15 February 2005 | Draft
Cardioid Attractor Fundamental to Sustainability
8 transactional games forming the heart of sustainable relationship
- / -
A. Marrying positive and negative
Interrelating positive-negative hybrids
Ordering classes of interpersonal relationship
Corresponding Taoist perspective: Ba Gua mirror
B. Cyclic dynamic perspective
Dynamics of a sustainable cycle
Psycho-social heat cycles
Disrupting the cycle
Coaction cardioid: interrelating the "games"
Mathematical functions of the cardioid
C. Marrying the cycles
Sustainability: interrelating the Carnot cycle with the Cardioid
Correspondence with inner and spiritual cycles
D. Relationship of the heart
Understanding a cardioid pattern of transactional relationships
"Disorders of the heart"
Dangerous consequences of ignoring the cycle
As explored in an associated paper (Being
Positive Avoiding Negativity: Management challenge -- positive vs negative,
2004), exhortations and injunctions to "be positive" are a common
feature of some religious groups, in the development of selling techniques,
in self-help therapies, in work group development, and in living with potentially
fatal illnesses. These are seen as a means of avoiding or defeating negativity
in those different contexts [more].
Here the focus is on highlighting the existence of a set of games, rather than
a single game, that potentially are all aspects of a sustainable cyclic system
that merits further attention.
The paper explores, in the light of a general systems perspective, the possibility
that this sustaining cycle can be understood in terms of the Carnot heat engine
cycle and the Coaction Cardioid cycle of Edward Haskell (1972), further developed
by Timothy Wilken (2002). It also explores whether any such generic cycle, relating
"positive" and "negative" conditions, can also be related
to the conditions identified by the Taoist Ba Gua mirror, and notably as a reflection
of the cycle of processes described in Taoist spiritual disciplines.
Interrelating positive-negative hybrids
It is very likely that it will prove fruitful to distinguish between several
forms of "positive" and "negative" -- to recognize valid
concerns but to avoid collapsing valid distinctions. A helpful pointer is provided
by Edward Haskell (Generalization
of the structure of Mendeleev's periodic table, 1972) in his work on
the coaction cardioid (see below), as introduced by Harold Cassidy. Cassidy
points out that:
In the cybernetic analysis of the more complex and organized systems we recognize
two distinct kinds of factors. There is the work component or components,
which we shall designate X, and the governor, or controller, which we shall
designate Y. Of course, the governor does work too (the strategic work), and
we have simplified the relationships very greatly. There will be cases of
a system made up of sub-systems, one controlling in some respects, not in
others, and so on. Let us stay with the simpler case. Now, the processes that
characterize X may, in the interaction with Y, be accelerated or in some way
enhanced ( + ), or may be unaffected ( O ), or may be decreased ( - ). Similarly,
the processes that Y undergoes. When the possibilities are cross-tabulated,
it becomes evident that there are nine and only nine of these qualitatively
different `coactions.' [glossary]
Haskell applies this insight to a range of systems, notably in the natural
environment (in Figure 1) but also in the social environment. In the case of
the different kinds of relations between animals in an ecosystem, the following
patterns then emerge -- of which 8 of the 9 are non-neutral. Note that there
are variations in the teminology of biological interaction, notably differing
from Haskell's usage [more
The dynamics of each of the 8 relationships might be described as a "game",
however asymmetrical or predictable the outcome (as with the "cat-and-mouse"
game of predation).
| Figure 1: Possible 8-fold Positive-Negative
||Y = "Control component"
Figure1 allows distinctions to be made between a form of "being positive"
such as "symbiosis" in which there is indeed mutual enhancement. This
contrasts with one such as "predation" in which one party (the controlling
one) in a transaction prides itself on achieving a "positive" outcome
at the expense of the other -- namely "feeding off" the other. This
is distinct from the situation of "parasitism" where it is the latter
party (the subordinated one) which successfully feeds off the former. The point
about such a table is its merit in avoiding confusion between different forms
of "being positive", some of which may be quite problematic because
their "negative" aspects are camouflaged by appearing to borrow some
of the desirable qualities of the mutuality of "symbiosis". Parasitism
and predation are not relationships of mutually beneficial mutuality. One party
effectively benefits at the expense of the other.
Haskell introduced the neologisms indicated in the above figure:
- allotrophy (derived from the Greek for "feeding the other"), namely one in which the activity of the system's work is not affected while that of the controller is unaffected
- allopathy (derived from the Greek for "making the other sick"), namely in which the activity of the system's work component is unaffected, but that of its controller is decreased
Similarly what might be considered most problematic is a form of (double)
"negativity", typical of "synnecrosis" in the table, in
which both parties lose energy through the interaction to the point of mutual
destruction. But again this should not be confused with other hybrid conditions
in which one or other may benefit unequally from the interaction. The table
is an indication of the possible range of interactions between positive and
negative as a form of psycho-social cybernetic system.
The table is particularly significant in that, as with environmental systems,
it is not the case that all "parasitism" or "predation"
should be eliminated from the pattern of psycho-social interactions -- however
much there is an expectation that the "Lion will lie down with the Lamb"
in a form of symbiosis. There are in fact conditions under which even synnecrosis
would appear to be appropriate -- as in decay processes necessary as precursors
to regeneration. The real challenge is to ensure an appropriate systemic balance
between the various forms of interaction - and, metaphorically, to avoid."throwing
the baby out with the bathwater".
Ordering classes of interpersonal relationship
Edward Haskell's insights have been very usefully (and extensively) adapted
by Timothy Wilken (The
Relationship Continuum, 2002) to an ordering of the spectrum of personal
relationships: adversity -- neutrality -- synergy. Wilken equates "synergy"
with "positive" and "adversity" with "negative"
therefore pointing to the relevance of his adaptation to the preoccupation of
this paper. Wilken's study reframes Haskell's above ordering in the following
table, where "win" equates with "positive" and "lose"
This raises the question whether a psycho-social system, any more than a biological
one, can be based on expectation of a win-win outcome for all parties under
all circumstances. How would life survive on the planet if there were not both
"winners" and "losers" in the feed chain.
The emphasis on "win-win" is an exemplification of the focus on the
positive (cf Hazel Henderson. Building
a Win-Win World: life beyond global economic warfare, 1996). There are
all sorts of reasons why this is a useful notion, and why there is useful mileage
in it. The Judgement Day of religions might even best be understood as the day
when every profoundly held belief system gets to say "I told you so" -- the
only twist being that we cannot understand how each needs to understand how
they were wrong in order to accommodate the rightness of others. It is at this
level (which in biblical terms "passeth all understanding") that win-win does
indeed hold in reality. It also holds as a useful slogan. But it is not clear
that it is with this notion that we can build a sustainable bridge between the
ideal and the practical levels for the following reasons:
- Ecosystems: Everybody is somebody else's lunch. It would be nice to assume
that we could design a sustainable social ecosystem where this does not hold.
We have not yet been able to demonstrate this.
- Learning: Kenneth Boulding (Ecodynamics: A New Theory of Societal Evolution,
1978) has a statement to the effect that we only learn through losing. From
this perspective a win-win society is a strictly non-learning society.
- Sacrifice: Careful review of social change shows that real change only occurs
through human sacrifice. Indeed it could be shown that most legislation is
passed only after people have died as a consequence of its absence. The careers
of most social innovators are marked by personal sacrifice. It can of course
be argued that they "win" in other ways.
The point is that there are several understandings of "winning". People respond
readily to the "gain without pain" interpretation. There is a danger that Henderson's
book will be used as justification for this perspective. Although there may
indeed be value in this, such appreciation will tend however to obscure other
levels of interpretation which may well be where the real breakthroughs lie.
Corresponding Taoist perspective: Ba Gua diagram
There is a striking similarity between Haskell's cybernetically-inspired presentation
and a widely-known classical Taoist presentation of the 8 basic trigrams of
the I Ching -- ordered into what is termed the Ba Gua (or Pa Kua) diagram.
This diagram of 8 "houses", also known as the Ba Gua Mirror, is the
basic tool for Feng Shui analysis, providing a practitioner with keys concepts
with which to analyze a situation [more
There are a very large number of (often highly ornate) circular representations
of the diagram available as images over the web. The following is a purely schematic
tabular version of that binary coding system. The full and broken lines signify
positive and negative respectively, each trigram (in the cells of the table)
therefore constituting the codification of a particular positive-negative (win-lose)
|Figure 3: Schematic representation
of 8-fold Ba Gua (Pa Kua) Mirror
Of greater relevance to the relationship between positive and negative (as
fundamental to Taoist insight) is the use of this Pa Kua framework as one of
the Chinese internal personal development systems intrinsic to qi
gong breathing exercises, meditation and a particular martial art: Pa
The martial art variant of Pa Kua is known for its evasive footwork, including
the characteristic circle walking and the spiraling, coiling, drilling, twisting,
and spinning movements, combined with powerful palm heel strikes. Pa Kua is
as much a martial arts combat style as it is a martial art taught for its health
benefits. There is every possibility that the dynamic relationship between expressions
of "positive" and "negative" in an interpersonal transaction
can be usefully understood as a martial art. Metaphorically, "evasive footwork"
is not something that is totally foreign to dialogue situations! The suggestion
above that "being positive" might be understood as one form of form
of game, from a transactional analysis perspective, can now be reviewed from
a Chinese perspective where it is both game and martial art.
It may well be that the discipline of Pa Kua facilitates the emergence of skills
in dialogue, or in responding to all complex combinations of positive and negative,
winning and losing.
In the Pa-Kua approach to Feng Shui, the compass is divided into eight directions,
each of which is depicted by a trigram (as above). Each of these directions
has a different significance, depending on the individual. Four of those directions
have a "positive" implication for the person, while the other four
have "negative" implications. The eight directions can be briefly
summarized as follows:
- Sheng Chi (Life Generating): Prosperity and wealth direction.
- Tien Yi (Heavenly Doctor): Health direction.
- Nien Yin (Relationships): Relationships direction.
- Fu Wei (Stability): Stability, tranquility and knowledge direction.
- Ho Hai (Mishaps): Accident or misfortune direction, relating to unforeseen
accidents and losses.
- Wu Gwei (Five Ghosts): Financial losses and theft direction.
- Liu Sha (Six Killings): Direction relevant in warding off malevolence.
- Chueh Ming (Total Loss): Loss and death direction. [more]
There is a vast body of Chinese literature exploring the philosophical and
practical implications of this and related schemas -- and their extension to
reflect finer distinctions in the pattern of psycho-social change through 64
Metaphors derived experimentally from the Chinese Book of Changes (I Ching)
for sustainable dialogue, vision, conferencing, policy, network, community and
lifestyle, 1997). Its merit is that it reframes the simplistic polarization
of "positive" vs "negative", extending it to include a complex
set of hybrid variants that are inadequately recognized by injunctions to "be
positive" or "avoid negativity" (cf Discovering
Richer Patterns of Comprehension to Reframe Polarization, 1998).
Given the theme of this paper, and the rich Taoist perspective, there is a
certain irony to the widely cited point that the Chinese symbol for "crisis"
is a compound of "danger" and "opportunity" -- "negative"
and "positive". This can readily be used to exploit a crisis inappropriately
(cf Victor H. Mair. How
a misunderstanding about Chinese characters has led many astray, 2005).
Perhaps the most dramatic example of such exploitation was the UK government
communications specialist who advocated releasing controversial government policies
on the occasion of the 9/11 crisis -- creatively taking the opportunity to ensure
that the public would be otherwise distracted [more].
Implicitly recognizing the function of different systemic conditions identified
in such tabular presentations, Bob Rusbasan (In
Praise of Negativity, 26 September 1999) argues:
Negativity is not always bad. We have to make decisions about all kinds of
things as we go through life, and that always involves balancing good and
bad factors. Say you are a young parent, deciding where to send your fifteen-year-old
daughter to school. You are considering two schools and have compiled a list
of positive aspects of each. The schools are running dead-even, and you can't
make up your decision. Then someone informs you that one of the two schools
has a big rape problem. Would you ask which one? Or would you stand firm on
your principle that negativity should not be a factor in your decision?
In envisaging ways of transcending positive-negative duality, it is worth noting
that negativity tends also to be associated with "dualism" and adversarial
"polarization", whereas it is then argued that "non-duality"
is associated with being positive.
Many disciplines of spiritual experience (whether Eastern or Western) distinguish
- Via affirmativa: The way of affirmation is an approach to
God through positive assertion about God's attributes. Many theologians claim
that the via affirmativa is inadequate without the via negativa,
because it, can speak only of the attributes of God, never of God's eternal
- Via negativa: This approach to God through negation, "the
negative way" or the "apophatic way", is a commonplace of all mysticism, based
on the insight that no predicates attach to God; no words of description may
be appropriately used. Stripping away any such delusions about God is understood
as a preparation for the truth -- eliminating all that is not God creates
the possibility of penetrating to the heart of the mystery. In Sanskrit the
sense of "not this, not that" is expressed by the phrase Neti
| more | more]
From the Eastern educational perspective of the Ananda
Marga Gurukul Network, Ac Shambhushivananda Avt (Cardinal
Human Values, 2002) argues that core values are primarily of two types:
vidya-related (those which lead to knowledge of divinity), avidya-related
(those which keep people tied to a material perspective, ignorant of divinity):
Our life is a constant effort to maintain a dynamic equilibrium between the
forces of vidya and avidya. Neither can we negate the avidya
which is the basis of our physical existence nor can we undermine the vidya
which propels and inspires us towards the divine stratum. Hence, we need a
new paradigm of values which lead us towards a healthy balance between vidya
and avidya, between centripetal and centrifugal forces, between introvertial
and extrovertial movement.
It might be usefully asked whether both a via affirmativa and a via
negativa are necessary to comprehension and response to any complex and
challenging relationship -- not just to that of "God". What indeed
is the "healthy balance" between the countervailing forces basic to
any form of sustainability?
The poet John Keats (Negative
Capability, 21 December 1817) is renowned for recognition of the essence
of maturity in terms of "negative capability". This is the capacity of "being
in uncertainties, mysteries, doubts, without any irritable reaching after fact
Dynamics of a sustainable cycle
If it is indeed the case, as demonstrated by Haskell in the case of ecosystems
of animal species, that sustainability is achieved by an appropriate mix of
of positive-negative (win-lose) conditions, the question is how this mix "works"
-- and why the most appropriate system is not entirely based on a "win-win"
condition. One insight is that the win-win condition is merely one particular
dynamic, a game, that is sustained within a set of games that are mutually dependent
for the viability of the system as a whole.
of Appropriateness, 1986) it was suggested that the interdependence
of these "game" dynamics was expressed through a cycle linking them
together. Understanding how such cycles of contrasting phases accomplish effective
transformative work in society may be facilitated by a thermodynamic metaphor.
The Carnot reversible cycle of heat and work, basic to the operation of any
heat engine. A heat engine
is a thermodynamic system that can undergo a sequence of transformations which
ultimately return it to its original state. The Carnot
cycle involves four successive and contrasting operations:
- expansion of a gas at a constant "hot" temperature: the
gas causes the piston to do work on the surroundings, driven by absorption
of heat from the high temperature source
- expansion without change in amount of heat: as a result of adequate
insulation, the gas continues to expand, doing work on its surroundings, but
associated with a cooling of the gas to the "cold" temperature
- compression at a constant "cold" temperature: with the
environment doing work on the gas, causing heat to be removed from the gas
into the low temperature source
- compression without change in amount of heat: as a result of adequate
insulation, the environment works on the gas to compress it and causing the
temperature to rise to the "hot" temperature..
The notion of a work cycle is introduced here because it is relatively clear
that a living system cannot exist in a condition of stasis. Living is synonymous
with one or more active work cycles through which energy is moved through feedback
loops to ensure integrity in the moment. The most obvious in mammals may be
the respiratory cycle. This energy may take the form of attention -- even vigilance.
The heat engine is driven by a particular difference between two conditions
-- to produce mechanical work by carrying a working substance through a cyclic
process. In the conventional heat engine this difference is temperature. Heat
is transferred to the sink from the source, and in this process some of the
heat is converted into work.
The suggestion here is that from a general
systems perspective this may be generalized to apply to other forms of difference
and other forms of work -- making it potentially relevant to new insights into
socio-economic cycles necessary for sustainability (notably in the light of
theory regarding the maximum efficiency of a Carnot cycle engine). The perceived
difference between "positive" and "negative" may also drive
such a cycle, possibly at an axiological level.
Psycho-social heat cycles
A variety of heat engines have been constructed. The question to be asked is
whether an analogous variety of "heat engines" could be usefully recognized
in psycho-social systems. For example, might the variety of such cycles correspond
to the variety of metaphorical uses of "heat" currently recognized:
- political heat ("diplomatic heat")
- heated market ("heated competition", "heated rivalry")
- dialogue heat ("heated dialogue", "heated debate", "heated
argument", "heated communication")
Of relevance to such explorations would be Marshall McLuhan (Understanding
Media, 1964), given the distinction he explored between "hot"
and "cool" media (cf Gordon Gow. Thawing
out Media: Hot and Cool, 1995; G E Stearn, McLuhan: Hot and Cool,
1967). A hot medium is one that extends one single sense in high definition.
Hot media are, therefore, low in participation, and cool media are high in participation
-- requiring completion by the audience.
Also of relevance is the work of Orrin Klapp (Opening and closing: Strategies
of information adaptation in society, 1978).
The concept of a work cycle is basic to thermodynamics -- and is exemplified
by the Carnot cycle. Elsewhere (Composing
and Engendering the Future, 2001), it was used to explore whether this
provides insights into a necessary dynamic relationship between past, present
and future in terms of the nature and focus of attention. This would be the
challenge in a cyclic shift between various positive-negative combinations.
Is there a sense in which living embodies some such cycle -- of which the the
heat engine is merely a limited material analogue? The heat engine cycle does
indeed have to relate past, present and future in order to sustain its activity.
The insights of such circulation may also be evident in the psycho-social attraction
of certain pattern dances -- presumably providing some kind of time-binding
resonance transcending past, present and future for participants.
Disrupting the cycle
Any attempt to isolate and prolong unduly the most effective work phase simply
jeopardizes the ability of the engine to continue operating -- as is illustrated
by the value of fallow phases in crop rotation. This may also be true of the
"win-win" condition. It is then quite inappropriate to view the non-work
phases as "inefficient". The operation of a task force (or meeting) of individuals
with distinct functions may also be interpreted as involving a cycle of phases
in which each function enters and leaves the limelight in turn. This is best
illustrated by the results of research by R
Meredith Belbin into the roles required for good teamwork. These have been
labelled as: chairman, company worker, completer-finisher, monitor-evaluator,
plant, resource investigator, shaper and team worker. A preponderance of any
one role type, especially the "most productive", jeopardizes both the appropriateness
of the group's work and its ability to renew itself and continue functioning.
The different levels of attention required in discussing the relationship of
distinct policies to policy cycles may be illustrated by the metaphors of walking
and dancing. In walking the right and left foot are moved forward alternately,
shifting the weight of the body from one to the other. Although in places of
difficulty attention may be focussed on one foot to the exclusion of the other,
the body can be more satisfactorily moved forward by focussing on the process
of walking, namely on the alternation between the two contrasting positions.
In a 2-party political process however, there is a necessary struggle between
the "right" and the "left", with no institutionalized awareness of what is achieved
by the process of alternation between them. There is little recognition of when
it is appropriate to relinquish a policy (increasingly framed as "negative")
in favour of an alternative (increasingly framed as "positive") and
then renew it to fulfil a new role. This may perhaps be more accurately compared
to the preoccupation of a drunkard, or a spastic, with the forward movement
of one leg (temporarily forgetting the need for the other).
Appropriateness of the 1st order may be compared to movement of a foot, whereas
2nd order appropriateness may be compared to the process of walking. Higher
orders of appropriateness may be compared to dancing and to a cycle of dances.
It is the movement between the steps, and the manner in which they are ordered,
which renders the dance meaningful. Focusing attention exclusively on any individual
step prevents the rhythm from emerging and thus obscures the meaning of the
dance. It is the rhythm which guides the self-organization of a dance, based
on the execution of the individual steps, whose importance can in no way be
neglected. The test of the appropriateness of any new mode is whether it embodies
a more "seductive" pattern in the sense of Jacques Attali (Noise:
The Political Economy of Music, 1985). In terms of 2nd order appropriateness
current policy initiatives -- and narrowly focused exhortations to "be
positive" -- may be compared to a drunkard's walk, a monotonous dance or,
more dangerously, a lock-step march.
The past century has provided widespread familiarity with engines, notably
combustion engines in motorbikes and other vehicles. The operation of the piston
cycle has entered collective consciousness in many ways -- as well as the distinction
between 2-stroke engines,
4-stroke engines, irrespective
of the number of cylinders. This suggests a line of inquiry as to whether thinking
itself can be understood as operating in cyles that might be usefully modelled
by such engines for many people. In this sense a basic cycle would alternate
between the extremes of any form of polarized thinking -- with each extreme
providing a turning point. One might be associated with the charge that drives
the cycle. Clearly this might be understood as a cruder pattern than that associated
with multiple cylinders -- if their operation could be integrated to reinforce
a common rotation. Of special interest in this respect are rotary engines (cf
the Wankel rotary engine).
Related to such understanding of an engine is that of gearing whereby rapid
rotation is translated into slower and more powerful rotation that can perform
certain kinds of work. Many forms of thinking might be associated with rapid
cycles. These need to be geared down to speeds that can mesh with operations
in the material world (see Conceptual
Birdcages and Functional Basket-weaving. 1980). This challenge might
be seen in relation to that of gearing down principle to practice.
Coaction cardioid: interrelating the "games"
As noted earlier with regard to Figure 1, Haskell's particular synthesis also
highlighted the interrelationship of the different conditions through a cycle
-- described as a coaction cardioid. This work has been extensively elaborated
by Timothy Wilken (UnCommon
In terms of Haskell's generalization of the periodical table pf checmical elements,
the cycle is the generally heart-shaped path of the radius vector in his Periodic
Coordinate System. This system provides a symbolic representation of the nine
possibilities whenever "parts" relate with other "parts"
to form "wholes" or unities, and whenever choices are made by the
"parts" within the "whole" or unity. As admirably explained
by Wilken (UnCommon
Science, 2002, pp. 141-145):
It is important to be mindful that the minus signs represent loss (of order)
and not negative integers. The plus signs represent gain (of order) and not
positive integers. And, the zeroes represent states of no change (of order),
rather than an integer with no content. Or, in the language of games: Lose,
Win, or Draw....Now if we are to depict what occurs as a result of the relationship
between X and Y, we need an initial reference device.
The initial conditions of X and Y can in each case also be represented by the
area of circles.
Then if we geometrically sum our circles, we get the "Initial co-Action Circle"
whose area represents the initial state of the "union" X and Y as a "single"
system.... It was considered a stroke of genius on Haskell's part to use this
Initial Co-Action Circle as the fourth axis of the Periodic Coordinate System.
This circle represents the state of the union at the beginning of a relationship.
It is the geometric sum of (X) and (Y) at the initiation of their co-Action.
This reference circle is made by sweeping a neutral Co-Action vector, ro,
around the ORIGIN.
How do you represent whether or not a relationship or co-Action has a synergic
or net (+) positive effect (increase in order), an adversary or net (-) negative
effect (decrease in order), or a neutral (0) or no effect at all (no change
in order). You must have a reference, what was the state of the system before
before the co-Action is initiated -- the condition of the individuals before
their relationship begins. This is the role of the third axis -- the (0, 0)
circle. Haskell sometimes called this the "scalar zero circle", sometimes
the Circle of Atropy. Perhaps an even better name might be the Circle of Neutrality.
This circle represents a net neutral relationship between (X) and (Y). But,
regardless what we call it, the area of this zero-zero circle represents the
geometric sum of X and Y's condition at the start of the relationship. This
represents the simple sum of their individual order before their interaction.
The cardioid cycle is then defined in relation (as seen in the figure below)
to a circle of unchanging order (or entropy). The coaction cardioid turns into
the zero-zero or scalar zero circle in the region of predominantly negative
coactions (inturning, in Greek, is entropy) -- toward Alpha (in Teilhard de
Chardin's terms). It turns out of the circle in the region of predominantly
positive coactions (turning out, in Greek, is ectropy) -- toward Omega (again
in Teilhard de Chardin's terms). The interactions, or "games", which
reduce the degree of order (increasing entropy) are then within the circle,
whereas those that increase the degree of order (decreasing entropy) lie outside
the circle. The cardioid describes the "path" between these different
conditions that is effectively associated with the sustainability of the system.
|Figure 4: Coaction cardioid (Haskell
Geometric representation of conditions in 8-fold Figure 1
[see also articulations by Wilken,
With respect to this geometric representation of Haskell's earlier tabular
version (Figure 1), Harold Cassidy notes:
As an example, in labor-management relations there is a profit-sharing arrangement
known as the Scanlon Plan. An essential feature of the Plan is to have a reference
period before it is put into operation, so that one will know whether there
is actually a profit or loss under the Plan and how much it is. The value
or range of a variable, measured at this time, would serve to place the (
O, O ) circle in the upper right half of the manifold, and of net ( - ) within
the reference ( O, O ) in the lower left. This yields a "Coaction Cardioid".
Along the Axis of Atropy bisecting quadrants 2 and 4, the magnitudes of x
and y are equal, but the signs are opposite, so the net coaction is zero.
To the right and above this axis is what the philosopher Braithwaite calls
the "cooperator's surplus". Once more we complete the philosophical
categories by calling attention to the "conflictor's deficit", as
we name it, in the lower left, net ( - ) part of the manifold [more]
Mathematical functions of the cardioid
As noted by Wilken (UnCommon
Science, 2002, pp. 159):
Haskell's Periodic Coordinate System presents syntropic, atropic, and entropic
process on a single model. Synergic co-Actions represent sytropic process.
Neutral co-Actions represent atropic process, and Adversary co-Actions represent
entropic process. To accomplish this Haskell synthesized three geometries
-- elliptic, plane and hyperbolic. He used Riemannian geometry to plot synergic
co-Actions, Euclidean geometry to plot neutral co-Actions, and Lobachevskian
geometry to plot adversary co-Actions.
Mathematically, as a curve, the cardioid
has properties that distinguish it in terms of membership of an exceptional
variety of remarkable curves:
- as special case of the epicycloid
for which the rolling circle and the rolled circle have the same radius.
- as degenerate case of the limacon
of Pascal, its orthoptic is the limaçon (see cardioid
is equal to itself (another cardioid).
- as conchoid of a circle,
- as pericycloid,
- as pedal
of a circle with respect to a fixed point on the circle,
- as (polar) inverse of a parabola,
- as cissoidal of two tangential circles,
- as catacaustic
formed by rays originating at a point on the circumference of a circle and
reflected by the circle,
- its catacaustic (with the cusp as source) is the nephroid
- as catacaustic, as well as a pedal, of the cissoid of Diocles
- its pedal is Cayley's sextic
- sinusoidal spiral
With respect to its generation:
- it is the envelope of the chords of a circle, between points P and Q, which
follow the circle in the same direction, where one point has the double speed
of the other. This construction is called the generation of Cremona.
- given a circle C through the origin, the cardioid is the envelope of the
circles with as diameter the line through the origin and a point on C.
(see also cardioid
enveloped by circles,
cardioid as envelope of tangents, cardioid
sliding inside nephroid, cardioid-evolute
and involute) [more
It notably figures at the centre of a Mandelbrot
In the following argument, for the sake of simplicity, the cyclic nature of
the cardioid is emphasized. A more interesting and relevant agument would need
to be made in terms of the role of a cardioid as a strange attractor -- notably
in the light of its emergence in relation to the Mandelbrot set [more].
In such a set the main cardioid part contains orbits with an attractor of period
1. However there are buds on the buds ad infinitum, all following that
same structure. Then the ends of all these double and re-double, eventually
ending in a spike. Every spike is composed of tiny Mandelbrots, similar to the
first -- but each has all of its parts with periods multiplied by m, the period
in its own cardioid body [more
The role of the cardioid in the Mandelbrot set is explored separately (Sustainability
through the Dynamics of Strategic Dilemmas: in the light of the coherence and
visual form of the Mandelbrot set, 2005). The relation of the Haskell
coaction cardioid with that of the Mandelbrot set is being explored by Kent
Sustainability: interrelating the Carnot cycle with the
Given the perspective of this paper, and notably from a general systems perspective,
it is possible that the Carnot cycle and the Cardioid cyle can be related. It
is also possible that any such generic cycle relating "positive" and
"negative" conditions can also be related to the features of Ba Gua
mirror, and notably as a reflection of the cycle of processes described in Taoist
spiritual disciplines (discussed below).
It is possible that the 4-phase Carnot cycle and the 8-fold coaction cardioid
can be interrelated, especially since their ideal representations do not necessarily
correspond to their distortions in practice. The purpose in doing so is to provide
metaphoric cues to what is understood about the positive and negative phases
of relationships -- and to build understandoing of the nature of the sustaining
|Figure 5: Carnot cycle (a), (b), (c), (d) -- Coaction cardioid cycle
Figure 6: Overlay of 4-phase Carnot on 8-fold Cardioid
| ecosystemic dynamic
added from environment
So, for example, is it the case that the qualities of a relationship are indeed
associated metaphorically in a significant manner with the juxtaposed terms
from the table above:
- symbiosis / synergy: expansive, constantly "hot" ("warm"),
sustained by environmental reinforcement, with action on the environment?
- synnecrosis: compression (mutually destructive), constant, "cold",
depeletion (draining) back to environment, impact of environment?
- commensalism / parasitism / allopathy: expansive, sustained warmth but cooling,
with action on the environment?
- amensalism / predation / allotrophy: compressive ("constraining"),
rising temperature, with action by the environment?
Correspondence with inner and spiritual cycles
Potentially relevant insights into the above cycle may also be derived from
the many spiritually-oriented disciplines of breathing (from the misleadingly
superficial to the incomprehensibly profound) which necessarily focus on the
respiratory cycle understood as divided into various stages of transformation
C G Jung, for example, associated these with a process of internal alchemy
[more]. He proposed a meditative
individuation process using the metaphors of alchemy and the archetypes of the
subconscious -- anima, animus, shadow and projection, plus many others. Curiously
it is Gregory Bateson in a section on Form, Substance and Difference
[more] of his book
Steps to an Ecology of Mind (1972) that relates the depth psychology
work of 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 [more].
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, 1970) [more].
The "circulation of the light" refers to the movement of energy along
a central pathway -- posing challenges metaphorically reminiscent of the design
and operation of high-energy particle accelerators, cyclotrons and synchrotrons
(for which there are many applet demos on the web) [resources].
The "light" is associated with contemplation suggesting that a form
of energy (awakened as the body becomes still) attains the qualities of the
light as the mind focuses upon or contemplates the spiritual ideal [more].
It is implied that this cycle necessarily embodies the insights associated with
the transformations of awareness encoded by the circular arrangement of the
I Ching (or the simpler Pa Kua arrangement of tyrigrams). Whilst breathing
has traditionally been valued as a metaphor in terms of "inspiration",
much remains to be discovered about how "inspiration" works for a
group and how "expiration" is to be understood as its cyclic complement.
Clearly the vast Asian literature on "heat" with regard to the health
of the body might usefully be reviewed in relation to the Carnot cycle.
The inner alchemy of Taoism uses the metaphor of wai-tan or "outer cinnabar"
which it superseded. Outer alchemy sought to achieve physical immortality --
perhaps to be usefully compared with the social project for "sust ainability"
-- through means of an elixir produced by alchemical means. The combination
of meditative breathing and sexual techniques of nei-tan echo the reduction
and recycling process through the process of circulation of essence through
the body, strengthening and replenishing of ch'i (vital energy) and resulting
in the formation of the sacred and immortal embryo or soul. Behind these processes
is the belief that, under the normal conditions, natural processes result in
a life cycle of seven states:
- the generative state of the womb;
- the state of birth and infancy;
- the state of childhood;
- the recognition of dichotomies following the division of yin and yang;
- the separation of the five elements, imbalancing one another and encrusting
- the predominance of acquired conditioning, governing emotions and desires;
- the complete domination of mundanity, with the extinction of positive energy,
leading to death
This natural cycle is reversed by the alchemical firing process in seven corresponding
stages through which immortality is achieved. The stages are:
- refining the self;
- recognition of the true mind;
- restoration of celestial awareness;
- assembling the five elements;
- merging yin and yang - formation of the golden elixir;
- unification of energy - formation of the spiritual embryo;
- transcending the world - incubation of the spiritual embryo.
Through this process reality is cultivated, restoring the self, which otherwise
gradually dies through the natural (entropic) process described above. Real
celestial positivity returns in the midst of total mundanity Inner alchemy may
thus be described in terms of processes whereby real knowledge (symbolized by
water) is retrieved from the overlay of artificial conditioning. The real knowledge
is then used to replace the mundanity infecting conscious knowledge (symbolized
by fire), thereby restoring the basic completeness of the primordial celestial
Expressed differently, the challenge of human development is that consciousness
is normally volatile, given to imagination and wandering thought. Real knowledge
then tends to become submerged in the unconsciousness, sinking into oblivion.
There is no appropriate integration of the two forms of knowledge which act
separately from one another. Through the alchemical process, these two forms
of knowledge are forced to interact. Real knowledge (water) stabilizes consciousness
(fire) and removes its volatility, while consciousness brings real knowledge
into action in life The task of alchemy is therefore twofold, to "empty the
mind" and to "fill the belly". The first is that of cultivating essence - ching
- displacing the mundane preoccupations of the human mind. The second is that
of cultivating life or vital energy - ch'i. When the "belly is full",
sane energy arises through accumulation of right action, and the energy of mundane
conditioning dissolves of itself.
An alternative representation of the task is that of discovering the flexibility
within strength and the strength within flexibility. Another is that of seeking
sense through essence and returning essence to sense, meaning that essence and
sense unite. The "firing process" is a metaphor employed in alchemical texts
for the order of practical spiritual work, namely the order of application of
effort in the cultivation of reality. The aim is to purify the vital energies,
which are said to have their physical aspect within the body, so as to unite
with the immaterial aspect active in the universe. Associated with this process
is the notion of a "crucible" which is subjected to the firing and within which
transmutation takes place, the human body in which ching and ch'i are melted
together to form shen-t'ai, a new being, the sacred embryo or soul.
This is the famous golden flower which opens when enlightenment is attained,
the transmutation as when the soul leaves the dying body and ascends to immortality
in heaven. It is through the firing process that the encrustations of the faculties
are burnt away to expose the awareness of the original spirit - shen.
This requires an appropriate combination over time of inward discipline, deflection
of externals, application of effort, gentle nurturing, and use and withdrawal
of energy. Originally the disciplines were mainly based on meditative techniques
involving control of breath as a means of developing shen-t'ai through
purification of ching to form ch'i and then of ch'i to
form shen. Finally the self is integrated with the universe as the mind
is purified and returns to nothingness.
The philosophical approach - tao-chia - thus described derives from
a more religious approach - tao-chiao. Through normal life the store
of ch'i is gradually exhausted. The aim of inner alchemy - nei-tan
- is to balance yin and yang so as to become one with the Tao; and to conserve
and strengthen the inner essence - nei ch'i - through the inner alchemical
firing process so as to restore the pure state of the essence as it was at the
moment of birth when the primal energy from which the universe arose entered
the body. It is this primordial ch'i which forms the mind - shen,
the body itself forming saliva - yu-chiang - and semen - ching.
Different meditative techniques have evolved with these ends in view, some
clearly viewing the whole process as spiritual while others adopting more physical
manifestations of the spiritual process to achieve the desired result, which
may be reduced thereby to a desire for physical longevity ("sustainability").
(i) Fang-chung shu, symbolic or actual sexual techniques aimed mainly
at exchange of energy through intercourse with a person of the opposite sex.
Attention is on experiencing Tao through participation in the creative process
reflecting the original creation of the universe. Or ching may be preserved
by suppressing ejaculation while bringing one's partner to climax. Attention
is on accumulating energy at particular "centres" through circulation of the
primordial essence within the body
(ii) Meditative techniques concentrating on the inner breath - ch'i
- while taking in pure air and exhaling contaminated air without loss of ch'i,
which is first harmonized - tiao ch'i. Other exercises may follow.
Meditative practices are employed in the process of creating channels for the
essence to circulate. There are two channels, first the hsiaou-chou-t'ien
(lesser celestial circulation, small round), descending from the heart through
the lower abdomen to the kidneys; and then the greater celestial circulation
(great round), involving the whole body and passing through a number of "centres"
from the base of the spine upwards - tu-mai (controlled path) - to the
top of the head, then down - jen-mai (involuntary path) - through the
face, chest and surface of the abdomen back to the base of the spine.
A particular concern of the practices of the Secret
of the Golden Flower is "protection of the centre". This is
effectively framed by the 8 outer conditions of the Ba Gua mirror and the 8
associated practices (or perhaps "games") of Pa Kua -- which may each
be understood as having a different function in protecting the centre. The centre
is present as the central "neutral" cell in the tabular representations
above. Of it that text relates:
The center is omnipresent; everything is contained in it; it is connected
with the release of the release of the whole process of creation. The condition
is the portal. The condition, that is the fulfillment of this condition, makes
the beginning, but it does not bring about the rest with inevitable necessity.
The meaning of these two words is very fluid and subtle. [more]
Understanding a cardioid pattern of transactional relationships
The suggestion here is that, whilst a Carnot heat engine cycle and the Cardioid
Coaction "cycle" can be fruitfully compared, the nature of the latter
is of a higher order of complexity. This is especially the case with regard
to its comprehension. It is for this reason that "parts" of it are
more readily understood in isolation -- as "games", such as "being
positive". These transactional games may indeed be linked as a cyclic process,
but perhaps only at a higher order of dimensionality. The geometry of how the
cardioid is generated merits further exploration as a focus for insight [more].
The cardioid may well function as a kind of value attractor (cf Human
Values as Strange Attractors: Coevolution of classes of governance principles,
1993). In this sense it is perhaps more fruitful to look further at the way
in which the different "games" as transactional patterns, define the
contextual cardioid pattern -- namely the sense in which all the games need
to be evoked in order for sustainability to hold.
Figure 7: Transactional game patterns defining a
(with traditional Taoist magic square numbering)
Figure 7 endeavours to hold phrases typically used to describe particular game
patterns in a mature interpersonal relationship. More specific variants could
be elaborated for dialogue relationships, environment/socio-economy relationships,
etc. Corresponding terms would be used to describe game patterns in an intergroup
relationship. In that case, for example:
- patterns 7 and 9 would be descriptive of the kinds of relationship that
hold between the UK and the USA
- patterns 4 and 6 would be descriptive of the capitalist ("predatory"),
and the socialist ("parasitical") relationship respectively
- patterns 3 and 1 would be descriptive of dysfunctional characteristic of
those between developed and developing countries, or between ethnic groups
- pattern 8 would be descriptive of "cycle of violence" relationships,
as in the Middle East (possibly in conjunction with patterns 3 and 1)
As a symbolic structure, the cells are numbered according to the Pa Kua pattern
of trigrams as a magic square in which all rows and diagonals sum to the same
total, in this case 15 (cf 9-fold
Higher Order Patterning of Tao Te Ching Insights: Possibilities in the mathematics
of magic squares, cubes and hypercubes, 2003). The structure might well
provide a framework for other insightful patterns (as a diamond and in relation
to the Tree of Life).
The level of abstraction associated with the cardioid as an attractor is such
that it raises the question of how the dynamics associated with that pattern
are to be recognized as a whole rather than through the sub-pattern transactional
games. It is perhaps Ron Atkin (Multidimensional Man; can man live in 3-dimensional
space?, 1981) who has best explored how knowledge and communication spaces
of this kind might be articulated on the invisibile complex boundaries which
can only be subliminally sensed by effectively "feeling the geometry" -- effectively
sensing the local constraining curvature of the transactional space (cf Social
organization determined by incommunicability of insights)
"Disorders of the heart"
It is tempting to explore the features of Figure 7 in terms of the "chambers"
of a heart through which it sustains circulation of the "blood" in
various forms of individual and collective relationships. This could be understood
in the light of the earlier discussion of the Carnot cycle (Figure 5)..
Given the symbolism of the heart -- in personal relationships (Cupid, etc),
spiritual relationships (Sacred Heart, etc), in caritative relationships, and
in relation to socio-political economy ("heart of the group", "heart
of the economy", "heart of the nation", etc) -- the "failure
of the heart" is worth examination in the light of the above cardioid of
- "Broken heart": is this understanding usefully associated
with a dysfunctional focus on the net synergy (or even the consummatory aspects)
of a relationship? Can the underlying cardioid pattern be understood as vulnerable
in different ways -- to different modes of "breaking"?
- "Bleeding heart" / "Hard hearted": are these
extremes indications of insights into other modes of dysfunctionality in a
- Heart failure (and disorders): is the incidence and nature of such
physical phenomena associated in any way with the variety of "disorders"
of the cardioid as a gnerative or holding pattern of socio-economic sustainability?
- "Heart deformation": can the cardioid pattern as an attractor
be usefully understood as "deformed" under certain circumstances?
With the USA as the cultural context in which "being positive" is
emphasized to a high degree, is it possible that there is a significant correlation
with the incidence of broken relationships and heart failure?
Dangerous consequences of ignoring the cycle
The management challenge with regard to "positive" may perhaps be
more usefully compared to that of a gardener. Typically particular plants may
need "more light" or "more shade", "wetter soil"
or "drier soil". The variety of plants and conditions have to be appropriately
interrelated in the concrete situation (notably as in the discipline of permaculture).
Issues of management may therefore include, metaphorically:
- when to encourage and appreciate an orchestra playing (on the sinking Titanic,
when disaster was inevitable), and when it is a distraction from concrete
action (Nero fiddling while Rome burns)
- when to encourage the ability of those with skills to identify and fix "dirty"
problems (overflowing sewers, multiple gunshot wounds), and when to appreciate
the "prissy" unwillingness to "muck in"
- when to acknowledge that there is a "problem" (as in the first
step in Alcoholics Anonymous: "I am an alcoholic"), and when to
focus on potential (irrespective of the problems)
The requisite perspective is then not to blindly favour "positive"
over "negative" but to understand when to favour one rather than the
other and how to sustain over time the cycle shifting between combinations of
"positive" and "negative". The practicalities of how to
do this with heat engines were basic to the miracle of the industrial revolution.
The challenge of how to design operational "value engines" and "axiological
Life-skills required for "self-management" call for analogous flexibility
in acquiring the grace and elegance associated with maturity in:
- knowing when and how to lose, notably in order to develop...
- knowing when and how to win
The dilemma is somewhat analogous to the choice offered between:
- allopathic treatments using pharamacuetical products targetted to a specific
illness (that may only be a symptom), namely without addressing the lifestyle
issues that will tend to re-engender that problem
- homeopathic treatment using products in the light of a whole-system perspective
that may not effectively address the immediate suffering
The dynamic between the various positive and negative combinations may perhaps
be best understood through archetypal popular tales -- so-called learning stories
-- or even TV soap opera. For example, consider the array of the 8 best-known
characters around Christopher Robin (Winnie
the Pooh Character Guide): Pooh, Piglet, Eeyore, Tigger, Rabbit, Owl,
Kanga, and Roo (see also Benjamin Hoff. The
Tao of Pooh; Anna Ludlow, Pooh's Little Book of Feng Shui, 2000)
[more]. These characters might
fruitfully be matched with the above-mentioned Pa Kua / Ba Kua directional classification
(possibly having Pooh at the centre and Christopher Robin as a character). Such
a correspondence has already been associated with the construction of the Beech
Hill Stone Circle in the heart of Ashdown Forest (Pooh's traditional home
The cycle may perhaps also be recognized in various classical tales based on
the notion of a ring (cf The
"Dark Riders" of Social Change a challenge for any Fellowship of the Ring,
2002). Possibly the most intriguing "ring" features in one of the principal
classic tales of China -- that of the Journey to the West written by
Wu Cheng'en in the 16th century (and translated by Arthur Waley, 1942). Otherwise
known as The Monkey King, this is the tale of Monkey as the mischievous
protagonist who takes on the heavens and the gods in a sequence of stories with
a group of companions.
The subtle dynamic of give-and-take is basic to many Eastern martial arts.
Increasingly these are translated into strategic management thinking (Chin-Ning
Chu, The Asian Mind Game:
unlocking the hidden agaenda of the Asian business culture - a wester's survival
manual, 1991; Scott Boorman,
The Protracted Game:
A Wei-ch'i Interpretation of Maoist Revolutionary Strategy, 1969) [more]
The curiously fundamental role of the cardioid in the above exploration suggests
that further reflection would be useful on:
- the cultural role widely played by the "heart" in relation to
true romance -- particularly with the implication that true romance is sustained
through a cycle that involves not only the synergy of consummation 'on which
the prime focus is conventionall placed), but also through a range of more
or less awkward, or even mutually destructive phases (described as characteristic
of a "mature relationship")
- the role played by the heart in various religions from the Aztec to the
Christian, and all that is associated with its "bleeding"
- the complex mathematical properties of the cardioid and what these may imply
for the degree of complexity required to both understand and achieve socio-economic
- the common visual image of the heart as "pierced" (by Cupid's
arrow) and the significance of that arrow in the coaction model and its cybernetic
- as a special case of the epicycloid,
the vertex of a cardioid is recognized as coinciding with a rose petal tip,
as defined mathematically, thus linking any such set of cardioids to the extensive
symbolism of the rose -- and raising the question of how the function of such
a set is to be understood in relation to the sustainability of any relationships
(and the place held by rose patterns of different complexity in distinguishing
various chakras in yoga)
Edward Haskell. Generalization of the structure of Mendeleev's periodic table.
In: E. Haskell (Ed.), Full Circle: The Moral Force of Unified Science. New York,
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Orrin Klapp. Opening and closing: Strategies of information adaptation in society.
Cambridge University Press, 1978
George Lakoff. Women, Fire, and Dangerous Things: What Categories Reveal About
the Mind University of Chicago Press, 1987
Harmen Mesker. The Eight Houses: a preliminary survey. 2002 [text]
Robert Munafo. Mu-Ency - The Encyclopedia of the Mandelbrot Set [text]
Swinton Roof. Attraction to Shape, 2002 [text]
R. G. H. Siu:
- The Tao of Science: An Essay on Western Knowledge and Eastern Wisdom. MIT
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Timothy Wilken. UnCommon Science, 2001 [text]