19 March 2005 | Draft
Psycho-social Significance of the Mandelbrot Set
a sustainable boundary between chaos and order
- / -
Annex 2 to Sustainability
through the Dynamics of Strategic Dilemmas
in the light of the coherence and visual form of the Mandelbrot set
Potential implications in terms of religious symbolism
Potential mytho-poetic implications
Potential experiential implications in terms of concentration
Potential implications for self-awareness, relationships and psychotherapy
Potential implications: fractal quasi-similarity of patterns
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
- Traditional posture of meditation: The similarity of the form of
the M-set with the representations of the meditating Buddha has been frequently
noted (see above). It is somewhat amusing that the conventional representation
of the M-set, with the "head" on the left, is that of the so-called
"reclining Buddha" -- and only some fractal browsers allow the M-set
representation to be reoriented. Such a shift in orientation does however
point to difficulties of comprehension -- given the conventional attribution
of the x and y axes in mathematics in general, and in representations of the
complex plane in particular.
It is of course the case that the graphical representation of the M-set, even
re-oriented, may vary according to preferences for colouring. This is a useful
reminder with regard to the distracting nature of representation. It is however
worth noting the striking resemblance of some representations of the M-set
with some representations of buddhic deities (notably Tibetan) with a flame-like
- Configurations of numerous Buddhas: One of the remarkable features
of the M-set is its fractal nature. As a consequence of a degree of self-similarity,
within the M-set there are to be found infinitely numerous replicas of the
buddha-like shape of its representation as a totality. However Tibetan Buddhist
mandalas may show a central Buddha-figure, surrounded by smaller secondary
Buddhas -- each of which may in turn be surrounded by even smaller Buddhas.
In this respect it is worth recalling that in Buddhism the Buddhas are traditionally
recognized to be "as numerous as the sands of the Ganges". They
would appear to be ordered in a manner reminiscent of the organization within
the M-set, as illustrated by the following extract from a Vajraguru prayer
on The Resolve to Practice Excellence:
The force of my resolve to practice excellence brings all victors
clearly to mind; emanations of my body, numerous as the atoms of all universes,
bow down in perfect obeisance before them. All Buddhas, numerous as the
atoms of all Buddha realms, stand in a single atom, surrounded by their
children. Similarly, this host of victors stands in every single atom throughout
the realm of totality. To them I direct my devotion and faith. [more]
This suggests interesting explorations of the number of Buddhas in relation
to the number of "iterations" required to resolve their isomorphs
as details within the M-set -- especially in the light of understandings of
the necessarily lengthy cycles of "reincarnations".
- "Fantastic" imagery: The jewel-like quality of some of
the "fantastic" detail of the M-set is also echoed in Buddhist texts
such as The
Lotus Sutra: The Mystic Powers of the Tathagata:
The living beings in their midst, the heavenly beings, dragons,
yakshas, gandharvas, asuras, garudas, kimnaras, mahoragas, human
and nonhuman beings, thanks to the Buddha's supernatural powers, all saw
in this saha world immeasurable, boundless hundreds, thousands, ten
thousands, millions of Buddhas seated on lion seats under the numerous jeweled
trees... Moreover, they saw immeasurable, boundless hundreds, thousands,
ten thousands, millions of bodhisattvas and mahasattvas....
As noted by Stanislaw Lem (The Cyberiad, 1975), and cited in a study
of the M-set:
Everyone knows that dragons don't exist. But while this simplistic
formulation may satisfy the layman, it does not suffice for the scientific
mind....The brilliant Cerebron, attacking the problem analytically, discovered
three distinct kinds of dragon: the mythical, the chimerical, and the purely
hypothetical. They were all, one might say, nonexistent, but each nonexisted
in an entirely different way....Suppose, for example, one organizes a hunt
for such a dragon, surrounds it, closes in, beating the brush. The circle
of sportsmen, their weapons cocked and ready, finds only a burned patch
of earth and an unmistakable smell: the dragon, seeing itself cornered,
has slipped from real to configurational space.
- "Heavens" and "Hells": Both Hinduism and Buddhism
postulate the existence of a multiplicity of hells and heavens -- or lower
and higher worlds -- variously characterized by continents, oceans and other
natural features. These may well be reminiscent of details encountered in
zooming into graphic representations of the M-set, possibly depending again
on the colouring technique used. Although perceived as "ridiculous"
to western rational thought, there is the intriguing possibility that the
identification and naming of these regions (or the myriad deities above) within
some belief systems is as legitimate as a naming procedure for dynamic features
of chaotic space as the naming of celestial objects by astronomers.
- Engendered features: Some religions, notably Hinduism and Buddhism,
stress the degree to which the features of the perceived world, including
"heavens" and "hells", engendered by the mind -- as a
product of the potential of "mindlessness" or "no mind".
The graphic detail of the M-set -- widely noted as variously reminiscent of
rivers, leaves and other natural phenomena -- is also effectively engendered
through the M-set.
- I Ching trigram arrangement (Pa Kua): As noted above
with respect to Taoism, the 8 basic trigrams of the I Ching (or The
Book of Changes) are conventionally displayed in a circular manner (known
as the Ba
Gua Mirror). As discussed below, the trigrams respectively correspond
to dynamics that are collectively fundamental to sustaining coherence -- as
indicated by the understanding of Tao.
With respect to the M-set, these distinct dynamics may be associated with
the features of different period, termed primary "bulbs", that are
attached to the cardioid region of the M-set representation. How this is done
depends on insights into the two classical arrangements of importance to Feng
Shui, the Ho tu or Lo Shu [more
A relationship between the Tao and the Mandelbrot set seems to have been noted
by Katya Walter (A
New (and very Old) Model for Nonlinear Computation, 1995), a Spanish
It has been explored to some degree by Arnold Keyserling (The
I Ching and the Five Stages of Creative Time, 1999):
Self organization can be understood as the capacity to create
information based on the Mandelbrot vector... considered as a continuous
creative process. The key is spontaneous improvisation in the moment --
acting in the Tao -- returning to Zero. This whole field has been thoroughly
explored for millennia by Chinese thinkers and so we will use their terminology.
The underlying principle of self organization, the factor creating a larger
identity, is called chi in Chinese philosophy (ki in Japanese).
(Strangely enough, this Chinese word is the same as the Greek word for the
Pythagorean symbol.) Chi appears in two aspects, wu chi, emptiness,
symbolized by the empty circle, and tai chi, plenitude, symbolized
by the ancient Chinese fractal...
Extending the trigram system to the 64 hexagrams of the I Ching, and
based on Mandelbrot's fractal theory, analogies have been explored between
the traditional Chinese energy system on the one hand, and the 64 units of
the genetic code or any other energy system on the other (see Cornelius
Celsus Foundation, Energy-Based
Deciphering of the Genetic Code). A more detailed comparison in terms
of both Mandelbrot and Julia sets has been made by Tony Smith (3x3
Octonion Matrix Physics Models).It might be argued that if
the focus of the I Ching on the variety of possible changes has any
merit, there would indeed be some kind of mapping onto the M-set and vice
versa (see also
Transformation Metaphors derived experimentally from the Chinese Book of
Changes (I Ching) for sustainable dialogue, vision, conferencing, policy,
network, community and lifestyle, 1997).
- Fractal patterns fundamental to occult representations of reality:
In addition to Buddhism (as noted above), other beleif systems may explicitly
indicate the fractal nature of reality.
- Kabbalah: In this Jewish occult system everything can be classified
in terms of, and reduced to, the ten Sefirot -- fundamental archetypal
principles or essences, qualities of the manifest and knowable Godhead.
Each of the ten Sefirot that together make up creation is itself composed
of ten Sefirot, each of which is in turn composed of ten. Each of the
Sefirot is thus divided fractal-wise into sub-sefirot, and those into
further sub-sub-sefirot, so the same applies with the doctrine of worlds.
- Astrology: In a careful articulation of the relevance of nonlinear
dynamics to understandings of astrology, Michelle Jacobs (Bringing
it Down to Earth: a fractal approach, 1995) argues:
If Chaos theory is so important to modeling the behavior
of complex systems -- that is, the behavior of the natural world, then
perhaps it has a similar role in the complex workings of astrology,
for astrology also reveals a subtle relationship between simplicity
and complexity; it too imitates life. Astrology and Chaos both provide
means of plotting the unfoldment of processes in time and they each
allow one to get a holistic overall view of nature.
But although Jacobs refers to the M-set, her argument is developed in
relation to strange attractors:
In Chaos theory, when we look at the large picture, apparently
random events can be shown to happen within patterns, with the Strange
Attractor representing the overall predictable state. And in astrology
we can easily see the birthchart as a Strange Attractor: It outlines
the pattern of the life and personality, but within that pattern there
are infinite variations on the theme. Truly, each sign or planetary
principle is in itself a strange attractor. Each contains a full potential
of self-similar or fractal expressions, but these expressions never
fall outside the attractor. An astrological planet contains all of its
possible presentations within the overall predictable form of the planetary
- Temple design: Sacred architecture is often premised on the insight
that it will reflect the subtlest of insights as a temple of the spirit [more].
- Cross (Christian): As noted earlier, the cross may be associated
with the axes that enable the graphic representation of the M-set in a complex
plane. The junction of the vertical and horizontal elements then lies at the
origin. It is interesting however how the choice of orientation is not arbitrary.
The seated-Buddha orientation is associated with an inverted cross -- a symbol
associated with satanic rituals. Such dramatic contrasts call for further
reflection (see below) on the content associated with the "real"
and "imaginary" dimensions of the complex plane and how "right",
"left", "above" and "below" are distinguished
-- and especially how values are projected onto them. Of great interest to
such an exploration is the work on arithmetical epistemology of Xavier Sallantin
(1975). His work on the nature of "evil" from that perspective also
merits careful attention (Le
problème du mal à la lumière de la cyberscience, 2001). Also of interest
is the relation of the symbolism of the inverted cross to that of the upward-pointing
sword. It is such challenges that point both to the role of the cross in relation
to the resolution of fundamental paradoxes and to the tensions held by the
cross and embodied in the dynamics encoded by the M-set.
- Sacred heart (Christian, Aztec): The cardioid, as noted
earlier, can be very explicitly associated with the "sacred heart".
Like the cross, it is also inverted in relation to the seated-Buddha orientation.
The cardioid as a complex mathematical object is of the greatest interest
in holding a very wide variety of relationships as noted in an earlier paper
Attractor Fundamental to Sustainability, 2005). As noted earlier,
any exploration of the dynamics of the geometric construction of the cardioid
suggests a rich pattern of associations supportive of insights associated
with any meditation on the "sacred heart". Of particular interest,
for example, is the cardioid evolute (see Figure 2) -- which, by contrast,
is conventionally oriented within the "seated" orientation of the
|Figure 2: Evolute of (inverted)
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
- Body as the temple of the spirit: "Do you not know that you are a
temple of the Spirit, whom you have from God, and that you are not your own?"
(1 Corinthians 6:19) and the need for cleansing it [more].
Emphasis may be placed on each "saved" person functioning as such
a temple. For Christians the body of Jesus was a temple of the Spirit (John
2:19-21). In the Essene Gospels, Jesus indicates: "the body is the temple
of the spirit, and the spirit is the temple of God. Purify, therefore, the
temple, that the Lord of the temple may dwell therein and occupy a place that
is worthy of him". Many religions, and martial arts, teach that the body
is the vessel, the "temple" of the spirit.
- Image of God in man: This is a theme widely explored [more
| more | more]
and especially by Robert C. Newman (Some
Perspectives on the Image of God in Man from Biblical Theology, 1984).
For Christians, for example: The imago Dei, the image of God in man,
is first mentioned in connection with man's creation on the sixth and final
day of creation -- "So God created man in His own image; in the image of God
He created him; male and female He created them." (Genesis 1:27-28)
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:
- Two dragons chasing a pearl: The decorative art of China, including
temples and traditional folk dances, focuses extensively on the theme of two
dragons. They are usually depicted facing one another in the air in eager
pursuit of a spinning pearl floating like an iridescent bubble between them.
This theme was a mark of books issued under imperial auspices. For Taoists,
the complex associations of that pearl include wisdom, yang energy, truth
and life -- even the everlasting life of those who perceive the truth and
attain enlightenment. The pearl can also be thought of as a symbol for universal
Qi energy -- the progenitor of all energy and creation. The two dragons are
associated with an understanding of eternity. The Chinese have a belief that
they themselves are "descendants of the dragon". Given their "protuberances
and jewelled scales", the two dragons can readily be associated with
the two sides of the M-set representation -- in pursuit of the circular 2-period
- Two dragons of Arthurian mythology: These supposedly represent the
two aspects of the British nation: the red one Celtic and the white Saxon.
- Ba Gua (Pa Kua) and associated dragons: The 8 basic trigrams of the
I Ching, in their traditionalcircular disposition (discussed
below), have also been associated with the nine classic dragons of China [more].
Each of these symbols can be understood as guarded by dragons, each with distinct
personalities and powers. As a set they protect the insights represented by
the cardioid region of the M-set.
- "Here be dragons": It is perhaps trivial to recall that
ancient maps had a tendency to mark the outskirts of maps of the known world
with phrases like "here be dragons". It is indeed the case that
what can be known, or "remembered", in any permanent sense might
be said to be represented by the boundary of the M-set.
- "Sword in the stone": This basic Arthurian myth regarding
the sign of true royalty might well be associated with the understanding of
how to distinguish, or grasp, the sword-like axes effectively embedded within
the chaotic stone-like structure of the M-set representation.
- "Withdrawing into the stones": This Irish Celtic myth recalls
the story of an elder race (the Tuatha Dé Danaan) that through their wisdom
were eventually able to "withdraw into the stones". There is now
extensive modern myth-making in relation to "intraterrestrials"
at another "vibratory level". Do some implications of mediational
insight suggest that people can withdraw into the "fabric of reality"
as embodied in the dynamics encoded by the M-set?
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
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
- "Oceanic awareness": This feeling, frequently reported
by investigators of alternative states of consciousness, might well be associated
with the cardioid portion of the M-set -- or quasi-similar sub-variants of
- Mindlessness: This condition (also termed "no mind"), to
which meditators of some spiritual disciplines aspire, might be associated
with the level of abstraction of the cardioid portion of the M-set. As expressed
from a Kriya Yoga perspective, for example, in commenting on the Bhagavad
Understanding of Gita happens in our bone marrow and blood-cells
through repeated chanting and contemplation. Gita invites us to be available
to the holistic consciousness of choiceless awareness, that is, to the 'ignorance'
which knows -- not to the 'knowledge' which is ignorant! Gita is the unique
world view that tolerates and requires holding together multiple positions
simultaneously so that religion holds whole mankind and does not degenerate
in bigotry and battle. The war in Gita is symbolic. It is the war between
wickedness and wisdom, between mindlessness and 'no-mind' with mind as the
bridge. Duryodhan is mindlessness, Arjuna is mind and Krishna is 'no-mind'
i.e. pure intelligence (chaitanya). Gita liberates us from our prior
self-preoccupied identities. Gita sets us free from disintegration to integration
(yoga), from reaction to action (Kriya Yoga), from longing to living
(swadhyay), from paradoxes to pure consciousness (Ishwara Pranidhan),
from Prakriti (inherent traits and tendencies) to Purusha (transcendental
truth of enlightened existence). Gita is the wisdom of sacrifice of the
fruits of action, the distinction of Gunas, the emergence of equanimity,
and the importance of non-doership. [more]
- Alchemy: Traditional representations of meditating practitioners
of Taoist "internal alchemy" (nei-tan) bear a striking resemblance
to the representation of the M-set. As discussed in an earlier paper (Cardioid
Attractor Fundamental to Sustainability, 2005):
It is possible that the representation of the M-set is to be usefully related
to the traditional concept of an alchemical "vessel" in which all
"base matter" can be dissolved -- the container for the "universal
solvent". Alchemy postulates the existence of such a universal solvent
as being capable of transforming base metals into gold and bestowing eternal
youth and therefore immortality on human beings. The universal solvent --
counterpart to the "philosopher's stone" -- is not ordinary water,
but "philosophical" water, the water of life, aqua permanens, aqua
mercurialis. For Robert Grinnell (Alchemy in Modern Woman, 1973)
it is basic to the transformative alchemical process of solutio which
facilitates the fluid, mobile basis of consciousness:
Curiously it is Gregory Bateson in a section on Form, Substance and
of his book Steps to an Ecology of Mind (1972) that relates the
depth psychology work of C G Jung to the thermodynamics of Sadi Carnot.
But it is in another book, translated by Jung's colleague Richard Wilhelm
(1929), that Jung comments on a fundamental cycle identified in a Chinese
text T'ai I Chin Hua Tsung Chih (The Secret of the Golden Flower)
-- more recently translated by Thomas Cleary (1991) -- which showed Jung
how to initiate this alchemy through the breathing cycle [more
| more | more].
This focus has also been compared to the Nestorian Gospel of St Thomas
The much-cited Chinese work discusses the "circulation of the light"
of awareness through various conditions during meditation [diagram]
reminiscent (if only in the metaphors used to describe them) of stages
of the Carnot heat cycle discussed above. The Taoist practitioners of
"internal alchemy" (nei-tan) refer to a continual circulation of
the ch'i (vitality principle) up the primary yang (positive), back
or Governor Channel (tu mai or du mo) and down the primary
yin (negative), front or Functional Channel (jen mai or ren
mo) of the body. (see Lu K'uan Yü, Taoist Yoga, Alchemy and Immortality,
For aqua permanens is a mode of the arcane substance; its
symbol is water or sea-water, an all-pervading essence of anima mundi, the
innermost and secret numinosum in man and the universe, that part of God
which formed the quintessence and real substance of Physis, at once the
highest supercelestial waters of wisdom and the spirit of life pervading
For Iona Miller (Chaos as the Universal Solvent, 1993):
In reducing all to pure water, the prima materia and the
ultima materia become synonymous. That primal consciousness state,
that creative and chaotic consciousness is the beginning of the operation
of "water", and its ultimate realization. It becomes easy to see why the
operation of water is the "root of alchemy." Through consciousness journeys
which liquify our rigid notions of self and world, we re-create the adventures
of the hero or heroine. The theme is the loss and recovery of identity.
- Chakra system: In Hinduism, its spiritual systems of yoga, in some
related eastern cultures, as well as in some segments of the New Age movement,
a chakra (from the Sanskrit word for "wheel, circle") is considered
to be an energy node in the human body [more].
The seven main chakras are described as being aligned in an ascending column
from the base of the spine to the top of the head (see fractal
representations). Each is associated with a certain color, multiple specific
functions, an aspect of consciousness, a classical element, and other distinguishing
Given the hypothesized organizing function of the M-set, it is interesting
to consider where these chakras are each located in relation to the geometry
of its graphical representation (in the "seated-Buddha" orientation):
There is a case for investigating the extent to which the position of these
chakras emerges from more complex mathematical features of the cardioid, its
associated curves, and especially from its generation. These might be understood
as significant associative pathways for those who embody this system.
(Root chakra) : Understood to be at the base of the human spine. On the
M-set representation it is most closely associated with the junction of
the inward-turned portion of the cardioid.
(Sacral / Hara chakra): Understood to be below the human navel. On the
M-set representation it is most closely associated with the axial origin
from which the cardioid is generated. Zazen as the study of the self,
is a particular kind of meditation, unique to Zen Buddhism, that is at
the "very heart" of the practice. Fundamental to that practice
is the centering of attention in the hara as the physical and spiritual
center of the body. In Hinduism, this is recognized as governing the function
of the gonads and the reproductive system. It is associated with the emotional
body, willingness to feel emotions and accept change. As the centre of
gravity of the body, this awareness is also fundamental to some Eastern
(Solar Plexus chakra) : Understood to be at the level of the human solar
plexus. On the M-set representation it is most closely associated with
the intersection of the vertical axis with the horizontal line between
the symmetrically positioned 3-period primary bulbs attached directly
to the cardioid.
(Heart/Lung chakra) : Understood to be at the level of the human heart.
On the M-set representation it is most closely associated with the intersection
of the vertical axis with the horizontal line between the symmetrically
positioned 5-period primary bulbs attached dfirectly to the cardioid.
(Throat chakra) : Understood to be at human throat. On the M-set representation
it is most closely associated with the junction of the cardioid with the
circular period-2 primary bulb above the cardioid.
- Ajña (Third
Eye chakra) : Understood to be between the human eyes. On the M-set representation
it is most closely associated with the centre of the circular period-2
primary bulb above the cardioid.
(Crown chakra) : Understood to be at the crown of the human head. On the
M-set representation it is most closely associated with the crown-like
features about the circular period-2 primary bulb above the cardioid.
- Types of concentration/meditation : In the terms, for example, of
the Yoga Sutras of Patanjali (1.17-1.18):
All objects are in one of four stages: Virtually all types, styles, methods,
or objects of meditation are included in one of these four stages, levels,
or categories (1.17):
Objectless concentration: The four stages (above) all have an object to which
attention is directed (samprajnata). Beyond these four is objectless
concentration (1.18), where all four categories of objects have been released
from attention (asamprajnata).
- Savitarka/Gross: relates to concentration on any gross object
while still accompanied with other activities of the mind, including meditation
on sensory awareness, visualized objects, the gross level of breath, attitudes,
syllables of mantra, or streams of conscious thought.
- Savichara/Subtle: relates to subtle objects, after the gross
have been left behind; the subtleties of matter, energy, senses, and the
mind are, themselves, the objects of meditation, inquiry, and non-attachment.
- Sananda/Bliss: emphasizes the still subtler state of bliss in
meditation. In this state, the concentration is free from the gross and
subtle impressions that were at the previous levels.
- Sasmita/I-ness: focuses on I-ness, which is even subtler, as
it relates to the I that is behind, or witness to all of the other experiences.
- Emptiness of mind: As with many spiritual disciplines (for which
there are extensive web references [more
Buddhism in its various forms (notably Mahayana Buddhism) places much emphasis
on the emptiness of mind and the ways in which that understanding is obscured.
Offering an intriguing association to the M-set cardioid, in The
Heart Sutra the Buddha reportedly states, for example:
...form is emptiness and the very emptiness is form; emptiness
does not differ from form, form does not differ from emptiness; whatever
is form, that is emptiness, whatever is emptiness, that is form, the same
is true of feelings, perceptions, impulses and consciousness.
In another example, the Tibetan Kalu Rinpoche reportedly stated:
You live in illusion and the appearance of things. There is a
reality, and you are that reality, but you don't know it. If you should
ever wake up to that reality you would realize that you are nothing, and
being nothing, you are everything. That is all. [more]
This could be understood in terms of the the human tendency to focus on, and
identify with, the experience of the sensible dynamics which the M-set "embodies"
-- but from a higher order of abstraction with which the individual may alternatively
choose to identify.
It is therefore interesting to note the Buddhist comparison between such understandings
of emptiness and the origin of numbers from emptiness. For Buddhists, emptiness
implies a sense of potential rather than nihilism. In Mathematics,
mind, ontology and the origins of number the recognition by Geshe
Kelsang Gyatso (1995) is noted that there are three levels of dependent relationship:
- Gross dependent relationship - causality - the dependence of phenomena
on their causes.
- Subtle dependent relationship - structure - the dependence of phenomena
on their perceived parts (including aspects, divisions and directions).
- Very subtle dependent relationship - the dependence of phenomena on
imputation by mind.
For Buddhists, these ideas are noted [more]
as being remarkably similar to the theory of the origins of mathematics, as
proposed by the mathematician John
von Neumann (1923) in the light of the theory of sets. He was the originator
of the architecture
used in most non-parallel-processing computers. Von Neumann suggested that
all numbers could be bootstrapped out of the empty set by the operations of
the mind. A set is a collection of things. An empty set is a collection of
nothing at all. An empty set can be thought of as nothing with the potential
to become something (that is to be become a set with at least one member).
Again, the three orders of abstraction might be fruitfully related to the
structure of the M-set.
From a perspective of hermetic philosophy (About
Spiritual Emptiness or the Void):
Emptiness designates a state of mind, an inner level of consciousness
based upon the renunciation of what one believes to be real, beyond all
comprehension or lack of comprehension. Emptiness is, therefore, a higher
level of the mind, an attunement with "nothing," i.e., with Pure Being that
has no reflection at all. This is the revelation that the abstract or higher
mind of a spiritual man receives. He is called "spiritual" because his mind
is open and in harmony with the whole of creation, or the Unknown God. Hence,
such a man is empty of a personal "interpretation" and is in touch with
the world of spirit. As he manifests the world of spirit within him, a new
one opens up before him, giving him a new understanding of life.
Such references point to many possibilities of using the M-set as a means
of holding the distinctions between various fundamental states of consciousness
associated with "spiritual" insight. For example, given the various
distinct modes of long term behaviour characteristic of the dynamical systems
mapped by the M-set:
- To what extent is the notion of iterations "escaping to infinity"
to be usefully associated with the typical forms of distracted everyday
thinking -- and in the most extreme form of attention deficit disorder?
In fact are the "number of iterations" before thinking does
escape to be understood as a measure of the degree of concentration?
- To what extent is asymptotic attraction to a single value (an attracting
fixed point) to be associated with "one-pointed" meditative
- To what extent are bounded systems characterized by periodic or cyclic
patterns of a certain frequency, to be considered as associated with forms
of meditation (sustained by repetitive chants) or, otherwise, by thoughts
going obsessively and habitually "round and round"?
- To what extent are chaotically bound systems (neither attracted to a
single value nor to a periodic cycle) to be associated with forms of concentration
that are neither attracted to a particular focus nor to a periodic pattern
of habitual reflection?
The extensive discussions regarding Buddhist insights into emptiness offer
many pointers to such a comparison [more].
Potential implications for self-awareness, relationships
- Self-referential awareness: A discussion within the Jung
Circle points to the implications of fractal generation of the M-set for
understandings of the self-referential nature of the psyche:
We can usefully regard this equation as the symbolic representation
of the dynamics of an archetype (perhaps the Self, since it is capable of
infinite magnification and reduction with no loss of detail?) Significantly,
it is an iterated equation. What does this mean? Well, simply put, its form
is a feedback loop (shades of the alchemical Uroboros, cyclic distillation,
or the self-cycling energy of the unconscious) in which the result of each
calculation (parallel with one's acquired wisdom?) is fed back into the
equation as the initial value of z. [more]
- Identity: The nature of the M-set offers a means of distinguishing
a higher order of identity and invariance from the various dynamics with which
it may readily be confused. It clarifies that identity may indeed be associated
with (a) attractors that are a fixed (and possibly obsessive) focus, (b) periodic
attractors, possibly associated with habitual patterns of behaviour, (c) patterns
of behaviour with a higher (chaotic) degree of order, and (d) dissipative
and unconvergent patterns of behaviour. However the M-set suggests a form
of psychic invariance that, whilst dependent on these dynamics, is indeed
of a higher and subtler order.
How the individual becomes aware of this subtle order is presumably what many
spiritual disciplines struggle to communicate and impart -- perhaps stressing
the significance of particular features or dynamics fundamental to emergence
of awareness of the M-set. This subtler sense of identity is presumably intimately
related to a personal form of "sustainability" -- through which
identity is sustained. It suggests that the "boundary" of the self
is best, and most subtly, defined by the chaotic boundary of the M-set, however
that is to be understood. In terms of spiritual insight, it is also worth
reflecting on the possibility that the array of features of the M-set may
perform functions analoguous to those of an aerial in entering into resonance
with yet subtler insights.
The relation between the J-sets and the M-set is succinctly indicated (see
by the statement that the J-sets lie in the dynamic plane, whereas the M-set
lies in the parameter plane -- in a 4-dimensional space. Adrien Douady, in
a widely quoted comment, notes that "You first plow in the dynamical
plane and then harvest in the parameter plane". In terms of identity
it might be argued that action is necessarily in the "dynamic plane",
but that identity emerges in the "parameter plane". The "mystery"
of identity lies in the transformation of awareness between these two planes
-- from the dynamic multiplicity of the one to the complex singularity of
the other. Any incapacity in this transformation may be closely related to
a lobotomy separating left-brain and right-brain functioning.
With respect to identity, Dick Oliver (Fractal Vision, 1992) notes:
When you begin to think fractal, the world is a very diffrent
place indeed. Each part of that world is defined not by its division from
others, but by its resonance with a greater whole. Separation becomes a
ridiculous idea when one's identity is gained by a unique transformation,
rather than a greedy withdrawal. Just as every part of the body synchronizes
with and permeates all others, my individuality is strengthened rather than
threatened through giving and cooperation.
The questions with respect to individual identity can be extended to those
of collective identity, whether at the family, community, tribal, ethnic,
national or global level. What does "resonance" mean with regard
to "citizenship", for example?
Any temptation to disparage the metaphoric implications of such an approach
to identity should take account of the teasing warning of Kenneth Boulding
(Ecodynamics; a new theory of social evolution, 1978):
Our consciousness of the unity of self in the middle of a vast
complexity of images or material structures is at least a suitable metaphor
for the unity of group, organization, department, discipline or science.
If personification is a metaphor, let us not despise metaphors -- we might
be one ourselves.
- Personal relationships: Given the directly felt significance of attraction
and love, there is a case for exploring the merit of the M-set mapping of
attractors (of different types) as a way of ordering understandings of relationships
between people. Clay Tucker-Ladd (The
Nature of Attraction and Love, 1996) provides a useful overview.
J. C. Sprott (Mathematical
Models of Love and Happiness, 2001) describes a two-dimensional linear
continuous-time dynamical model of the love/hate relationship between two
individuals and extends the dynamics of this simple model through some nonlinear
dynamics, notably modelling love triangles with chaotic solutions.
The question is whether forms of love can be suitably distinguished in
terms of types of attractors. For example, in the distinctions made by Eric
Fromm (The Art of Loving, 1956):
- Motherly love: As an unconditional, all-protective, blissful
love that cannot be controlled by the receiever, is it appropriately associated
with the fixed point attractors of the cardioid region? Its absence is
associated with despair, perhaps of those regions beyond the M-set boundary
that are "escaping to infinity".
It is currently considered as threatened by narcissism and possessiveness.
- Fatherly love: As a conditional love, earned through "good
behaviour", is it usefully associated with the primary period-2 bulb?
- Brotherly love: Are the periodic dynamics of different patterns
of relationships to be usefully associated with the bulbs of varfious
periods? Fromm argues that brotherly love, "which underlies all others",
is currently threatened by the reduction of human beings to commodities.
- Erotic love: Currently debased by its separation from brotherly
love and the absence of tenderness.
- Love of God: Might be understood in terms of the total dynamic
to which the M-set points. This could be consistent currently regression
"to an idolatric concept of God".
- Self-love: Understood to be that form of love without which we
cannot love others -- currently considered to be threatened by selfishness.
As a form of self-reference, this this might usefully be related to the
iterative process through which initial values are fed back.
There is indeed an irony to the possibility that the infinite range of
dynamics of "affairs of the heart" may be, in a specially significant
way, interrelated by the cardioid of the M-set representation. The evolution
of "love of god" from its matriarchal form in nature religions,
through a patriarchal form in monothesitic religions, points to the challenge
of comprehension of the signifiance of the M-set in this context -- as implied
by the via negativa of western mystics, and the nondualistic approaches
of eastern religions.
It is of some interest that the concept of "family values" may
indeed be related to the dynamics circumscribed by the M-set. The chaotic
boundary zone may well correspond to those relationship dynamics of which
it is said that one has to work at "making a relationship work".
To the extent that the challenges of ensuring sustainble global governance
are isomorphic with those of sustaining family values, there is clearly
a danger in deriving simplistic concepts of global governance from simplistic
understandings of the dynamics associated with family values -- as exemplified
by the chaotic boundary of the M-set.
- Depth psychology: From the perspective of depth psychology and the
collective unconscious, the iterative dynamic through which the M-set is generated
can perhaps be usefully associated with the concept of the Eternal
Return and archetypal dynamics. As argued by John R Van Eenwyk (Archetypes
and Strange Attractors: the chaotic world of symbols, 1997):
If 'the backbone of fractals' is 'feedback and the iterator,'
this may be at the heart of what Mircea
Eliade called 'the myth of the eternal return.' Could iteration and
the eternal return be referring to the same thing?
The M-set may be thought of as pointing to the form and nature of the outcome
of the individuation process -- distinguishing precursor conditions that it
might mistakenly be assumed to be, or on whose dynamics it depends. Of particular
interest is the bilateral symmetry in the "seated orientation" ("real"
axis corresponding to the "spinal chord"). This results from the
understanding of "positive" and "negative" in relation
to the "imaginary" axis -- to the right and left. This raises the
possibility of associating the two halves as complementary in psychological
terms, namely the challenge of integrating the shadow, as articulated by C
G Jung [more]
-- and the "peaceful" and "wrathful" deities of Tibetan
Buddhism. The M-set is a way of expressing and holding the tension between
those two halves. Of course, given the existence of "shadow cabinets",
through which opposition parties articulate their views in parliaments, it
points to a more general challenge of integration, including that of the "shadow
The eternal return begins in tensions of opposites (present and future,
actual and potential, sacred and profane), manifests itself in fractal imagery
(transcending categories and demonstrating self-similarity across scale
through recapitulations of the original act of creation), is sensitive to
initial conditions (demonstrated by the wide variety of myths and rituals
of different cultures), and iterates (the eternal return). The oscillatory
dynamics (tensions of opposites) that generate myths and rituals enliven
them as well, by bringing up new possibilities. Thus, creation occurs over
and over again....So, the eternal return is an iterative dynamic: it allows
the present to be fed back into the original equation. While all archetypal
processes generate feedback dynamics, the eternal return is the epitome
of all such aspects of archetypal processes. It is the archetype of archetypal
dynamics, so to speak.
- Four quadrant perspective of Ken Wilber: Considerable atterntion
has been focused on the achievements of Ken Wilber (A
Theory of Everything: An Integral Vision for Business, Politics, Science,
and Spirituality, 2000) in ordering the range of insights into understanding
individual integration and personal development in a social context. He notably
groups a comprehensive list of twelve "schools" of consciousness (ranging
from cognitive science to psychosomatic medicine and bioenergy) as subsets
of four quadrants of existence (An
Integral Theory Of Consciousness). Central to this perspective is
a model based on two axes, giving rise to those four quadrants:
The quadrant model is enriched by concentric circles centred on the origin
and indicative of successive stages of evolution or development. Thus in each
quadrant there are ten or more developmental levels, such as from atoms to
brains and from prehension to vision-logic. Complementing the attention that
the quadrant model has attracted, a variety of reservations and criticisms
have been expressed [more |
| more | more]
- Individual -- Collective:
- Interior view (subjective) -- Exterior view (objective)
- Subjective: On the Left are positioned the "interior",
unobservable things -- what individuals/groups think (and feel).
- Objective: On the Right are positioned the "exterior",
observable phenomena -- what an individual/society does (associated
with nature and science).
- Quadrants: The interaction of these two axes gives four personality
types, to which codes are commonly attributed:.
- Upper Left (UL): Intentional (individual subjective), notably associated
with sincerity, integrity and trustworthiness (and the arts, beauty
and the self).
- Upper Right (UR): Behavioural / Neurological (individual objective),
notably associated with correspondance, representation and propositional
- Lower Right (LR): Socio-economic (collective interobjective), notably
associated with systems theory web, structural-functionalism and social
systems mesh.(and morals)
- Lower Left (LL): Cultural (collective intersubjective), notably
associated with cultural fit, mutual understanding and a sense of
rightness (and the good).
The question here is whether the understanding of Wilber can be meaningfully
associated with the organization of the M-set, as graphically represented.
Once again, the assumption of this exploration is that there should at least
be a geometrical transform between the mappings they respectively constitute
-- especially since the polarities identified by Wilber are fundamental to
the dynamic of society.
In the case of the axes of the M-set, the distinction made (in mapping its
emergence in the complex plane) is between "real" and "imaginary"
-- with a concept of "positive" and "negative" in each
case. It could be argued that "interior" can be understood as "imaginary",
and "exterior" as "real". The problem with this is that
Wilber places them on the same axis, rather than on orthogonal axes. With
respect to "individual" and "collective", these might
variously be understood as associated with "positive" (for those
stressing individualism) or "negative" (for those stressing collectivism
and community) as illustrated in the following table.
||Individualism perceived as selfish
(as typical of Asian cultures, socialism, etc)
||Individualism perceived as the focus
and justification of social developmenti (as typical of western cultures)
||Collectivism as perceived as a constraint
on personal freedom and development (as typical of western cultures)
||Community as perceived as the key to
well-being (as typical of many non-westerncultures)
Again it would appear that the significance of "individual" and
"collective" is confused in relation to contrasting understandings
-- in this case of "positive" and "negative". The Wilber
and M-set mappings can possibly be reconciled by recognizing that Wilber's
axes effectively run through the quadrants created by the M-set axes. In other
words, rotating Wilber's axes by 45 degree offers a means of exploring their
significance in relation to the M-set.
But this possible "fix" in fact points to a more fundamental issue
that is of significance to further exploration of both mappings -- as with
the "positive" and "negative" connotations of the orientation
of the Christian cross in relation to the axes in the "seated-orientation"
of the M-set representation (see above), there is a dynamic to their valuation
that calls for further attention. This dynamic is associated with the a
priori issues explored by mathematical epistemologist Xavier Sallantin
(Xavier Sallantin. L'épistemologie de l'arithmetique, 1976)
and by the direction in which an I Ching hexagram is "read"
when presented with other hexagrams in any circular "arrangement"
(eg out=top or in=top). The point to be explored is whether there are one
or more cognitive steps between recognizing the dimensions of the complex
plane in terms real (positive and negative) and imaginary (positive and negative)
and the cognitive significance subsequently attached to any fundamental dimensioning
of complexity in terms of individal-collective and exterior-interior.
One commentator, Andrew P Smth (Wilber's
Eight-Fold Way How Many Sides Does a Holon Have?, 2004), points to
some of the difficulties noted in Wilber's work:
In his most recent work, the four aspects have undergone another
round of breeding. In the new generation of holons, every one of the original
four aspects has an inside and an outside.... One might have thought that
the terms interior and exterior would adequately convey these notions, but
it turns out, according to Wilber, that interiors have both insides and
outsides, as do exteriors....
From the perspective of his own model, Peter Collins (An
Integral Mathematical Stage Model of Perspectives, 2003), notes in
a contribution to the Ken
Wilber Forum regarding the M-set (Fractals
2 - Recursion and Complexity):
Several writers have noticed that Wilber's four-quadrant model conflates
two different meanings of the individual vs. social dynamic. One meaning
used by Ken is the distinction between individual and social holons. The
other meaning is the individual (or agentic) vs. social (or communal) aspect
of any holon, individual or social
This mistranslation of the true relationship between whole and
parts (and parts and whole), I would consider the biggest problem in conventional
science. Putting it more formally it represents gross mistranslation of
complementary opposites (in vertical terms). Even Ken Wilber consistently
mistranslates this relationship. For Ken development is synonymous with
transcendence (i.e. the holarchical view of evolution where "lower" level
parts are transcended and included in "higher" level wholes. However development
is equally synonymous with immanence (i.e. the partarchical view where "higher"
level wholes are made immanent and included in "lower" level parts). However
this complementary view of development is largely missing from Ken's writings.
In a unique and careful exploration of the psychological significance of the
M-set, Collins focuses his discussion on the challenge of shifting from the
(conventional) quantitative appreciation of it to a holistic one. He usefully
Psychological reality is indeed based on constant recursive procedures.
One of the most important manifestations of this is the manner by which
perceptions and concepts continually interact in experience. Every phenomenon
has both perceptual and conceptual aspects. Indeed this applies intimately
to such mathematical entities as numbers....
Collins concludes with the vital point that: "all the important features
of the Mandelbrot Set have psychological qualitative equivalents".
Vision-logic for example exemplifies this at the rational linear level...
Here a dynamic iteration procedure continually takes place. We start with
concrete phenomena. These are then transformed in conceptual terms leading
to new starting phenomena in experience. And the process continues in this
manner leading to an ever-changing interpretation of reality. So a psychological
feedback mechanism is inherent in the very manner we experience phenomena....
Thus we now see that the psychological feedback process involves a continual
movement as between (horizontal) perceptions and (vertical) concepts. The
starting values are one-dimensional. In other words the perceptions are
understood in linear rational terms. However the transformed values are
two-dimensional. In other words the decisive qualitative shift from concrete
(particular) to formal (holistic) understanding implicitly involves intuition
(which combines polar opposites in experience and is two-dimensional).
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:
- Spiritual insight: Given the quasi-similarity, it becomes evident
how difficult it may be to distinguish between a spiritual experience with
a "Buddha" -- that is effectively a sub-sub-sub variant of the whole
-- and one that corresponds to an experience of the whole. The challenge lies,
in the graphical metaphor, in the ability to "zoom out" beyond the
limited experience to be able to view, encompass, or identify the whole. Without
that, the experiencer may be "trapped" within the frame to which
"browsing" has led -- perhaps righteously to be contrasted with
even more partial experiences, if "zooming in" is possible.
Clearly communication with others, who have encountered a quasi-similar experience
elsewhere in the "spiritual fractal", may be somewhat meaningful
-- but fraught with potential misunderstanding. Dogmatic statements regarding
the identical nature of such experiential patterns may then be righteously
- Personal relationships: Again a pattern associated with "being
in love" may in fact reflect a sub-sub-sub portion of the pattern as
a whole. As such it may nevertheless be compared somewhat meaningfully with
the experience of others (as in romantic encounters and fiction). But it may
again be difficult to distinguish between levels of experience -- each being
able to make inappropriate assumptions about the nature of the experience
of the other.