Engendering Invagination and Gastrulation of Globalization (Part #9)
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The central argument above challenges the adequacy of conventional "logical" explanations inadequately supported by visual or other renderings attentive to the challenge of comprehension. Hence the inspiration of the forms and processes of nature. It is in this sense that the physical form of the globe embedded in the toroidal dynamics of an electromagnetic field offers further support for reframing "globalization" beyond the spheroidal. Integrity is then associated with a larger pattern as illustrated in the following figures.
Fig. 7: Reframing the context of globalization Indication of relationship between spheroidal and toroidal in terms of electromagnetic fields 
Origin of the Earth's Magnetic Field (Reproduced from Google Knol) 
Earth's magnetic field (Image by NASA) 


Fibonacci spiral: "Global", understood as based on a sphere, is however defined geometrically by use of pi (π)  as is any sense of circle or cycle (as discussed above). The toroidal form into which the sphere may be transformed is based on π ^{2}. This association with pi is intimately related to the cognitive sense of integration of the forms respectively represented. This has long been recognized in design and considerations of aesthetic appreciation. It is fundamental to the design of many integrative symbols and architectural forms intended to convey a sense of coherence.
Seemingly missing from any design repertoire restricted to pi is the wellrecognized psychoactive power of design based on the golden ratio (golden mean) proportion. It is this ratio which is associated with the sense that proportions are "right"  that they "fit" appropriately. The ratio is often denoted by the Greek letter phi, usually lower case (φ). This is absent from equations defining the torus as such but is evident in helicoidal and vortical dynamics which may be intimately related to the torus (Enabling Governance through the Dynamics of Nature: exemplified by cognitive implication of vortices and helicoidal flow, 2010). As noted there, whilst difficult to "describe", spirals in the form of whirlpools, whirlwinds, or tornados are widely recognized in terms of their global dynamic. Spiraling structures, such as representations of the caduceus or DNA, are acceptable despite their complexity (Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007).
As a proportion, φ is fundamental to the Fibonacci spiral (mentioned above) with which design considerations for global governance might be fruitfully associated, as previously discussed (Tao of Engagement  Weaponised Interactions and Beyond, 2010; Designing Global Selfgovernance for the Future, 2010). The possibility to be explored is whether phi (unlike pi) is basic to a fom of "conceptual packaging" constraint which is fundamental to the formulation and credibility of viable governance designs.
In this context, what is being "packed" are categories, conceptual clusters or holons as understood in the argument above as forming a "blastosphere". "Globalization" at that stage is then a process of "packing". The argument earlier is that this packing process reaches a point of psychosocial instability in its spherical "inflation". The instability can be mitigated or postponed by reclustering holons into a smaller number of larger holons  effectively detaching attention from the detail of the original holons, limiting it to the "general" and avoiding the specific. However this detachment progressively introduces a new form of instability, namely loss of credibility from the perspective of the more "local" holons. The "global" perspective thereby becomes increasingly unreal and incomprehensible from a "local" perspective.
The Fibonacci spiral may then serve as an aid to understanding this packing constraint in relation to the critical point at which the "blastosphere" is forced to undergo transformation through "invagination" and "gastrulation". In considering this possibility it is appropriate to recall that the Fibonacci spiral is associated with a diversity of patterns in nature, most notably as seen in the nautilus shell. If the accumulation of holons in the "blastosphere" is understood as building up in a manner mapped by construction of a Fibonacci spiral, the question is at what stage in that accumulation (understood in terms of the development of that spiral) does the process of packing become unstable?
How, for example, might this relate to Dunbar's number  a theoretical cognitive limit to the number of people (hypothesized at 150) with whom a person can maintain stable social relationships  notably through "social grooming" (as discussed in Annex B).
Fig. 8: Construction of Fibonacci spiral (numbers in both images indicate the length of sides of squares, not the number of "boxes" within each square) (reproduced from Adaptive Hypercycle of Sustainable Psychosocial Selforganization: designing a mapping of a Chinese metaphorical pattern language, 2010) 
Fig. 8a: Initial steps in process of construction of the spiral, based on a succession of combinations of squares (detail of the image on the right)

Fig. 8b: Insertion of connecting curves into the framework of image on the left (only steps 1 through 8 shown on left) 


The progression may be framed in terms of explanation of successively higher order, possibly understood as of greater dimensionality (across "planes" a, b, c, d, e, etc corresponding to labels in the image above):
 a: 1st order: goodevil, goodbad (stressing the problematic "other", a duality to be eliminated from a fundamentally unitary system)
 b: 2nd order: a 4fold explanation holding polarity dynamics and distinctions (Jung, AQAL, etc)
 c: 3rd order explanation:
 d: 4th order explanation
 e: 5th order explanation
The transition curve from one explanatory order to another is achieved through the geometrical metaphor of a "pivot" point, the centre point of the curve by which the transition is "encompassed". These are marked in the above diagram (A, B, C, D, E, etc) corresponding to the explanatory order.

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Fig. 9: Golden spiral A logarithmic spiral whose growth factor is related to the golden ratio (φ) Reproduced from Wikipedia to show the degree of approximation of the (green) whirling rectangle construction of (Fig. 8, above) to the (red) golden spiral 

Cognitive implication: from pi to phi: A key question is what understanding of "globality" and "globalization" is fruitfully and meaningfully carried by the circle (πbased), the sphere (πbased), the torus (π ^{2} based), and the spiral (φ based). Under what  potentially dynamic  conditions is transition between these forms appropriate?
Why, for example, has the nautilus shell been adopted as a more relevant symbol of continuing development and sustainability (cf the symbol of The New Zealand Curriculum Framework and the Nautilus Institute for Security and Sustainability)? By contrast, why are the circle and sphere considered appropriate representations of completed "global" integration  however questionable? And why does the torus feature in subtler, and even more paradoxical, symbols of integration (Ouroboros, halo, etc)?
In this light, what is the cognitive quality or modality implied by the formula defining these forms? How are they to be experientially decoded  in ways relevant to psychosocial organization? It is with respect to these cognitive challenges that the fascinating formal insights offered on various sites are frustrating (cf. Ron Knott, Phi's Fascinating Figures, 2010; Gary Meisner, Pi, Phi and Fibonacci Numbers). What are the cognitive implications to be associated with φ^{2} or with φ^{3}, for example?
The nature of the cognitive challenges can however be usefully highlighted through the experiences associated with distinct games (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005). Sets of games of different complexity  and typically with one or more "others"  may be ordered in terms of the Fibonacci spiral, as discussed separately (Tao of Engagement  Weaponised Interactions and Beyond: Fibonacci's magic carpet of games to be played for sustainable global governance, 2010).
Proximity and least energy: As noted by Lisa Zyga (Scientists find clues to the formation of Fibonacci spirals in nature, Physorg.com, 1 May 2007):
While the aesthetics and symmetry of Fibonacci spiral patterns has often attracted scientists, a mathematical or physical explanation for their common occurrence in nature is yet to be discovered. Recently, scientists have successfully produced Fibonacci spiral patterns in the lab, and found that an elastically mismatched bilayer structure may cause stress patterns that give rise to Fibonacci spirals. The discovery may explain the widespread existence of the pattern in plants.
The summary discusses the experiments of Chaorong Li, et. al. (Stressed Fibonacci spiral patterns of definite chirality, Applied Physics Letters, 2007 and Triangular and Fibonacci number patterns driven by stress on core/shell microstructures, Science, 2005) and specifically the authors' understanding that:
Patterns that evolve naturally are generally an optimized configuration for an assembly of elements under an interaction...We conjecture that the Fibonacci spirals are the configuration of least elastic energy. Our experimental results provide a vivid demonstration of this energy principle. This is the best support for this energy principle of phyllotaxis (or 'leaf arrangement,' often credited to D'Arcy Thompson) before a rigorous mathematical proof is available.
If Dunbar's number can be understood as a psychosocial instance of a more general constraint in packing cognitive constructs, the progressive packing (as a consequence of accumulation) might be ordered by a psychodynamic analogue to some such "configuration of least elastic energy". Following the design principles of Christopher Alexander  and until it reaches a critical degree of instability  this would correspond to recognition of a sense of satisfactory appropriateness (Comprehension of Appropriateness, 1986). The question here is whether that stage is typically associated with a stage in the evolution of the Fibonacci spiral as illustrated above in Fig. 8. As the inhabitant, designer and constructor of that spiral, the nautilus mollusc has periodically to "change gear" and shift into a new phase.
Assuming the squares in Fig. 5a correspond to recognizable holons (categories, friends, nations, friends / followees, music, plants, authors, wines, perfumes, celebritites, etc), in relation to the Dunbar range, the pattern offers the following possibilities for instability through their accumulation in the psychosocial "blastosphere":
 1 + 1 + 4 + 9 + 25 + 64 = 104
 1 + 1 + 4 + 9 + 25 + 64 + 169 = 273
The "packing" of conceptual constructs (as holons) into the "blastosphere", or onto its surface, could be explored in topology in terms of proximity space  the axiomatization of notions of "nearness" that hold settoset (as articulated by Frigyes Riesz), as opposed to the better known pointtoset notions that characterize topological spaces (S. A. Naimpally and B. D. Warrack, Proximity Spaces, 1970). The quest for least energy configurations is known as the Thomson problem (in the case of electrons), or more generally as the Generalized Riesz Problem. From a psychosocial perspective, such configurations might be related to proxemics, namely the study of the set measurable distances between people as they interact, as introduced by biologist anthropologist Edward T. Hall.
"Shells" of "globality"? The possibility of instability with progressive "globalization" is also remarkably modelled by the buildup of the electron shells of atoms, as represented in the periodic table of chemical elements. Of interest is the cognitive appreciation of "shell" as a geometric metaphor  shared with the nautilus "shell"  but especially the concentric nature of such shells with the increase in complexity of the atom. Usefully to be understood as "degrees of globality"?
One effort to generalize the pattern to encompass the psychosocial domain is that of Edward Haskell (Generalization of the Structure of Mendeleev's periodic table, 1972). It is also discussed in relation to a possible periodic table of human sciences. An argument for using this pattern in understanding the constraints on cognitive organization has been developed separately (Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order, 2009).
Of relevance are mathematical efforts to deduce the form of the periodic table (Denis H. Rouvray et al., The Mathematics of the Periodic Table, 2005) and even the implications for a (selfreflexive) possibility (Towards a Periodic Table of Ways of Knowing  in the light of metaphors of mathematics, 2009).
Of particular interest in this periodic metaphor are the constraints on the complexity of atoms, with the build up of electron shells. With respect to such instability (and Dunbar's number), in oneuse of the metaphor (Periodic Pattern of Human Life: the Periodic Table as a metaphor of lifelong learning, 2009) the extraordinary fact is noted that:
The total number of those elements confirmed is currently 111, with unconfirmed claims made with regard to elements up to 122 (see Timeline of chemical elements discoveries). The oldest person in history, whose age has been verified, is Jeanne Calment (18751997)  122 years. Consideration has been given to the extension of the Periodic Table beyond the seventh period, with an eightperiod table suggested by Glenn T. Seaborg in 1969  with elements up to 210 hypothesized. These concerns parallel those of life extension into a similar number of years, notably using strategies for engineered negligible senescence.
Also of interest is the seemingly controversial proposal of JeanClaude Perez (Mendeleiev Periodic Table Prediction Equation, 19972008). He sought a single mathematical equation which would organize the information of the most heterogeneous table of science  generating and predicting its structure. Perez integrated this with explorations of a possible numerical structure of DNA, genes and genomes, the golden ratio and Fibonacci numbers laws, and subsequently proposed an Equation of Life (2008), as summarized in book form (Codex Biogenesis; les 13 codes de l'ADN, 2009).
One interesting insight in the case of chemical elements is the probable existence of socalled islands of stability, namely the possibility of elements with particularly stable "magic numbers" of protons and neutrons. The numbers currently recognized include 2, 8, 20, 28, 50, 82, 126. This would allow certain isotopes of some transuranium elements to be far more stable than others; that is, to decay much more slowly  with halflives of possibly even of the order of millions of years. Methusalah? Enduring global civilization?
With respect to this argument, these possibilities merit considertation in the light of the Dunbar number  a theoretical cognitive limit to the number of people (hypothesized at 150) with whom a person can maintain stable social relationships  notably through "social grooming" (as discussed in Annex B). Might "shells of globality" correspond in some way to the "magic numbers" constituting "islands of stability"? The extensive representation of possible isotopes, suggests a means of analyzing and representing possible stable configurations of groups of different size (see the Wikipedia Table of nuclides and Table of nuclides combined).
Again however, it is the experiential comprehension of instability in "globalization" which is readily obscured by any such potentially adequate "periodic" description  the focus on spherical shells and pi. With respect to any set of holons, the key issue is how many are recognizable in the sense highlighted by Dunbar's cognitive constraint  as a partial repackaging and circumvention of the constraint of George Miller (The Magical Number Seven, Plus or Minus Two, Psychological Review, 1956)?
Again this points to some kind of selfreferential constraint potentially embodied in a helicoidal structure  the function of phi in the Fibonacci spiral. It is especially intriguing that the number of cells in a blastosphere is asserted to be typically 128, namely 2^{7}  consequent on a 7fold process of cell division. Simulations of that process already show a degree of instability at 2^{6}  namely 64, upheld as significant in the Chinese binary coding system of the I Ching (appreciated by both Jung and Pauli, as discussed below). It is possible that the Dunbar constraint varies to a degree with the conditions of stability  as potentially modelled by the higher shells of the periodic table.
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