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Joy in the Present
      

19 March 2005 | Draft

Sustainability through the Dynamics of Strategic Dilemmas

in the light of the coherence and visual form of the Mandelbrot set

- / -

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 M-set
Potential implications: orders of abstraction and "explanation"
Meshing mathematical and experiential understanding
Psycho-social significance of the M-set (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 audio-visual techniques
References

Annex 4: Features of Mandelbrot and Julia Sets (not completed)

Introduction

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 M-set). 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 M-set, 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 M-set 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 well-trained in mathematics as well as in a social or human science.

Relationships between "incommensurables"

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 & 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 M-set, 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 -- so-called 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 self-similarity 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 zn+1= zn2 + c. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set (J-set) is connected and not computable. It is the particular instance that is discussed in what follows (and referred to as the M-set). [more]

Although the M-set 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 M-set 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 non-trivial example of a holomorphic parameter space. Given the significance of pi in defining a circle as a simpler object, the generation of the M-set by iteration may be compared to the iterative calculation of pi (cf Alex Lopez-Ortiz. 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 M-set, is: zn+1= zn + sin(zn), especially if initialized to zn = 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 'multi-dimensional' mental space of an individual or a 'swarm' of individuals - a non-material ("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 meaning-exploration 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.

Thesis

Dissipative systems, and the M-set, offer a language through which to explore and identify viable patterns of sustainable relationship between essentially incompatible modes of behaviour or anti-thetical modes of thinking. It is these which are typically fundamental to the strategic dilemmas in pyscho-social 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:

  • stable (perhaps exemplified by the "constancy of the heart"),
  • unstable (exemplified by nonviable projects of every kind),
  • a form of periodic stability (exemplified by cyclic patterns of interaction),
  • chaotic variety (as with the many weird and wonderful, "improbable" relations between real people).

This approach offers a pattern language to explore the complexities of the periodic resolution to strategic dilemmas -- the space of not-this, not-that (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 M-set) as characteristic patterns of great variety. The set of all such patterns (the M-set 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 psycho-social 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 non-dynamic 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, "M-set" might more usefully be understood as the "Memorable set" or the "Mnemonic set". But the challenge to comprehension -- through "iterative re-membering" 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 3-dimensional 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

Methodological approach

The following points endeavour to provide a rationale for the approach taken:

  • Everything in the technical description that follows is the coherent expression of one "thing"
  • As a fundamental description of dynamic relationships, it is in a significant sense already "known" to the reader
  • As such it is characteristic in some way of human living and being
  • This is despite the curious mathematical technicalities through which it is described and which, as an artificial language, may be difficult to comprehend
  • To the extent that it is in some way already intuitively known and recognized by the reader, it may resonate with archetypal symbols and patterns of quasi-similar form from different cultures
  • Such resonance can be usefully distinguished from popular enthusiasms for "fractal thinking"
  • There is a case for exploring the more rigorous mathematical descriptions to determine whether particular features and properties support valuable insights relevant to challenging psycho-social issues and dilemmas

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.

Challenging aspects of this exploration

This exploration offers an intriguing challenge in attempting to render comprehensible some rather subtle insights. For mathematicians the M-set 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 psycho-social 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 right-brain exercise. The mathematics may offer a left-brain 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 right-brain 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 "non-mainstream 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.

Dissipative systems and their illusory continuity

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 587-588):

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 M-set. 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).

Structure of the visual representation of the Mandelbrot set

In order to offer a framework for any more detailed discussion of some of the technicalities of how the M-set 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 M-set, Figure 1 is rotated 90o 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 180o 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 Mu-Ency - 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.

Interpreting features of the M-set

For detailed descriptions of features of the M-set, see the web resources in the above table.

The M-set 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 M-set is characterized by filaments or tendrils within which some new cardioids appear, not attached to lower level "circles". [more]

  1. Complex plane / numbers:
    2. The M-set is not represented graphically on a plane in a normal 2D space. The nonlinear dynamics to which it points can only be effectively represented on a complex plane. Mathematically one dimension is then "real" and the other "imaginary". These dimensions are discussed with respect to the axes, to be followed later by their possible psycho-social implications. Complex numbers (instead of "real" ones) are required to define the position of a point on a complex plane. Each point on that plane represents a single complex number of the form x + yi, where y is the distance left or right from centre line (negative when left, positive when right) and x is the distance above or below the centre line (negative when below, positive when above) and i is the root of -1. The widespread interest in the M-set derives from the simplicity of the iterative formula giving rise to such a variegated object. If the formula was only based on "real" numbers it would give rise to an uninteresting picture.

  2. Axes: Conventionally the "real" dimension is represented on the horizontal axis (x-axis), whereas the "imaginary" is represented on the vertical axis (y-axis). It is important to recall Palmer's articulation above regarding the illusory quality of what is being described. The axes are given their significance by arbitrary convention regarding "x" and "y", "vertical" and "horizontal", "positive" and "negative". In the representation used here, the vertical axis (above the horizontal) holds the positive values of real numbers, the negative below; the horizontal axis holds the positive imaginary values on the right, the negative on the left.

    The axes of the complex plane on which the M-set is represented may be usefully compared to the experiential significance of being "crossed". This expression tends to be used to describe the encounter with a mode of behaviour that is inconsistent with the logic which one normally used. It reflects a different mode of organization. This distinction might, for example, be used to describe the relationship between "right" and "left" in politics, or between "mainstream" and "alternative" development strategies -- or even between "female" and "male".

  3. Points: The axes of the M-set permit various complex numbers (having a real and an imaginary component) to be positioned in relation to one another in a systematic manner. Specially named points include:
    • Origin: This is the point at which the axes cross, defined as (x=0; iy= 0) and on the basis of which the M-set is generated. It might be usefully compared with the birth place of person, one's home, the starting point of the dynamics between two irreconcilable positions, etc. It may be related to the centre of gravity of the body (hara), as in martial arts.
    • Complex points: These are the positions that complex numbers taken up in terms of the axes. They effectively mark the condition of a nonlinear dynamic relationship, such as the status of an argument or a relationship relationship
    • Fixed points: This is the point at which transformation ends, despite any further iteration. In conditions of nonlinear dynamics, behaviour in a person's life may be governed by:
      • attracting fixed points: where behaviour tends to be attracted to a single fixed point of focus (family or job), holding the notion of eventually "returning home" or "coming back". In the complex plane, the origin (0) and 1 are considered special fixed points.
      • repelling fixed points: where behaviour is repelled by a single fixed point (a place, a person, a perspective, etc)
      • attracting periodic points: where behaviour tends to alternate between attraction to two or more such fixed points (family and job and sport)
      • repelling periodic points: where behaviour tends to alternate between repulsion by one or more such fixed points (family and job and sport)
      • Lyapunov-stable fixed points:
      • neutral (nonhyperbolic) fixed points: ****
      Infinity is also treated as a fixed point since complex numbers near infinity (far from 0) stay near infinity (far from 0).
    • Critical point: This is the starting point through which the dynamic function is tested to determine whether it results in connected or disconnected sets. It might be understood as the critical point through which the coherence of a discussion or an initiative is tested. It might also be understood in terms of kairos as the dramatic moment at which the future outcome of an interaction is effectively mapped out -- a moment of "destiny".
    • Centre of gravity: The M-set is symmetrical around the real axis, on which the center of gravity is therefore located. Its imaginary coordinate is therefore 0. [more]

  4. Iterative generation: Iterative generation of the M-set happens in the "dynamical plane", with the set of all possible parameters is the "parameter plane". For most functions there are areas of the parameter plane (i.e. certain parameter values) for which the iteration exhibits the same properties as the quadratic function by which the M-set is generated, and those regions of the parameter plane contain shapes that look like the M-set.
    Given the manner in which "iteration" is associated mathematically with fractals, it might be assumed that it is unrelated to natural phenomena. In fact it is exceptionally common in that it simply means using the current state of a system to create the next state, and repeating this process many times. It is a feature of any standard office procedure, of driving a vehicle, of selling to a client, or of courtship behaviour. In each case it is typically impossible to predict with any certainty -- however well-controlled the process -- the outcome of that process.
    • Function of M-set: The simple function used to generate the M-set is as follows: zn+1= zn2 + c. In it, z starts out with both the real and imaginary parts set to zero (namely z = 0 + 0i). Then c is initialized to the complex number representing the point to be calculated -- its real portion is its vertical distance from the centre of the plane, its imaginary portion is its horizontal distance from the centre of the plane. The result of each iteration is fed back into the function.
    • Outcome: The iterative process is characterized by a "dynamic interaction" between z and c. If z is larger than one, when it is squared it jumps outwards -- breaking towards infinity. But if c is located in the opposite direction, then when c is added in the function z is pulled back. On the other hand, if z is smaller than one, squaring it makes it even smaller. In this way c will push z inwards or outwards -- unpredictably towards 2 or falling back to 0.
    • Function of J-set: Exactly the same function is used as for the M-set. Now, however, z is initialized to the current point, and c is initialized to a seed value -- another complex number typically taken from the M-set. Through the iterative process, the fate of all possible seeds for that fixed value of c are considered -- with those those seeds whose orbits do not escape forming the filled Julia set of x2 + c. For each different value of c, an entirely different Julia set is generated. Given that there are an infinite number of values for c, there are an infinite number of J-sets. Each J-set can be zoomed into to any level level of magnification. The filled Julia set is a picture in the dynamical plane, not the parameter plane.
    • Iterations: The maximum number of iterations (N) used in testing points in the generation of the M-set can be selected as desired, for instance 100. Larger N will give sharper detail but take longer. In effect iter