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19 March 2005 | Draft

Psycho-social Significance of the Mandelbrot Set

a sustainable boundary between chaos and order

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Annex 2 to Sustainability through the Dynamics of Strategic Dilemmas
in the light of the coherence and visual form of the Mandelbrot set


Introduction
Potential implications in terms of religious symbolism
Potential mytho-poetic implications
Potential experiential implications in terms of concentration and meditation
Potential implications for self-awareness, relationships and psychotherapy
Potential implications: fractal quasi-similarity of patterns

Introduction

The concern in this Annex to Sustainability through the Dynamics of Strategic Dilemmas is to indicate features associated with the Mandelbrot set (hereafter the M-set) in order to point to their significance in configuring complex experience -- rather than in describing natural phenomena, as is normally the case. The assumption is that the features offer templates for innovative thinking in response to highly divisive strategic and value dilemmas. An assumption is also made that the mind is uniquely capable of undertaking operations that explore the features of complex spaces such as those with which the M-set is associated. Note that contextual arguments and references are provided in the main paper.

Such an assumption is reinforced by the documented abilities of inventor Nikola Tesla (1856-1943). He is known for having envisioned, built and tested complex electromechanical devices in his mind (without blueprints) -- with a degree of success, even to sensing when they were out of balance. For this reason he did not even build prototype models. As he stated in his autobiography: "The moment one constructs a device to carry into practice a crude idea, he finds himself unavoidably engrossed with the details of the apparatus." Furthermore: "As he goes on improving and reconstructing, his force of concentration diminishes and he loses sight of the great underlying principle." Tesla's inventions invariably assembled together without redesign and worked perfectly. (Nikola Tesla: Humanitarian Genius )

There is therefore every possibility that the essential properties of the M-set, or particular features, have been discovered through non-mathematical disciplines -- notably those concerned with the development of meditative concentration techniques. This possibility is highlighted below. The possibility has even be documented in a delightful trap for the unwary (Ray Girvan, The Mandelbrot Monk, 1999).

The concern of this argument might therefore be expressed in terms of responding to the question as to how someone -- with such a degree of insight -- might then choose to communicate those insights more widely. And how would they do this in the absence of the requisite mathematical techniques or corresponding computer graphics abilities?

Would they seek to hang the insights onto: the structure of the human body (the chakra system); the human heart (the "sacred heart" of the Aztecs and the Christians); the cross (of Christianity); a mandala; a container or vessel (as in alchemy); an arrangement of trigrams (the Ba Gua mirror); temple design (sacred architecture); etc? In other words is there some kind of a structure that effectively acts as an "index" to such insights -- whilst in itself pointing to a level of subtle experiential coherence that interrelates them without subsuming any of them? What form might such a "Rosetta stone" of the subtlest experiential insights take?

The question is therefore the degree to which the following features of the M-set can be meaningfully and usefully internalized as a template for valuable new forms of thinking.

Potential implications in terms of religious symbolism

Figure 2: Evolute of (inverted) cardioid [more]
Evolute of (inverted) cardioid

In much religious symbolism there is an effort to "hang" subtler understanding onto the features of the body -- as with the classic image by Leonardo da Vinci of the outstretched body across a circle, as a measure of all things (known as Vitruvian Man). There are numerous references to the subtle isomorphism of God and human:

Potential mytho-poetic implications

Insights distinguishing levels of abstraction, of the same subtle order as the M-set, may have been deliberately (or inadvertently) anchored in cultural artefacts such as myths, legends, leys and the like. Whether they are termed "archetypes" or not, their fundamental nature and importance is nourished by the societies by which they are valued for the order they offer.

Examples of the possible relationship between such cultural artefacts and the form of the M-set include:

In each case, these myths might also be understood as an ability to function at the level of abstraction mapped by the domain of the M-set, again to be understood as a form of stone.

Potential experiential implications in terms of concentration and meditation

It is possible that the M-set usefully points to the challenges of concentration and meditation. Indeed it is possible that it was those with such skills that were instrumental in formulating myths to anchor their insights as an aid to wider understanding.

Potential implications for self-awareness, relationships and psychotherapy

Potential implications: fractal quasi-similarity of patterns

The precise fractal ordering of the M-set, and the quasi-similarity of the patterns interrelated by it, raise interesting questions about the level of abstraction of the patterns perceived in any of the above cases.

This is most evident in the experiential nature of the following two cases:

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