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8th March 2006 | Draft

Interrelating Cognitive Catastrophes in a Grail-chalice Proto-model

implications of WH-questions for self-reflexivity and dialogue

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Annex to Conformality of 7 WH-questions to 7 Elementary Catastrophes: an exploration of potential psychosocial implications



Introduction

As a human response to the perception of a cognitively chaotic situation, WH-questions (when, where, which, how, what, who, why) might be considered to lend themselves to analysis with the tools of catastrophe theory as developed by Rene Thom and others. Thom had developed differential topology into a general theory of form and change of form as a mathematical way of addressing the work on morphogenesis done by C.H. Waddington in the 1950's. Thom's Classification Theorem culminates a long line of work in singularity theory. The term "catastrophe theory" was suggested by C. Zeeman (1977) to unify singularity theory, bifurcation theory and their applications. The crucial theorems rigorously establishing Thom's conjecture were proven by Bernard Malgrange (1966) and John N. Mather (1968). Its essential concern is change and discontinuity in systems (cf Robert Magnus, Mathematical models and catastrophes). WH-questions may be considered as triggered and formulated in response to discontinuity -- when habitual adaptive responses to change are inadequate.

It is possible therefore that the set of WH-questions may in some way be mapped onto elementary catastrophes. This is partially suggested by mathematical techniques of conformal mapping where, for example, the "cognitive flow field" around one known shape (as with an elementary catastrophe) might be mapped onto the flow field around a particular WH-question -- preserving the "angles". Conformal mapping notably makes use of complex variables as combinations of real and imaginary numbers. [applet]

This exploration develops aspects of earlier work on WH-questions (Functional Complementarity of Higher Order Questions: psycho-social sustainability modelled by coordinated movement, 2004; Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness, 2004).

Cyclic patterning of WH-questions: vital cognitive self-reflexivity in a "Kekulé resonance" model

Fundamental to psychosocial dynamics, however, is the relationship to what is defined as a significant "other". For adults this is typically a person of the opposite sex. The set of WH-questions is called into play in framing, articulating and comprehending the dynamics of the relationship. Whilst the physical triggers for attraction (or indifference) are the subject of widespread comment and explanation, as are the reproductive instinct and the associated emotions, the mysterious attraction driving the relationship remains a focus of questions.

A widely accepted framing of the attraction is that it in some way involves a process of completion -- beyond that associated with intercourse. There is a case for exploring the possibility that the attraction is partly driven by the encounter between a set of questions and a set of answers. The encounter has the potential of revealing to each those hidden facets of themselves of which otherwise they could not become conscious. The other becomes a key to self-reference and self-reflexivity through which synergy emerges. Lucas describes this in terms of bidirectional causality. This revelation is reflected in the dependence on an other to view parts of one's physique that are necessarily invisible to one's own eyes -- in the absence of a reflecting surface whose role the partner then performs. The "outer" attractors succeed in part because of the access they offer to the "inner" -- an illusory access because they only model them as the "map", not not the "territory". Is self-reflexicity in any way to be understood in terms of the geometry of "inversion" (see below)?

This dynamic relationship between polar opposites -- the interactions and interdependence of opposing worlds and forces (spirit and matter or heaven and earth) -- is traditionally symbolized by the geometry of two circles intersecting to create an almond shape known as the "mandorla", the "vesica piscis" or the associated fertility "fish symbol", the primitive Christian "ichthys" [more | more | more]. Although it figures in Christian paintings, curiously the Catholic Encyclopedia makes no mention of the vesica piscis (except to distance itself from its relevance to freemasonry), limiting itself to discussion of the symbolism of the fish. The vesica piscis might be understood as a symbol for the resolution of any divisive "two culture" challenge, since it has been celebrated for both its spiritual symbolism and for its mathematical significance -- since the Pythagoreans [more].

In terms of the chakra model, various psycho-spiritual disciplines stress the need for an appropriate relationship between the chakras --each "unblocked" to permit an appropriate flow of energy. For the individual this can be framed as ensuring a relationship between the lowest and the highest chakras -- potentially to be understood here as between the when-question and the why-question, appropriately posed. As noted earlier the applet of Lucien Dujardin (Catastrophe Teacher: an introduction for experimentalists -- parabolic umbilic, 2005) is particularly suggestive of some such understanding. Why are particular forms (sexually) attractive? What is the attraction of some particular forms? What is attractive (to some) about the process of pursuit itself (cf the geometry of "pursuit curves")?

This situation highlights the mysterious mirroring role of the other in completing any such cycle -- understanding that the sense of otherness may be projected onto a living person or onto some other focus of devotion. In archetypal terms the cycle may be understood as an embodiment of the Ouroboros -- the snake biting its tail. It is the capacity to reframe creatively one's sense of origin -- the personal Big Bang of one's own engendering -- as the ultimate when-question of one's personal cosmology. The reframing is achieved through a paradoxical coupling with the ultimate why-question -- completing the cycle -- that gives an existential sense of perspective (cf Dean Brown and Wenden Wiegand, Law of Nothingness, 2003: "The beginning of everything and the end of everything is the void"). In the symbolism of alchemy, the Ouroboros embodies the circular nature of the alchemist as a self-reflexive opus -- in whom incommensurable opposites, such as male and female, or conscious and unconscious, together engender a previously incomprehensible whole of higher dimensionality.

Possible self-reflexive cycles of WH-questions

Model A

Model B

.

How

.

.

?

.

Which

.

What

When

.

Why

Where

Who

Where

Who

When

Why

When

What

.

?

.

.

How

.

One approach to the contrasts between Model A and Model B in the table is to benefit from the insight reprersented at the time by the Kekulé benzene model. This involves recognition of the alternation between two ring structures -- resulting in a structure that is neither one nor the other and consequently named a resonance hybrid. In the case of benzene, neither form really exists -- explained using a higher level of theory than conventional molecular bonding. The single bonds are formed with electrons in line between the carbon atoms Kekulé was himself famously inspired by the Ouroboros archetype in making his discovery -- after allegedly dreaming of a snake seizing its own tail.

Illustration of resonance hybrid structure

Illustration of resonance hybrid structure

This approach suggests the possibility of a special integration or "union" of the generic "when-question" into the "why-question" in the above cycles to form a single 6-fold cycle of WH-questions -- a super-ordinate pattern of dialogue (combining the 3 cuspoids and the 3 umbilics). More importantly it addresses a criticism of the catastrophe theory approach from the perspective of complexity theory. The argument is that in real dynamic systems the individual catastrophe models should not in fact be considered separate -- a one-problem-at-a-time methodology -- but should be treated as interdependent. This then concords with the reality of kinetic intelligence and the dynamics of decision-making in the moment.

Integrating "seed insights" in this way (and as discussed in relation to letterforms) strengthens the case for careful examination of the fundamental epistemological importance attached by Buddhism to the repeated use of sound in mantras, and notably the 6-seed-syllable Om Ma Ni Pad Me Hum of Indo-Tibetan Buddhism -- the most used mantra in the world [more more]. This is understood as serving to bypass the distractions of speech in responding to the apparent catastrophes of reality: "Through mantra, we no longer cling to the reality of the speech and sound encountered in life, but experience it as essentially empty. Then confusion of the speech aspect of our being is transformed into enlightened awareness." (Kalu Rinpoche)

Om Mani Padme Hum
(Tibetan script)

Om Mani Padme Hum

Are such combinations of letterforms to be understood as a visual representation of a form of sound mapping of viable "pathways" through various cognitive catastrophes? Seemingly no attempt has yet been made to render the complex forms of catastrophes comprehensible through sonification (cf International Community for Auditory Display, Sonification Report, NSF, 1999; Bruce N. Walker, Psychophysical Scaling of Sonification Mappings, 2000; Frances L. Van Scoy, et al, The Sound and Touch of Mathematics: a Prototype System, 2001; Haixia Zhao, et al, 'I Hear the Pattern' - Interactive Sonification of Geographical Data Patterns, 2004; John Dunn & Mary Anne Clark, Life Music: The Sonification of Proteins, 2004). The 10th Conference of the International Community for Auditory Display included a panel on Approaches to Sonification for Listening to the Mind Listening (2004) that presented sound representations of EEG recordings.

Are the letterforms in any way to be associated with Thom's own schematic representation of his "archetypal morphologies"?

Interrelating the three umbilic catastrophe forms: a "Grail chalice" proto-model

Given the challenge to comprehension, and the elusiveness of such forms, especially intriguing from a symbolic perspective is the manner in which the three umbilic shapes (hyperbola, ellipse, parabola) can be brought together into a single classical form -- a chalice.

"why" parabola Parabola Chalice typical deep cup
"who" ellipse Ellipse minimal stem with a knob (nodus or pommellum)
"what" hyperbola Hyperbola wide base
 

2-D Cross-section

Example of a chalice

This form is readily comprehensible as an image -- even though its multidimensional significance remains elusive. The steps involved in integrating the geometry underlying the separate catastrophic forms are then:

  • orient a parabola vertically (around the y-axis), open at the top
  • position an ellipse of minimum eccentricity symmetrically beneath the parabola and tangential to its vertex (noting that this operation cannot be based on any explicit property of the ellipse, unless the eccentricity is reduced such that it is a sphere)
  • ensure that a hyperbola is oriented such that it is open along the x-axis (positive and negative)
  • position the hyperbola beneath the ellipse (such that it shares the same y-axis with the ellipse and the parabola)
  • ensure that the upper curves of the hyperbola are tangential to the ellipse -- and the lower branches correspondingly
  • terminate the upper curve on each side of the hyperbola at the tangential points to the ellipse
  • rotate the combined form around the y-axis to form an object in 3D that is readily understood to take the form of a classical chalice.

Mnemonic significance

Further consideration could be given to the following, if only for mnemonic purposes:

  • although the object exists in 3D, in terms of the dimensionality of the catastrophe umbilics this is merely a particular subsection of a form that is much less amenable to comprehension -- it could be understood as having additional spatial and/or temporal attributes or "frills"
  • on rotation:
    • the focal point of the parabola is unaffected within the paraboloid -- since the parabola itself is "one-pointed" and the point is "centred"
    • the two focal points of the ellipse form a notional ring within the ellipsoid, centred on the vertical axis common to the parabola and its focal point (unless the ellipse takes the form of a sphere)
    • the two focal points of the hyperbola form a notional ring outside the hyperboloid, also centred on the same vertical axis
  • the process of constructing the "chalice" as an integrated object points to a number of challenges which can be described by geometrical analogy:
    • orientation: it is to be expected that any shared sense of orientation within a common framework would not be easily achieved in practice
    • proportion: it is to be expected that the relative importance of each would be differently understood, resulting in dimensions and proportions difficult to interrelate
    • symmetry and minimal eccentricity: in the case of the ellipse, for example, this could well be a challenge to ensure given the importance differently attached to the two focal points
  • the alternative approach to hyperboloid construction using tilted wires (see discussion) is suggestive of polarized understandings and their necessary twistedness, especially if the rigid wires are treated as "axes of bias" (whether in inter-sectoral dialogue, inter-cultural dialogue, understandings of meaningful work, or lifestyle preferences) as adapted from the cognitive bias exploration of W T Jones (The Romantic Syndrome; toward a new methodology in cultural anthropology and the history of ideas, 1961)
  • the four lower order question-catastrophes (in the light of the table above) might be considered to form a 4-quadrant (or 16-celled) "table" lying "beneath" the "chalice" and on which it may then be placed at the common point of the quadrants
  • the three components of the "chalice" may be used to distinguish information (hyperboloid), knowledge (ellipsoid), and wisdom (paraboloid) and their associated questions: "what", "who" and "why", respectively -- whilst emphasizing the connectivity between them, and to the "data" table beneath them

Explanatory frameworks

For purposes of communication and reflection, this form then lends itself to a variety of explanatory frameworks or "stories":

  • the symbolic significance of the real challenge in practice of appropriately interrelating the geometry of the three forms -- and the significance of only partial success, or a distorted, disproportionate result
  • in geometric terms the hyperbola, ellipse and parabola are part of a family of 4 curves (including the circle) called conic sections [more | more]
    • viewed from above, the "chalice" has a circular cross-section, possibly with concentric circles associated with features of the three component forms
    • the curves are notably distinguished by their eccentricity (e): circle (e = 0), hyperbola (e = >1), ellipse (0 < e <1), parabola (e = 1); for a hyperbola, the larger the eccentricity, the more it resembles two parallel lines, for an ellipse, the more elongated it becomes.
    • limiting cases: just as the circle is the limiting case of the ellipse when the two foci coincide, so the parabola (e = 1) is the limiting case of both the ellipse and the hyperbola, when one of the foci is moved to infinity; a hyperbola may be thought of as a kind of ellipse that is split in half by infinity.
    • the curves are all produced when a 2D plane cuts a hollow (right circular) 3D cone at different angles and positions. An interactive demonstration by James White (Exploring Conic Sections. In:: 3 Dimensional Graphics) shows the way in which a cone interacts with a plane in order to study the transition from ellipse to parabola to hyperbola as the plane parameters are varied, whether in 2 dimensions or 3.
    • the (isoptic) curve produced by varying the tangents to a parabola, such that the angle of their intersection is a constant, is a hyperbola; if that angle is pi/2 (namely an orthoptic curve) then for both an ellipse and a hyperbola the curve is a circle
    • the curves can be considered as forming a "story" of successive transformations from one to another -- from the hyperbola with two "separate" foci, through the ellipse with two "contained" foci, to the parabola with a single focus; these offer a useful mapping of possible relationships between "self" and "other".
  • in optical and radiation terms (notably as metphors for communication challenges and opportunities):
    • parabola: a radiant point at the focus will reflect or refract off the parabola into parallel lines; the paraboloid is conventionally used (in automobiles and antenna) for its properties in projecting a beam of light or radio waves placed at its focal point -- but could be used to receive such a beam to bring it to the focal point (as with an antenna). Such a property could be usefully related to the preoccupations of the "vision" metaphor in formulations of collective strategy, and the typical corrective measures which are the concern of "opticians" -- as well as extremes calling for corrective surgery, such as "cross-eyed".
    • ellipse: rays from one focus will reflect to the other focus.
    • hyperbola: rays coming from one focus will refract to the other focus
  • in terms of geometrical projective transformation of the the conic section curves, there is a strong case for reflecting on the mnemonic significance of associated curves through:
  • in any exploration of design solutions to the relationship between hyperbola, ellipse and parabola (preferably using interactive java applets to explore their 3D variants), one constraint could be to use the line between the foci of the hyperbola as the base of a triangle, whose sides then each pass through a focus of the ellipse to intersect at the focus of the parabola (NB: in 3D the the triangle is a cone).
  • in reflection on the significance of viable design solutions,

Symbolism

In terms of their significance as symbols, again for mnemonic purposes:

  • in considering the transformations from hyperbola, through ellipse to parabola (or their 3D equivalents), the two curves of the hyperbola need to move together and "through" each other in order to form an ellipse; during this process they may be understood as forming two overlapping circles centred on the two foci -- the vesica piscis (discussed above)
  • the properties, and elusiveness, of the "chalice" portion resonate with the myths, legends and symbols of the Holy Grail [more]
  • the object and its axis are together reminiscent of the classical symbolism of the "chalice" and the "blade" (Riane Eisler, The Chalice and the Blade: our history, our future, 1988)
  • the challenge of polarization is potentially well-framed by the "chalice" (as might be expected from its symbolism and "healing" function), notably through the properties of the hyperboloid
  • to the extent that the elusive form is a container of complex processes defined by complex numbers, the y-axis could be used for imaginary numbers and the x-axis for real numbers (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005)
  • the formal resemblance of the parabolic umbilic to a fountain (above), has resonances with myths associated with the Grail chalice as notably stressed by the Order of the Grail: "Again, and this is very important to meditate and work with inwardly, the Grail that which precisely sustains and validates all things... involves knowing oneself in relation to the Fountain of Life".
  • given the association of the paraboloid with why-questions, the theme of this exploration, it could indeed be considered a suitable container for "whyne" -- appropriate for celebration at a roundtable of the Council of the Whys!

Toward a new typology of dialogue -- based on the "Grail chalice" proto-model

If the challenge of dialogue is to interrelate the three elements (paraboloid, ellipsoid, hyperboloid) appropriately, a simple positonal notation might be used to distinguish the types of dialogue, their emphases, and the potential challenges they may face. The following symbols are used (the significance of the first in logic is indicated as a matter of interest):

Indicative types of dialogue

Model A

.

Model B

.

Model C

.

set union

.

.

X

.

.

.

.

.

O

.

.

O

.

.

.

.

.

≥≤

.

proper subset

,

.

set intersection

O

≥≤

Indicative examples:

  • Model A: Cup positioned on Stem above Base. Desirable ("normal") dialogue dynamic (notably in the Council of the Whys)
  • Model B: Base (rotated 90 degrees) positioned above Stem; Cup offset to the left (below Stem) and tilted rightward
  • Model C: Cup inverted, offset to left; Base offset to right; Stem dysfunctionally between them; all elements set low
  • etc

The notation could be extended to cover situations in which the elements were of relatively different size (bigger or smaller) in relation to one another, and variously conjoined or disjoined, resulting in dysfunctional "cups" (and dialogues). This example does not include the case in which the ellipsoid could be of greater eccentricity -- distended vetically or horizontally. More thought could be given to the relation between the various axes.

There is the case for developing an applet to explore the various ("skewed") conditions of a dialogue, and the possible transformations between them in the light of the geometry.

By focusing on the "Grail chalice" in a dialogue situation, a satisfactory arrangement of the "table" is assumed. Typically considerable effort goes into the arrangerment of tables in strategically important "roundtable" dialogues (eg summits, etc). There is considerable expertise in such diplomatic table arrangements, placement and the appropriate etiquette associated with them. The same cannot be said regarding the "Grail chalice" questions and the many potential dysfunctional configurations illustrated by the notation of the table above:

  • What ("base"): here the issue in a dialogue is one of naming, framing and patterning issues and strategies -- preferably renaming, reframing and repatterning
  • Who ("stem"): here the issue in dialogue is identification and affirming of who "we" are, the bonds between "us", and what "we" represent -- preferably through new insight into the coherence of that identity at a more fundamental level at which it can be reaffirmed
  • Why ("cup"): here the issue is the understanding framing the dialogue -- whether destiny, vision, principles, values or some other justification

An especially intriguing line of exploration is associated with the receptive capacities of a parabolic antenna. Given the stress placed on achieving a "common vision" in social groups, this may be considered as an intuitive realization of the need for the kind of coherent receptivity achieved with a parabolic reflector to handle radiation in "invisible" parts of the electromagnetic spectrum. The group dynamic design challenges of configuring an array of perspectives might then be usefully related to those of constructing such an antenna [more]. As with a circular mindmap [more] -- this contrasts with the tendency to configure the elements of any vision in a bulleted laundry-list (or tree-structure) of values rather than in any design with a focal point. Arguably laundry-lists and tree structures seek to align an array of visions in an identical manner that is analogous to a flat mirror -- optically incapable of bringing light to a focus. A paraboloid array designs in the necessarily different alignments required of the component perspectives in order to achieve coherence and focus. The question to be explored is whether more could be achieved through the use of "paraboloidal mindmaps". Of course this analogy raises questions as to the multidemensionality and "origin" of the vision that is brought to a focus by such a design!

As what amounts to an ongoing dialogue, an organization or community might also be described in terms of this "Grail chalice" model -- as a measure of its distance from "Camelot"!


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