathematics, the bedrock of rationality, stands on the brink of bifurcative nonsense. Recursive functions and algorithms in computer programming are notoriously difficult to debug and in some cases impossibly so perhaps.
It is fruitful to look for "levels of abstraction" from experience through to the Mset. But such a ladder is itself problematic as implied above. Furthermore, there is a body of literature stimulated by feminist scholars (cf Carol Gilligan, 1982, 1990) that questions use of "levels" and points to use of a configuration of complementary modes that may be variously accessible (see Learnings for the Future of InterFaith Dialogue : Insights evoked by intractable international differences, 1993). Such cognitive modes might be understood as attractors of different types. The configuration might be understood in terms of the Mset pattern. Some exploring fractals also question this asymmetrical viewpoint. Although a fractal image of the "lower" parts are contained within the "higher" whole, remarkably the "higher" whole is equally contained in the "lower" parts.[more]
In addition, "below" the experience of dynamics, there is the question of how one participates in those dynamics  entering into them nonabstractly. And, "above" the abstraction of the Mset, there is the question of how one engages with it and embodies it. Both extremes are beyond the maths (and may merge together forming a "cognitive torus", as with the Ourobouros). The Mset may be effectively understood as an experiential standing wave.
These considerations may require a decision from the reader as to how to approach suggestions in this paper. The broader issues may indeed make it "too flaky" for mathematicians and decisionmakers, and too formal for experiential people. But as such it does hold the dynamic of the dilemma their unrelatedness constitutes. Given the ambition of the paper in relation to strategic dilemmas, it is to be expected that this would be in some measure reflected in how its content is explored. Those with a relative dominance of leftbrain over rightbrain would seek any order offered by the mathematical abstractions in response to chaos, whereas those with a relatively dominant rightbrain would be more persuaded by the aesthetic continuities of the patterns and what they imply for participative experience.
Julie Thompson Klein (Interdisciplinarity and complexity: An evolving relationship. 2004) concludes her review of the challenges to more integrative thinking as follows:
Because of the relevance of fractals to many fields, they tend to raise questions about the limited specialized boundaries of science, thus facilitating a more integrated approach. As such hey require a different type of understanding than is typically associated with scientific understanding. The conventional approach is still based on the rational paradigm (which is limited in many ways). Fractals are claimed to require a deeper holistic appreciation involving both reason and intuition. [more]Contests of legitimacy over jurisdiction, systems of demarcation, and regulative and sanctioning mechanisms continue, and perceptions of academic reality are still shaped by older forms and images. Yet, boundaries are characterized by ongoing tensions of permanency and passage. Simplified views of the complex university only add to the problem of operational realties that outrun old expectations, especially older definitions that depict one part or function of the university as its "essence " or "essential mission "... Repeating the same metaphors... adds to the confusion, impeding understanding of new knowledge, new relationships, and nonlinear, nonvertical perspectives that are multidimensional and multi directional. A wider range of physical and topological or architectural metaphors are being used to describe relations of elements that make up innovations and their contexts  dimensions, joints, manifolds, points of connection, boundedness, overlaps, interconnections, interpenetrations, breaks, cracks, and handles. And ... we might add ... a Mandelbrot set.
As noted above with regard to "incommensurables", the stimulus for this investigation was associated with the the challenge of dealing with irreconcilable perspectives, notably a focus on the "positive" (as the "good") and an avoidance of the "negative" (as the "bad"). Using the axes of the complex plane, to position perspectives reflecting different kinds of "positive" and "negative", reframes the dynamics of the dramatic polarization on which much attention is un fruitfully focused. Furthermore this framing of the dynamics that characterize the encounter between polarized perspectives indicates the possible existence of various zones that merit greater attention:
The question is whether the Mset is indeed indicative of a zone of stability relevant to understanding of other paradoxically opposed, valuecharged perspectives, such as:
In other words, does the Mset then perform a kind of "keystone" function sustaining a space. This challenge of balancing polarities has been explored in previous papers, notably in the light of the metaphor provided by tensegrity structures (Implementing Principles by Balancing Configurations of Functions: a tensegrity organization approach, 1979; Transcending Duality through Tensional Integrity: From systemsversusnetworks to tensegrity organization, 1978), and subsequently explored by management cybernetician Stafford Beer (Beyond Dispute: Invention of Team Syntegrity, 1994).
In each case, the question is whether the tensions between the valuecharged strategic polarities can be fruitfully dissociated into "real" and "imaginary" components such that the dynamics engender a sustainable boundary vital to psychosocial coherence  without collapsing the dramatically opposed perspectives that characterize the polarity.
In a strategic context, "real" is associated with factual data. But as is evident in practice, proponents of opposing initiatives have divergent interpretations of "real" and of the weight to be attached to different "facts", held to be "true". Each is then free to accuse the other of responding to "imaginary" interpretations  and this tends to be very sharply stated in debate (caricaturing the opposition with terms such as "dreamers", "deluded", "unrealistic", etc) regarding what is "false". In a sense each sees the opposition as responding to an unreal "image" of reality. It is the dynamics of disagreements of this nature that need to be held with a framework of requisite complexity  transcending relativism  in order for governance to articulate strategies that are sustainable.
There is a certain irony to the tendency of strategic proponents to plead for more "facts" (monitoring, research, etc) prior to action  or to call for more "imaginative" thinking to respond more effectively to new kinds of crises or the inadequacies of previous strategies.
The argument here is that the kind of sustainability that would be sustainable  rather than being itself a victim of these dynamics  is at a level of abstraction to which the Mset usefully points. As a framework, it in no way denies the existence of the dynamics between constituencies with different understandings of what is real and what is imaginary. Rather the recognition of the Mset depends on those dynamics  just as the 2D polarities within a tensegrity are essential to the emergence and viability of the 3D structure resulting from their configuration.
In this light the question becomes how to recognize and distinguish the strategic elements contributing to recognition of such an Mset. The need is to offer clearer understanding of the role of "real" and "imaginary", recognizing that "real" to one group may be "imaginary" to another. This reinforces the point made with regard to transforming the axes between "real" and "imaginary", or between "positive" and "negative".
Considering once again how such distinctions would be made in the absence of the cartesian understanding of axes, it is worth reflecting again on the notation used in the thinking basic to the I Ching, namely the system of trigrams configured in the Ho Tu or Lo Shu arrangements of the Ba Gua mirror basic to the discipline of Feng Shui (as discussed earlier). There the focus is on "directions" rather than axes. But clearly two distinct digrams (rather than trigrams) would provide an adequately complex notation to distinguish "positive" and "negative" on a "real" axis, with a second two to distinguish "positive" and "negative" on an "imaginary" axis. In this connection, it should not be forgotten that it was the exposure of Gottfried Wilhelm von Leibnitz to the I Ching that inspired his development of the binary coding system.
The nature of the relationship between "real" and "imaginary" can be further considered in the light of the Chinese categories of "yang" and "yin" which are not fruitfully treated as "opposites". All relationships based on yin and yang are considered as relative. Mutual interaction must be considered, therefore, nothing can be defined as strictly yin or strictly yang. Yin and yang are symbolically represented by the LiangI (two symbols). The YangI is represented by a continuous straight line and the YinI is represented by a broken line. The conditions to which they refer cannot be considered as permanent states. There is always dynamic movement which is encoded through combinations. The first group of these is called SzuHsaing. These four figures (digrams) are formed by combining the YinI and the YangI. The SzuHsaing represent the maximum number sets that can be formed by combining two differing elements in sets of two.
This development is framed in a muchcited passage in the I Ching that relates these abstractions to the strategic challenges of human affairs:
In the Changes there is the Supreme Ultimate (T'ai Chi), which produced the Two Forms (yin and yang). These Two Forms produced the four emblems (SzuHsiang), and these four emblems produced the eight trigrams (Pa Kua). The eight trigrams serve to determine good and bad fortune (for human affairs), and from this good and bad fortune spring the great activities (of human life).
As noted earlier, the question is then how to relate such digramatic codes to Mset axes in the light of greater insight into the contrasts between the Ho Tu and Lo Shu arrangements and understandings of "real", "imaginary", etc in the Mset mapping. There is a case for exploring these arrangements in the light of use of polar coordinate mappings of the Mset, notably in relation to the rotation number of "bulbs" (cf Linas Vepstas: DouadyHubbard Potential; Mandelbrot Bud Maths; Parameter Ray Atlas. 2000) [more].
If it was also necessary to distinguish conditions that were "positivereal" from "negativeimaginary", for example, then indeed trigrams would be necessary. Finer distinctions (as with the compass directions SSW or NNE) could then be made by adding an extra line position to the digram notation. The 64 hexagrams of the I Ching ("The Book of Changes") provide the notation for a Chinese approach to the dynamics that emerge as the Mset from the mapping onto the complex plane. There is an extensive body of literature exploring the significance of the mathematical underpinnings of the I Ching [more  more  more], notably that of Stephen M Phillips (I Ching and the Eightfold Way). It is appropriate to reflect on the justification for the incorporation of hexagrams into Version 4.0 of the Unicode standard [more]
In contrast with the mathematics of the Mset, this Chinese system was designed to embody qualitative value contrasts (rather than purely quantitative value contrasts) and was notably used in the clarification of strategic options by the emperors of China. The I Ching was in fact required reading for the Chinese civil service for about 1,000 years. One might well ask what tools of comparable complexity and scope are currently used with respect to global governance.
Of particular interest is the polarity between "objective" approaches ("real") and "subjective" approaches ("imaginary"). This is notably evident in the "objective" attitude of mathematicians to complexity  in comparison with the "subjective" attitudes associated with the psychosocial phenomena noted above. However it is the "real" nature of the 40 religious conflicts around the world  driven by a sense of "positive" ("good") and "negative" ("evil") in this "imaginary" dimension  that can be contrasted with the "unreality" of mathematics to those engaged in those conflicts.
Both Physicist David Bohm (Wholeness and the Implicate Order, 1980; The Undivided Universe: an ontological interpretation of quantum theory, 1993) and those referring to his work, tend to cite the fractal nature of the Mset. In this light, the strategic challenge of "real" and "imaginary", from a psychological perspective, is helpfully discussed by Nicholas Williams (Psychological Entropy, 2003):
Nonlinear dynamics challenges our preconceptions to the very core  in fact it reverses our basic expectations about the world. If we did not know better, we would naturally assume that positive feedback (amplification of errors) can lead to nothing but meaningless static, zero information in other words, but what the study of nonlinear systems shows us is that the reverse can happen  instability can lead to the system regrouping on a more highly organized level, not the other way around. There is an information jump, not a drop.... We think of linear systems, on the other hand, as being progressive in the sense that they are continually moving in the direction of increased information content. The reverse of this is true  no new information ever comes into a system exhibiting linear change, they are utterly tautological and never say anything new.
The psychological implications of this are immense  we are forced to confront David Bohm's spooky vision of the rational (i.e. linear) mind as a dealer in illusions, forever offering up visions of progress, new ways of looking at the world which are actually only "reshuffled" versions of the same old thing. The "no win situation" that is neurosis arises precisely out of this disguised tautology because all the "solutions" that our rational mind offers up are in fact reshuffled versions of the original problem. The (purely rational) attempt to come up with and apply a solution is the very root of the problem that the neurotic mind seeks to eradicate  the attempt to fix the problem is the problem, not the problem itself. [emphasis added]
Notably with respect to the challenges of sustainability, at every level of society, the above argument raises the question as to whether the Mset offers a means of addressing the valuecharged strategies that are currently so divisive in public debate. The nature of these strategic dilemmas was documented in relation to the concerns of the 1992 Earth Summit (Configuring Strategic Dilemmas in Intersectoral Dialogue, 1992). The possibility of configuring such dilemmas was explored, using the strength of opposition between polarized perspectives as a design feature, as a means of framing a new kind of space (Configuring Conceptual Polarities in Questing: metaphoric pointers to selfreflexive coherence, 2004). The work of Stafford Beer reflects this possibility, notably his articulation of the practice of adaptive "problem jostling", or "problem jostle", for team syntegrity. In this light, there is a case for exploring the relation between the complex dynamics of a spherical tensegrity structure and its fractal organization in seeking equilibrium (cf Donald E. Ingber. Tensegrity II. How structural networks influence cellular information processing networks, 2003).
From the perspective of management cybernetics, a key principle is that of the requisite complexity necessary for the management of a complex global society. This principle is known as Ashby's Law. As the most complex mathematical object known, and given the understanding of mathematics as the science of relationships, the Mset could therefore be understood as the most complex relational object that could prove to be a suitable candidate in that respect. There is otherwise the danger that "sustainability" will be sought  and purportedly found  at a level of abstraction at which it cannot be sustained.
It should not be forgotten that the principal managementrelated arena to which the fractal perspectives of chaos theory so far have been applied is that of the financial markets (see resources at Orlin Grabbe. Chaos and Fractals in Financial Markets, 2003; Fractal Finance; Chaos, Fractal Theory in the Financial Markets). Fractal equations are wellsuited to the wildly random world of financial trading, where price fluctuations have been resistant to traditional mathematical models. This is also an application which Benoit Mandelbrot has himself explored (The (mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward by Benoit B. Mandelbrot and Richard L. Hudson, 2004).
There appears to be no reference to the relevance of the Mset to strategy in other strategic domains. With respect to the concept of sustainability, again research seems to have had the same preoccupation (Benoit B. Mandelbrot. Fractal financial fluctuations; do they threaten sustainability?). Joanne Tippett (A Pattern Language of Sustainability Ecological design and Permaculture, 1994) refers to Mandelbrot's work but does not elaborate its relevance.
But, with respect to city planning, for example, Verna Nel (Complex Adaptive Systems and City Planning) argues that:
The Mset is however considered to be of much greater strategic relevance by Moshe S. Landsman (Toward a Fractal Metaphor for Liberation of Palestinian Women, 2001) in a discussion of stages of liberation where he considers that the fractal metaphor plays a significant part in both understanding the process and in planning intervention strategies. He considers that the fractal characteristics of Mset have at least a metaphorical potential for enhancing conceptualization of multilevel social processes. Among them are the following:Globalisation implies that countries and cities are tangled in webs of connectivity. Simple reductionist explanations are no longer adequate. A new paradigm, appropriate for the new order is required. This paper argues that the metaphors of complex adaptive systems can meet the demands of city planning both in a global context and in South Africa today.
1. The level of intervention is not as important as the strategy. Since structural and functional changes will eventually interact and cause changes on other levels of the process, the interventor.. may begin at the level that is simply the most convenient...
2. As hinted above, the strategy of intervention, which enhances multilevel change, is one that fosters awareness of parallel processes above and below the level of intervention and attempts to address them as well. If they cannot be addressed at this particular point in the process, they should at least be monitored, as changes at one level can teach us lessons at another.
3. Not only are changes at one level portentous for another, but any phenomenon occurring at one level of intervention gives up potentially valuable information for understanding another. Therefore, when the interventor reaches an obstruction at the present level of intervention, observation of processes at other levels may often uncover ways of dealing with the obstruction.
4. The fact that fractals are borderline phenomena may foster conceptualization of strategies for social change. Nonfractal changes in the terrain may be an indication that we have achieved meaningful social change. On the other hand, meeting the same fractal terrain may tell us we are not out of the woods, and may be going around in circles.
This suggests that the Mset may have wider implications for social change in situations fraught with strategic dilemmas. How to understand sustainability under such dynamic conditions may be intimately related to the challenge of understanding the Mset. For example, if the multiplicity of conceptual models (through which the dynamics of change are envisaged) were to be understood as Jsets, and represented by them, what significance would be associated with the corresponding Mset  and how would it be understood?
As noted by Julie Klein (above) with respect to interdisciplinary approaches to knowledge generation of relevance to complexity, incoherent advocacy of distinct models, as currently practiced, fragments strategic initiatives and isolates their proponents (cf Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004). There is clearly a case for reviewing such disjunction in terms of the language of attractors and repellors  and event horizons.
The difficulty in seeking to apply strategies based on such models is that few of them take account of the existence of other models  understood by their proponents as reflecting more adequately the priorities of alternative strategies. The existence of competing models engenders a dynamic in the dialogue relating to governance, especially at the global level. This dynamic is seldom based on rational discourse. In fact it is typically characterized by irrational argument and can usefully be described as constituting a nonlinear dynamic system.
The question is whether the Mset offers pointers to mapping the dynamic between alternative strategies. Given the structure of the Mset, this might then both distinguish and interrelate strategies that could be described (after an "iterative" succession of budgetary cycles) as characteristically:
The argument above suggests that it is precisely the dynamic between the "real" and the "imaginary" dimensions of such strategies that engenders the dynamic stability mapped by the Mset. The question is whether these dimensions satisfactorily hold the "reality" of radically opposed proponents in the light of the "imaginary" characteristics that they attribute to those whom they oppose (or by whom they are opposed).
Such concerns have been fundamental to initiatives to profile the many thousands of problems, strategies and values of international constituencies (Simulating a Global Brain: using networks of international organizations, world problems, strategies, and values, 2001). In particular such profiles endeavoured to capture understandings of the problem (or the strategy) as "real"  in contrast with "imaginary" or in some way misleading. It is such differences in perspective that are the fundamental drivers of global dialogue.
In mapping dynamics onto a complex plane, the Mset suggests the value of mapping the dynamics between the complete range of human activities onto such a surface. This contrasts with current approaches  even when based on mapping those activities onto what amounts to a generalization of the periodic table (Functional Classification in an Integrative Matrix of Human Preoccupations, 1982; Alternation between Development Modes: reinforcing dynamic conception through functional classification of international organizations and concerns, 1982). This was partially inspired by Edward Haskell's original work (discussed earlier).
Seismologists, meteorologists, economists, chemists, hydrologists, and every kind of engineer, have been confronted with visual patterns that were more elegant than predictable. There is a trap to the graphical representation of the Mset and enthusiastic explorations of the strange imagery. This is partly indicated by the muchcited statement of Alfred Korzybski (Science and Sanity  an introduction to nonaristotelean systems and general semantics, 1933): The map is not the territory [more]. In distinguishing the Mset from Jsets, this phrase is especially apt.
The Mset is above all significant to individual and collective navigation of a complex reality  to the extent that it can be embodied in a meaningful way (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002). How such "embodiment" is achieved is a preoccupation of many of the symbol systems and disciplines referenced above. It also reflects the concerns explored by enactivism (notably that of Francisco Varela et al. The Embodied Mind: Cognitive Science and Human Experience, 1991).
At its simplest, "looking at" the Mset representation sets up a knowerknown polarity without seeking to reframe the associated dynamics. The point is well made in the Chinese tradition by the Ba Gua Mirror (see above). This uses the 8 complementary trigrams to frame a mirror in which the observer is confronted with the real challenge to understanding  the Delphic "know thyself".
"Looking at" should be challenged by "sensing the geometry" as discussed earlier. One approach to this is through shifting to a 3D representation of the Mset (cf Ralph Abraham. Complex Quadric Dynamics: A Study of the Mandelbrot and Julia Sets, 2001). Another approach is by understanding the use of the facilities of Mset fractal browsers as cognitive operations in their own right (cf (Computer Use as Philosophy in Operation: Metaphors of the Inner Game, 2003).
It is also useful to recognize the extent to which strategic thinking is trapped into the linearity of textual explanations and verbal discourse  even though it may be endeavouring to encompass nonlinear dynamic phenomena. One response is to animate representations of institutions and their programmes (cf Animating the Representation of Europe, 2004). A more integrative approach is however called for  justifying exploration of the potentials of the Mset.
A quite different approach is through sonification of the Mset, notably the production of "Mandelbrot music". For example, David Spondike (Hearing the Mandelbrot Set) presents an experiment in sonifying the Mset noting:
There are many ways in which the Mset can be mapped to sound. These examples demonstrate but one. The purpose here is twofold. 1. To demonstrate the feasibility of sonifying (as opposed to visualizing) scientific data, and 2. To demonstrate the possibility of finding musical structures in the Mset. To compose music implies creative manipulation. This "composition" was kept to a minimum in these examples, perhaps lessening the musical qualities of the examples, but in an effort to preserve objectivity toward the scientific data.
There are a number of other experiments in giving musical form to the Mset [more  more]. Related arguments are presented with regard to giving musical form to traditional Chinese conceptual coding systems (Musical Articulation of Pattern of Tao Te Ching Insights: Experimental sonification based on magic square organization, 2003).
The case for sonification of scientific data has been articulated by the US National Science Foundation and the International Community for Auditory Display (Sonification Report: Status of the Field and Research Agenda, 1997). For example, Davide Rocchesso (Audio effects to enhance spatial information displays, 2002) has given consideration to use of sonification in representing the complex plane. There are many experiments in relation to "fractal music" [cf Fractal Music Lab].
It is worth reflecting on the role of overtone chanting, notably in Tibetan Buddhism, as a means of articulating higher forms of order [more]. There are extensive web references by those concerned with the possibility of extraterrestrial intelligence (notably in the SETI community) to the role of Mset representations  especially in the light of crop circles purportedly of that form [more  more  more]. Clearly the ability of a civilization to recognize the full significance of the Mset could be considered an "entry qualification" for such communication (see Communicating with Aliens: the Psychological Dimension of Dialogue, 2000).
As noted in an earlier paper (Enhancing the Quality of Knowing through Integration of EastWest metaphors, 2000), the embodiment of knowledge, in the light of the chanted hymns of the Rg Veda, has been explored by Antonio de Nicolas (Meditations through the Rg Veda, 1978), using the nonBoolean logic of quantum mechanics (Heelan, 1974). The unique feature of the approach is that it is grounded in tone and the shifting relationships between tone; it is through the pattern of musical tones that the significance of the Rg Veda is to be found. As de Nicolas indicates:
There is a case for exploring how this classical perspective is to be combined with that of enactivism in the light of possible articulation of experiential dynamics through the Mset."Therefore, from a linguistic and cultural perspective, we have to be aware that we are dealing with a language where tonal and arithmetical relations establish the epistemological invariances... Language grounded in music is grounded thereby on context dependency; any tone can have any possible relation to other tones, and the shift from one tone to another, which alone makes melody possible, is a shift in perspective which the singer himself embodies. Any perspective (tone) must be "sacrificed" for a new one to come into being; the song is a radical activity which requires innovation while maintaining continuity, and the "world" is the creation of the singer, who shares its dimensions with the song." ( p. 57)
Ralph H. Abraham:
Marcia Birken. Analogy, Mathematics, and Poetry: Bibliography, 2003 [text]
David Bohm. The Undivided Universe: an ontological interpretation of quantum theory. Routledge, 1993
David Bohm. Wholeness and the Implicate Order. Routledge, 1980
Kenneth Boulding. Ecodynamics; a new theory of social evolution. Sage, 1978
Antonio de Nicolas. Meditations through the Rg Veda. Shambhala, 1978
Evgeny Demidov. The Mandelbrot and Julia sets: Anatomy [text].
Robert L. Devaney. An Introduction to Chaotic Dynamical Systems. Westview, 2003.
Jonathan J. Dickau. The Mandelbrot Set and Cosmology. 1997 [text]
Vladimir Dimitrov:
Stephen L. Field. Recovering the Lost Meaning of the Yijing Ba Gua. 1999 [text]
Christine Fielder and Chris C. King. The Sensitivity of Chaos (in Sexual Paradox: Complementarity, Reproductive Conflict and Human Emergence), 2004 [text]
Kathleen Forsythe:
Edward Haskell. Generalization of the structure of Mendeleev's periodic table. In: E. Haskell (Ed.), Full Circle: The Moral Force of Unified Science. New York, Gordon and Breach, 1972 [text]
Patrick A. Heelan. The Logic of Changing Classificatory Frameworks. In: J A Wojciechowski (Ed). Conceptual Basis of the Classification of Knowledge. K G Saur, 1974, pp. 260274
Cameron L Jones. The Fractal Dimensions of Storytelling and New Media, 2001 [text]
Anthony Judge:
Virpi Kauko. Visible and Nonexistent Trees of Mandelbrot Sets, 2002 [text]
Linda Keen. Complex and Real Dynamics for the Family. λ tan(z), 2001 [text]
Linda Keen and Janina Kotus. Dynamics of the Family. λ tan(z), 1997 [transference of properties from the dynamic plane to the parameter plane] [text]
Karsten Keller. Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set. Lecture Notes in Mathematics #1732. SpringerVerlag, 2000.
Chris C. King. Fractal and Chaotic Dynamics in Nervous Systems. Progress in Neurobiology 36, 1991, pp. 279308 [text]
Orrin Klapp. Opening and closing: Strategies of information adaptation in society. Cambridge University Press, 1978
Julie Thompson Klein. Interdisciplinarity and complexity: An evolving relationship. E:CO, 6, 2, Fall 2004, pp. 210 (Special Double Issue) [text]
George Lakoff. Women, Fire, and Dangerous Things: What Categories Reveal About the Mind. University of Chicago Press, 1987
Moshe S. Landsman. Toward a Fractal Metaphor for Liberation of Palestinian Women. Radical Psychology, Spring 2001, Vol. 2, Issue 1. [more]
Mikhail Lyubich. The Quadratic Family as a Qualitatively Solvable Model of Chaos. 2000 [text]
Bill Manaris, Charles McCormick, and Tarsem Purewal. Can Beautiful Music be Recognized by Computers? Nature, Music, and the ZipfMandelbrot Law, 2002 [text]
Benoit B. Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman, 1983
Benoit B. Mandelbrot and Richard L. Hudson. The Misbehavior of Markets: A Fractal View of Risk, Ruin and Reward. Perseus Publishing, 2004
Carole McKenzie and Kim James. Aesthetics as an aid to understanding complex systems and decision judgement in operating complex systems. E:CO Special Double Issue Vol.6 Nos.12 Fall 2004 pp.3239 [text]
Harmen Mesker. The Eight Houses: a preliminary survey. 2002 [text]
Iona Miller. Chaos as the Universal Solvent: ReCreational Ego Death in Tantric
Consciousness. Chaosophy, 1993, Ch. 3
John Milnor. Periodic Orbits, External Rays and the Mandelbrot Set: An Expository
Account. (postscript,
mirror of StonyBrook preprint). [text]
John Milnor. Pasting Together Julia Sets: A Worked Out Example of Mating, 2000 [text]
Robert Munafo. MuEncy  Mandelbrot Set Glossary and Encyclopedia, 19872004 t [text]
J K Norem and E C Chang. The positive psychology of negative thinking. Journal of Clinical Psychology, 2002, 58, 9931001
Robert L. Oldershaw. MSet as Metaphor, 2001 [text]
Kent D Palmer. Steps to the Threshold of the Social: the mathematical analogies to dissipative, autopoietic, and reflexive systems. Apeiron Press, 1997 [text]
F David Peat. The Philosopher's Stone: Chaos, Synchronicity and the Hidden Order of the World. Bantam Books, 1991.
Stephen M Phillips:
M. Romera. Atlas of the Mandelbrot set, 1999 [text]
Swinton Roof. Attraction to Shape, 2002 [text]
Xavier Sallantin. L'épistemologie de l'arithmetique. (Communication aux Seminaires internationaux d'épistemologie de l'Abbaye de Senanque, Sept. 1976). Paris: Laboratoire Bena de Logique Generale 1976. (A longer version of this paper appeared as Pt. 3 of L'Epreuve de la Force. Cahiers de la Fondation pour les Etudes de Defense Nationale. No. 2, Octobre 1975)
Dierk Schleicher. Attracting Dynamics of Exponential Maps. 2003 [text]
Michael Schreiber. Fractal Maps of CyberMarkets. (Theories and Metaphors of Cyberspace) [text]
R. G. H. Siu:
Larry Solomon. The Fractal Nature of Music, 2002 [text] (The topic of this essay is the manifestation of fractallike structures in existing music, rather than the creation of new music)
David Spondike. Hearing the Mandelbrot Set [text]
Will C. van den Hoonaard and William W. Hackborn. Chaos as Metaphor for the Study of Social Processes in the Postmodern World: A Bahá'í Illustration, 1994 / 2002 [text]
John R Van Eenwyk. Archetypes and Strange Attractors: the chaotic world of symbols. Inner City Books, 1997
Linas Vepstas:
Lucy Wang and Lea Oksman. Magnificent Fractals: of measures, monsters and Mandelbrot. Yale Scientific Magazine, Spring 2003 [text]
Len Warne. A Meditation on the Mandelbrot Set, via Walt Whitman [text]
Timothy Wilken. UnCommon Science, 2001 [text]
Nicholas Williams. Psychological Enropy, 2003 [text]
T R Young. Chaos theory and symbolic interaction theory: Poetics for the postmodern sociologist. Symbolic Interaction. 1991, 14, 321334.
T R Young. Chaos and social change: Metaphysics of the postmodern. The Social Science Journal 1991, 28, 289305.
For further updates on this site, subscribe here 