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Produced in relation to The Buckminster Fuller Challenge 2010, organized by The Buckminster Fuller Institute, in support of the development and implementation of a strategy that has significant potential to solve humanity's most pressing problems.
Richard Buckminster Fuller (1895-1983) is renowned for his innovations as designer, inventor and futurist. He is most widely known for his invention of the seemingly improbable geodesic dome. One of his key initiatives was the elaboration (in collaboration with E.J. Applewhite) of an understanding of synergetics (Synergetics: Explorations in the Geometry of Thinking, 1975; and Synergetics 2: Explorations in the Geometry of Thinking, 1979). With his particular use of language and neologisms, he defined this as
A system of mensuration employing 60-degree vectorial coordination comprehensive to both physics and chemistry, and to both arithmetic and geometry, in rational whole numbers... Synergetics explains much that has not been previously illuminated... Synergetics follows the cosmic logic of the structural mathematics strategies of nature, which employ the paired sets of the six angular degrees of freedom, frequences, and vectorially economical actions and their multialternative, equieconomical action options... Synergetics discloses the excruciating awkwardness characterizing present-day mathematical treatment of the interrelationships of the independent scientific disciplines as originally occasioned by their mutual and separate lacks of awareness of the existence of a comprehensive, rational, coordinating system inherent in nature.
He was instrumental in encouraging a systemic global approach to use of energy and resources (Operating Manual for Spaceship Earth, 1969) and the educational World Game which has now been developed into the Global Simulation Workshop (Thomas T. K. Zung, Buckminster Fuller Anthology for the New Millennium, 2002).
In what follows a distinction is made between Fuller's focus on:
It is the last two emphases that are explored here, especially in the light of the expression "geometry of thinking" in the title of the two volumes of his magnum opus. Whilst his own interpretation of these may indeed be held to be implicit in that work and in other writings (I Seem to Be a Verb, 1970), most reference to his work focuses on the first and second emphases. It is the second emphasis that is typically the inspiration for proposals submitted to the annual Buckminster Fuller Challenge of the The Buckminster Fuller Institute "in support of the development and implementation of a strategy that has significant potential to solve humanity's most pressing problems".
The argument in what follows is that it is the cognitive and existential implications from which the more tangible, material proposals emerge -- however much their understanding is conditioned by evident design constraints. It is these intangible factors that hold the key to understanding why psychosocial systems have proven to be more than inadequate in designing viable global responses to global challenges -- sustainable in their own right rather than as idealistic proposals for global sustainability. In this sense the more tangible preoccupations are secondary and derivative -- however viable and successful they may be as isolated initiatives, typically presented together as a list, rather than systemically configured.
There is therefore a case for exploring Fuller's Synergetics: Explorations in the Geometry of Thinking as though it might indeed concern what the words imply -- but in the light of richer understandings of psychology, epistemology and ecological philosophy than are evident in his writings. What might the "geometry of thinking" then imply if understood in this way?
The argument here follows from a long-term interest in the psychosocial implications of the design principle developed by Fuller and underlying the geodesic dome, namely that of tensional integrity (tensegrity). This permits the construction of structures with an integrity based on a synergy between balanced tension and compression components. These implications are discussed in a set of documents, most notably From Networking to Tensegrity Organization (1984). A particular interest is the possibility of enabling virtual organizations and conceptual structures through web technology ***. Such possibilities have been explored to some degree by management cybernetician Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994).
As previously cited in reviewing Fuller's understanding (Synergetics: Examples of Integrated, Multi-set Concept Schemes, 1984, referring to sections in his two volumes):
For Fuller's collaborator (Synergetics as defined by E.J. Applewhite):
Synergetics is Fuller's name for the geometry he advanced based on the patterns of energy that he saw in nature.... Fuller felt that the old classic approaches did not describe the way nature actually behaves. For instance, Euclid's lines were thought to go off to infinity. Fuller says lines are vectors of energy and he rejected the notion that anything physical could be extended indefinitely.... His geometry hinges on the tetrahedron, the simplest structural system within insideness and outsideness: he advances it as the most economical way to measure space and to account all physical (and metaphysical!) experience. This is what he calls synergetics: an empirical mathematical system in which geometry and number mesh without fractions. It gains its validity not from classic abstractions but from the results of individual physical experience.
It is clear that Fuller himself strongly related geometry with epistemology and the thinking process, even implying that this accorded with a sense of energy processes in thinking.
Fuller's use of language and neologisms (exemplified above) is notorious for its impenetrability, even by those sympathetic to his initiative and appreciative of its tangible outcomes. But this challenge to comprehension is in fact central to any "geometry of thinking" and it can be argued that he does not address this adequately with respect to the fundamentals of his approach. This is not to deny his attention to education with respect to global allocations of resources and the like -- or his very lengthy seminars endeavouring to communicate his understanding. It could be argued that the failure to address this dimension adequately -- within the framework of any "geometry of thinking " -- is fundamental to the limited consideration of his insights at the level at which they might prove most relevant to his demonstrated global concerns.
Whilst it may indeed be the case that some have the intellectual facility, or particular intelligence, which enables them to engage fruitfully with geometrical abstractions, this is not common. The tangible products of these abstractions in the form of the regular polyhedra and semi-regular polyhedra do indeed offer a kind of illusion of comprehension -- perhaps an intuition of its implications. Their symmetry naturally facilitates this.
There is the intriguing possibility that the increasing complexity of polyhedra may be understood as a form of representation of increasing dimensionality -- and therefore the challenges and possibilities of their comprehension through their symmetry.
The mathematician who has best addressed the challenge of comprehension is Ron Atkin (Multidimensional Man: can man live in 3-dimensional space?, 1981). He illustrates this by the challenge of comprehension in relation to experience "within" the geometry of a triangle -- especially with regard to the perspective necessary to comprehend the triangle as a whole. Hence the subtitle of his book. It is this "fourth" perspective which offers a link to what amounts to the "point of departure" for Fuller, namely the tetrahedron (Threshold of Comprehensibility: a fourfold minimal system? 1983).
Other mathematicians have explored the challenges of shifting perspective from 1, though 2, to 3 dimensions. The cognitive challenge has long been delightfully articulated in such mathematical fiction as Edwin A. Abbott's Flatland: a romance of many dimensions (1884), Charles Howard Hinton's An Episode on Flatland: or how a plain folk discovered the third dimension (1907), A. K. Dewdney's The Planiverse (1984), Ian Stewart's Flatterland (2001), and Rudy Rucker's Spaceland (2002). The 1884 novel has recently taken the form of an animated version (Flatland, 2007) to highlight the challenges otherwise.
The concern with comprehension is also central to the study of multi-term systems by John G. Bennett (General Systematics, Systematics, 1, 1, June 1963) who, like Atkin, represented them using polygonal (rather than polyhedral) configurations. His related interest in the "energy" associated with such systems was consistent with his professional responsibility for research on coal, as discussed elsewhere (Reframing Sustainable Sources of Energy for the Future: the vital role of psychosocial variants, 2006). The insights have been further developed by Anthony Blake (The Supreme Art of Dialogue: structures of meaning, 2008).
A degree of concern with comprehension is evident in the adaptation to (psycho)social organization by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994) of Fuller's work on tensegrity -- and the subsequent development of syntegrity (Allenna D. Leonard, Team Syntegrity Background. 2002; J. Truss, et al. The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003).
The point to be stressed however is that Fuller's more general arguments in terms of the tetrahedron and polyhedra fail to address the capacity to comprehend such structures. The need to do so is seemingly bypassed by the evident ability to represent and to construct them -- implying that this achieves the degree of comprehension of any cognitive challenge they represent -- if his basic argument is to be taken seriously. It is not clear who can claim the capacity to think in dimensions appropriately articulated by even the simplest regular polyhedra -- the icosahedron or the dodecahedron, for example. Exploration of syntegrity focuses primarily on the icosahedron.
This point may seem to be irrelevant if the degree of comprehension is indeed adequate to the design and construction of complex geodesic domes -- effectively very complex polyhedra. The point may seem to be even more irrelevant with the proven capacity of mathematicians to detect ever more complex symmetry groups (of which polyhedra are the simplest form). For example:
(made with Stella Polyhedron Navigator)
|Representation of E8 from Wikipedia
|Logo of the United
Nations Climate Change Conference (Copenhagen, 2009)
However it could be argued that the ability to develop formalizations concerning many dimensions involves a degree of "kidding oneself" (or others) regarding the capacity to understand the "systems" and wholes they are thereby held to represent. Operationally the assumption of success is achieved through engaging in a "concrete" iterative process over time -- a succession of operations or procedures in which the parameters are slightly changed. A geodesic dome may, for example, be constructed in this way -- as with elaborating an "ex-planation" from which some kind of overview (of the plane) is obtained.
The argument is exemplified by the fact that mathematicians exploring high orders of symmetry now accept that any coherent "proof" of a theorem relating to them may be hundreds of pages in length. That for the so-called "enormous theorem", depends on many distinct contributions, totalling some 15,000 pages in length -- and far beyond the capacity of any single individual, however specialized or dedicated. Mathematicians do not seek to comprehend the many "dimensions" in such theories. They are merely letters or numbers in equations.
For someone (like the writer) challenged by even "four dimensions", the elegance of any visual representation (such as those above) implies the possibility of a form of comprehension which is as elusive as any harmony sought in the reconciliation of disciplines, faiths or ideologies in response to the challenges of global governance. If, as is implied, the challenges of global governance call upon more than three dimensions, clearly how people are empowered to engage with such complexity merits insights which may indeed benefit from Fuller's "geometry of thinking". In the case of the above structures, for example, the challenge has been (naively) considered elsewhere (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Dynamics of Symmetry Group Theorizing: comprehension of psychosocial implication, 2008).
The assumption made here is that the tangible emphases of Fuller (and of those who assume them to be adequate to the challenges of global governance) are overly simplistic and inadequate precisely because they ignore the underlying insight implied by Fuller's own analysis -- and possibly the source of his own conscious creativity. If indeed a viable system can only be understood in terms of four dimensions, represented minimally by a tetrahedron, then attention is required to comprehension of more complex polyhedra. What kinds of thinking do they imply -- as they increase in complexity?
The relevance of the challenge of comprehension to global governance is evident in the fact that many people, if not most, consider that they are "right" -- whether individually or through the groups and belief systems with which they are associated. The problem is that they each tend to disagree with each other, often violently -- namely with others who also perceive themselves to be "right" (Jonathan Haidt, The Righteous Mind: why good people are divided by politics and religion, 2012). It is from the dynamics of this "system" that some form of coherence is expected to emerge for effective global governance -- perhaps through the hypersimplification that any democratic majority is "right" and that the minority is then "not-right", or possibly when some elite considers itselfs "right", irrespective of the understanding of a democratic majority..
The question is how any "geometry of thinking" can assist in enabling coherence under such circumstances. Many have assumed that it is possible to present or impose a particular pattern of order -- with whom all are expected to agree and to consider "right" -- although it is readily apparent that such agreement is not forthcoming (despite current unprecedented use of forms of individual and collective violence, deemed "right" in their turn).
In the metaphorical spirit of the cognitive linguistics of George Lakoff and Mark Johnson (Metaphors We Live By, 1980), the sense of being right can be interpreted as the sense of being "upright" on the surface on which one stands -- being "upstanding", holding a "standpoint" in relation to what one "understands". The challenge of any global perspective is that being upright is what each of the 6 billion people on the world can do anywhere on the surface of the globe, provided it is recognized that people are then variously angled relative to each other, with some on the other side of the globe even being "upside down" -- although not from their own perspective.
The metaphor offers ways of understanding why those distant from where I am "upright" appear smaller -- and may even disappear over the horizon into invisibility, insignificance and irrelevance, whatever their perspective. This perspective, compared to their understanding of what is held to be "normal", may appear to justify their being labelled from elsewhere as "abnormal" and "extremist", having "fallen off the edge of the map".
Curiously an effort has been made to claim that globalization is effectively a flattening of the Earth (Thomas Friedman, The World Is Flat, 2005). The problematic aspect of this perspective, recognized as a "flat earth mentality", has been criticized elsewhere (Irresponsible Dependence on a Flat Earth Mentality-- in response to global governance challenges, 2008). However, if each perceives themselves to be upright on a plane -- "normal" with respect to that plane -- then these planes are all effectively tangential to the surface of a "globe" -- whatever the cognitive significance of "plane" and "globe". The "plane" might be considered what is readily comprehensible, with "ex-planation" offering a measure of non-planar understanding, whether or not this is indeed comprehensive and "global". As noted above, Fuller appropriately challenges the notion that anything physical could be extended indefinitely to infinity -- whether a line or a plane.
If the perspectives of all those variously upright in this way are to be honoured, somehow they need to be positioned around the globe -- irrespective of how unusual their orientation may be from my own position. The plane with respect to which I am right and normal, as a tangent to the globe, may then be understood as intersecting other tangential planes of different orientation -- which collectively are configured as approximations to the surface of the globe. Together they are a polyhedron of 6 billion faces, with fewer faces to the extent that more agree with one another on what is right -- "democracy with a human face"? If each such plane is understood to be like the piece of a jig saw puzzle, the challenge might then be how to fit all the pieces together. In the plane this is studied as tiling patterns or tesselation. Here the implication is the need for a spherical tiling -- a spherical jig saw, as suggested by the famous Rubik's cube.
This geometry is relatively comprehensible, even if the implication of different orientations is not -- as reflected in clashing disciplines, religions and civilizations. It is however possible that if humanity does indeed live in more than three dimensions -- Atkin's question -- that only even more complex surfaces can appropriately reconcile the differences in orientation and the associated paradoxes. Arguably it is more than clear that simple tilings, plane or spherical, are not adequate to the challenge of global governance -- or answers would already have been found long ago. Claims that a sphere are adequately indicative of global representation are then clearly inadequate, if not extremely naive.
Other surfaces of great interest, on which people could be variously "right" and "normal", are the torus, the Mobius strip and the Klein bottle -- before even considering the multidimensional shapes considered of such fundamentaL significance by mathematicians and physicists. These also point to the question of whether everyone being "right" together could be better represented by them standing on the inner surface of a globe -- and therefore with a convergent orientation towards the centre, rather than the divergent orientation associated with standing on the outer surface. This is of course not consistent with physical experience, to the extent that is relevant.
The argument here is that Fuller's "geometry of thinking" is extremely useful to the discovery of configurations of adequate complexity to the cognitive challenges of global governance -- starting with the simpler polyhedra, or the tensegrities based on them. It has not been mined to that end. The issue is how to "decode" the geometry as a means of holding comprehension of distinct perspectives and interrelating them -- and the challenge of understanding another apparently strange perspective, namely any other way of being "right".
One common approach to avoiding the challenges of multidimensional geometry is through clustering styles of thinking, whether as lists or in simple tables. The sense of being "right" is then associated with a particular cluster having a shared mode of thinking or engagement with reality. This is most evident in lists of psychological types or forms of intelligence (Systems of Categories Distinguishing Cultural Biases, 1993).
Differences of appreciation may then be "ex-planed" by associating people or groups with different types. To the extent that such typing is reframed as "profiling" or "stereotyping", such "ex-planation" does not effectively address the dramatic dynamics of the relationship of one type to another in practice.
Lists of clusters may however be configured into a two dimensional table or matrix, as with the Myers-Briggs Type Indicator. Consideration may be given to the complexification of such representation into a periodic table (Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order, 2009) . Classical approaches, such as the organization of the hexagrams of the I Ching, may also be understood in this light. Also of interest is the configuration of types into a circular form, as with astrological types, or with the Chinese Ba Gua.
Any such tabular representation may itself be subject to geometrical transformation to introduce the possibility of other integrative implications. The table may be rolled into a cylinder, with the ends joined to form a torus -- with its own significance. Two tori may also be linked (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics Transforming a matrix classification onto intertwined tori, 2006). Geometrically a torus is closely related to the sphere which is Fuller's primary focus. Types and categories may also be configured spherically, even at the most basic level of accounting (Spherical Configuration of Categories to Reflect Systemic Patterns of Environmental Checks and Balances, 1994; Spherical Accounting: using geometry to embody developmental integrity, 2004).
There is however a related challenge of comprehension which needs to be considered. Whilst the metaphorical geometry of the previous paragraphs offers a way of understanding in principle the issue of different ways of being right, it does not address the curious phenomenon that the creative thinkers inspired by geometry typically cannot "stand" each other and find each other's efforts irrelevant or worse. This is a classic, well-documented problem in the development of mathematics but is evident in any other comprehensive approach, most notably that of religions (even where they attach fundamental significance to geometric symbols).
The argument is confirmed by Fuller himself who seldom, if ever, cites others who have ventured into that domain -- or preceded his own ventures. The disagreements he had about priority in the discovery of "tensegrity" are only one indication. The phenomenon has been discussed elsewhere using the metaphor of "body odour" (Epistemological Challenge of Cognitive Body Odour: exploring the underside of dialogue, 2006). It has been specifically explored by contrasting two approaches to "integral futures", that of Ken Wilber and that of David Lorimer (Self-reflexive Challenges of Integrative Futures, 2008).
A natural consequence of a creative vision of an integrative framework is to see it as all-encompassing, rendering other considerations irrelevant or misleading, whether or not they are explicitly subsumed. This is of course typical of belief systems, ideologies and of many disciplines -- and of "scientific revolution" as discussed by T S Kuhn and Karl Popper.
Rather than the mutual citation pacts characteristic of some academic endeavour -- potentially questionable as "incestuous" -- it is effectively a case of mutual non-citation pacts, deliberate marginalization and the like (Considering All the Strategic Options -- whilst ignoring alternatives and disclaiming cognitive protectionism, 2009). The possibility of coherent global strategies emerging from such a context, and attracting widespread support, is clearly problematic, whether or not they are of requisite variety to be sustainable in cybernetic terms.
Fuller is clearly exceptionally conscious of the place of the "self" in any global system, as illustrated by the following:
It is however quite unclear that these cognitive insights have been explored in subsequent development of his work. In part this is a consequence of his special jargon -- a typical characteristic of innovators with respect to modes of knowing.
Fuller's reference to 12 "others" is clearly of potentially fundamental significance to the common symbolism of the unreconciled Abrahamic religions that are at the root of so much global discord in a context of faith-based governance. (Root Irresponsibility for Major World Problems: the unexamined role of Abrahamic faiths in sustaining unrestrained population growth, 2007; Generic Reframing of the 12 Tribes of "Israel": "We have met the Zionists and them is us", 2009).
Is this a failure of cognitive geometry? If the issues of archetypal symbolic and spiritual 12-foldness have yet to be effectively addressed, why is there any expectation that secular analogues, in the form of roundtables and jury panels, should operate effectively in support of global governance? How is it that the exceptional genius, William Sidis (1898-1844) is widely quoted as saying: I attach great value in the working out of my theories to the help given by polyhedral angles of the dodecahedron which enter into many of the problems. Is there a cognitively unexplored implicit heritage from the Greek dodekatheon (and its Roman analogues) -- now primarily embodied in the trade marks on luxury goods? Is there an important sense in which cognition is dependent on a "dodecameral mind" (Union of Intelligible Associations: remembering dynamic identity through a dodecameral mind, 2005)? Curiously one of the shapes hypothesized for the universe is a Poincaré dodecahedral space.
The challenge is that each such innovator has somehow arrived at a unique insight from which claims are made with regard to its capacity to encompass the whole. This is necessarily an "inconvenient truth" for others exposed to it, or other innovators who may follow (An Inconvenient Truth -- about any inconvenient truth, 2008). The problem is central to the challenge of any new global strategy which purports to render obsolete, or subsume, any previous strategy. The question is "where" is any such insight positioned within a "geometry of thinking", and how is it configured in relation to other such insights?
The question is therefore how to render explicit such paradoxical reflexivity and how to embody it in the light of the arguments of Francisco Varela (Laying Down a Path in Walking: essays on enactive cognition, 1997; The Embodied Mind: cognitive science and human experience, 1991) and George Lakoff and Mark Johnson (Philosophy In The Flesh: the embodied mind and its challenge to western thought, 1999). Such approaches have perhaps been most succinctly summarized by Jennifer Gidley (The Evolution of Consciousness as a Planetary Imperative: an integration of integral views. Integral Review, 5, 2007) to the effect that:
However, the growing awareness of a potential planetary crisis has highlighted the significance of finding new ways of thinking, if humankind is to move through our current complex challenges. This critical imperative appears to be mobilizing researchers from a wide range of disciplines to broaden the notion of evolution of consciousness beyond its biological bounds.
The international community makes a case for "tolerance" with little articulation of any practical utility. In the aftermath of the financial crisis of 2008, it was argued that the inequality created by huge bankers salaries is a price worth paying for greater prosperity (Public must learn to 'tolerate the inequality' of bonuses, says Goldman Sachs vice-chairman, The Guardian, 22 October 2009). This is reminiscent of the tolerance expected by a US Ambassador to the United Nations, Madeleine Albright when questioned on whether the sanctions against Iraq (killing more children than at Hiroshima) was appropriate. Albright replied: "I think this is a very hard choice, but the price -- we think the price is worth it." (We Think the Price is Worth It, Fair, 2001).
The simple question in terms of any creatively elaborated "geometry of thinking" is how does it allow for those who do not think in the same way -- who disagree? This is the core challenge of global governance -- and of democracy. How can allowance be made for disagreement in such a geometry? However "tolerance" is also a design term fundamental to construction of all kinds. But how is "tolerance" to be understood in a "geometry of thinking"?
There is no map of how the unique "cognitive odours" are positioned in relation to one another -- of how the mutually offensive "stinks" ensure distance between innovators. In effect there is no effort to process or represent disagreement -- except in efforts to facilitate its elimination through reconciliation in agreement. Is this text itself an invitation to agreement?
And yet again, Fuller's articulation of tensegrity offers insights into how disagreement might be configured, namely how perspectives of different orientation might be configured without seeking to ensure that there is only one orientation (Using Disagreements for Superordinate Frame Configuration, 1992).
Interestingly, tensegrity structures are robust precisely because of their dynamic "tolerance" to destabilization. But how might that translate into any cognitive configuration?
There is a curious implicit assumption that if everyone agreed with what is presented by some as being "right", then all global strategic challenges could be resolved. There is relatively little recognition that differences of perspective are vital to the sustainability and survival of any system. In cybernetic terms this has been expressed as the requisite variety, or requisite complexity, to manage a complex system. In psychosocial systems, however, this variety may often be framed as anathema -- as a recognition of those who are not "on program" or "singing from the same hymn sheet", having failed to grasp the message. In some belief systems they may simply be framed as "unbelievers" against whom any sanction is appropriate. How is this perspective to be integrated?
Consistent with Fuller's focus on three and four in constituting a viable system as described geometrically, others have struggled with the articulation of psychosocial systems in the light of three and four categories, in various combinations. The work of Carl Jung has been especially significant, notably through his efforts to interpret a range of symbol systems from a variety of cultures in this light -- some with geometrical representations. This work was more specifically developed by Marie-Louise von Franz (Number and Time, 1974). Typically however any associated representations have been in two dimensions through polygons. This is also the case of the AQAL system of the Integral Movement with its focus on four quadrants of a circle.
Fuller has done much to extend the representation into three dimensions but in doing so has essentially focused on the geometry and its tangible significance and has rendered implicit the psychosocial implications. It is Stafford Beer and his collaborators who endeavoured to give functional significance to polyhedral representation of psychosocial systems -- with his focus on the icosahedron through syntegrity. Associated with this work has been the further development of his management cybernetics into the elaboration of a viable system model (VSM). However the psychosocial implications of this are much diluted, except in the extension into knowledge cybernetics by Maurice Yolles (Knowledge Cybernetics: a new metaphor for social collectives, Organisational Transformation and Social Change, 2006; Exploring Cultures Through Knowledge Cybernetics, Journal of Cross-Cultural Competence and Management, 2007).
Such ventures are a long way from any recognition of the psychosocial significance of more complex polyhedra in more than three dimensions (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007). Given the level of disagreement which prevails in the global system, the question is how complex might a polyhedron need to be to be able to map such complexity and the many orientations it implies? And how then to comprehend it?
The consequence of such failure is well illustrated by the traditional Eastern tale and image of the seven blind men and the elephant. How many perspectives are required for a global understanding? How to determine how many may be lacking for such a perspective? The simpler example of the two perspectives necessary for stereoscopic vision makes clear the nature of the challenge described by as the need for "poly-ocular vision" by Magoroh Maruyama (Polyocular Vision or Subunderstanding, Organization Studies, 2004). However this focus on the vision metaphor obscures the more general challenge of the need for quite different modes of knowing as exemplified by the challenge of integrating a set of polysensorial understandings (Strategic Challenge of Polysensorial Knowledge: bringing the "elephant" into "focus", 2008). This is consistent with the argument of Chris Lucas that generally multiple objectives or parameters have to be met or optimised before any 'master' or 'holistic' solution is considered adequate (Practical Multiobjective Optimisation, 2006).
The cognitive challenge may be clarified through nuclear fusion as a metaphor -- where the technical challenge is to configure magnets so as to confine and contain plasma such that it does not touch the walls of its container when heated to extreme operational temperatures. The question is what is the configuration of perspectives necessary to contain collective attention globally without it becoming inappropriately attached to the forms by which it is contained -- if such containment is to be transformative (Enactivating a Cognitive Fusion Reactor Imaginal Transformation of Energy Resourcing (ITER-8), 2006)
This same challenge of cognitive fusion might also be described from an Eastern perspective in terms of a configuration of an appropriate variety of perspectives adequate to the containment of ch'i. The complete set of 64 conditions encoded by the hexagrams of the I Ching hexagram might be understood in this light. Both understandings offer insights into the challenge of global governance.
In each such case it is not simply the structural configuration -- as is clear from tensegrity. It is a question of the dynamics of the structure. It is the contrast between inadequate assumptions regarding the adequacy of "genetics" as a source of comprehensive explanation and the recent recognition of the importance of the dynamics of epigenetics. It is a question of how the different modes of understanding "dance" together. The question may be what is the epigenomic map (Ian Sample, Epigenome Decoded: 'A new frontier for genetics', The Guardian, 15 October 2009). Or perhaps the challenge is an "epimemetic map"?
Again this recognition is partly represented through Fuller's detailed articulation of the transformation between different configurations, notably in terms of the vector equilibrium (Vector Equilibrium and its Transformation Pathways, 1980). But again there is almost no trace of the psychosocial implications of this dynamic in the subsequent development of his work.
Although Fuller's work is seemingly based on a cognitive understanding of self, as mentioned above, the dramatic challenge in any psychosocial system between subjectivity and objectivity is apparently absent. And yet it could be argued that its articulation is present in a most insightful manner through his extensive discussion of the geometry of the relationship between polyhedral duals. However its potential psychosocial significance is again "buried".
The images below indicate how duality might be encoded in the objectivity of "points", interrelated by "lines" of argument triangulating them into a configuration -- thereby creating "sides". But subjectively, the "sides" may be held to be "points", interrelated by a different pattern of "lines" -- thereby creating a different set of "sides". One mode of understanding is the invert of the other. In Jungian terms it might be understood as an implicit shadow or one "hidden" within an "other".
|System duality represented
by transformation between polyhedra
Icosahedron morphing into Dodecahedron
(made with Stella Polyhedron Navigator)
Representation of such images by software enables the transformation of one to the other to be followed -- in geometrical terms (Stella: Polyhedron Navigator). There is a real challenge to ensuring any cognitive continuity between subjectivity and objectivity, especially collectively in a psychosocial system. In drama it is described as the process of enantiodromia, whereby one is transformed into its opposite. This may be observed on a larger scale where a political system hostile to the values of another progressively acquires the values of the other. On the occasion of the UK Tory and Labour annual conferences in 2009, much was made of the Tories taking on the "red" values of Labour and of Labour taking on the "blue" values of the Tories. It might be argued that the US has adopted the level of invasive surveillance of its population that it previously disparaged in USSR society -- whilst the latter has taken on a range of dubious capitalist values it previously disparaged.
The geometric representation highlights the sense in which if one of the pair is an "objective" system, the other is an anti-system associated with the "subjectivity" whereby that objectivity is called into question. This might be said to be the nature of the relationship between "mainstream" global initiatives lauded for their "objectivity" and "alternatives" disparaged for their lack of objective coherence. However, even in the case of any "objective" system, perspectives emerge amongst those who perceive the system not to be what it is claimed to be -- recognizing underlying dynamics, etc. This is most obviously evident in "objective" efforts to rationalize complex organizations -- possibly resulting in disaffection amongst personnel who experience matters otherwise (an extreme example being the staff suicides in France Telecom, a major scandal in 2009).
A "geometry of thinking: must necessarily encompass these extremes and the dynamics between them.
The argument above uses the richness of Fuller's synergetics to stress that attention is required as much to what is not elicited from that work, by himself and those who have followed, as to what is evident and a widely appreciated inspiration. In that sense, as a comprehensive model, the more general argument is that it does indeed model integrative thinking through the geometry -- but some (if not a greater proportion) is effectively "hidden" or "buried" within the model. The model embeds or embodies that of which it is not conscious. It effectively implies a (cognitive) dual which is implicit and to that extent absent.
Like each comprehensive endeavour, it is its own metaphor -- using the phrase of biologist Gregory Bateson (Mary Catherine Bateson, Our Own Metaphor; a personal account of a conference on the effects of conscious purpose on human adaptation. 1972). In explaining why "we are our own metaphor", Bateson pointed out that:
One reason why poetry is important for finding out about the world is because in poetry a set of relationships get mapped onto a level of diversity in us that we don't ordinarily have access to. We bring it out in poetry. We can give to each other in poetry the access to a set of relationships in the other person and in the world that we're not usually conscious of in ourselves. So we need poetry as knowledge about the world and about ourselves, because of this mapping from complexity to complexity. (pp. 288-289)
In this sense a model is not what it seems. Its hidden dual holds a complementary pattern of insight. It is only together that they are effectively as comprehensive as is claimed for the objective interpretation. But comprehending what has not been rendered explicit is a challenge because it is "about" the process of comprehension -- of "thinking about the geometry".
Due to the progressive interlocking of accumulated patterns into nested meta-patterns, as a solution to human processing capacity limitations, there is a form of directed, convergence onto a progressively clarified ultimate meta-pattern, towards which learning tends asymptotically, in that final (en)closure is never achieved (except possibly as an essentially transient, private, transcendental experience). Bateson describes this ultimate pattern as:
The pattern which connects (all living creatures) is a meta-pattern. It is a pattern of patterns. It is that meta-pattern which defines the vast generalization that, indeed, it is patterns which connect." (Mind and Nature; a necessary unity, 1979, p. 11)
Of possible relevance in terms of poetic sensibilities, in the light of Bateson's comment, is the concept of a clinamen. This is a constraint that guarantees an enormous flexibility of meaning and is an act of memory.
Model builders have a marked tendency to set cognitive processes in definitional "stone" -- to reify and to petrify (Reorganizing Knowledge and Unfreezing Categories, 2009). Matters are in fact worse when it is "the other" who gets set in stone and petrified -- by whom one is then "petrified" in turn, hence "terrorism". Any dynamic engagement with such a "stone" -- dialogue -- is then ridiculous. Some of the challenges are delightfully highlighted by the following points.
|The Hazards of System Building
Matthew Melko, System Builder
(Presented at the Foundation for Integrative Education Conference, Oswego, New York, 1969;
reproduced in Main Currents in Modern Thought, vol. 269 no. 2)
Summarizing points made above, any model that claims to be comprehensive -- a cognitive Theory of Everything -- would seem to require or imply in some way:
These may be fruitfully related to criteria emerging from other ways of looking at the challenge of a meta-model.
|Criteria for an Adequate Meta-model, 1971|
Ordering of a dimension
Dynamic and evolutionary
One approach is to distinguish any criteria by a succession of patterns, as in the following exploration of articulations governed in terms of from 1 to 20 factors variously challenging to comprehension. These are derived from Examples of Integrated, Multi-set Concept Schemes (1984) as explained in Patterns of N-foldness; comparison of integrated multi-set concept schemes as forms of presentation (1984).
|Distinguishing Levels of Declarations of Principles, 1980|
5:Constraints on existence
6:Coherence through renewal
7:Modes of change
9:Implementation of a transformation process
10:Endurance of a form
11:Empowerment and importance of a form
12:Harmoniously transformative controlled relationship
14:Cycle of development processes
and development of form
16:Values and assumptions
17:Relationship potential of a form
18:Inadequate transformation attempts
20:Significance of mutually constraining forms
As the primary discipline of relationships, the wide range of mathematical disciplines might be mined for the underlying or generative metaphors from which they each emerged (Towards a Periodic Table of Ways of Knowing -- in the light of metaphors of mathematics, 2009). Clues as to how to think about the emergent insight of anything "meta" may be sought in a variety of "disciplines" and "ways of knowing".
Alternative Conceptual Realities:
clues to the dynamics of enacting new paradigms through movement, 2002
Movement and Attitude Control
Movement and Attitude Control
"Ascent" and "Escape"
from kinetic intelligence and sports psychology
Clues from animal locomotion
Clues from animal locomotion understood generically
Clues from Christian vices and virtues
Clues from yogic perspectives on afflictions of the mind
Clues from Buddhism
Clues from the streetwise and from nonviolence
Clues from the martial arts
Clues from psychotherapy and game-playing
Clues from dialogue
the clues framing any static perspective
Clues to integrating movement through kinetic intelligence
Clues from catastrophe theory, force dynamics and manoeuvering
Clues from navigation of multi-media and virtual reality environments
to "ascent" from Christianity
Clues to "escape" from Buddhism
Clues to "ascent" and "escape" from Theosophy
Tuning and playing category arrays: methodological challenges
Patterns of aesthetic associations
The particular focus of that paper and its sections (listed above) is on the possibility of "re-reading" the clues from spiritual traditions, in the light of the disciplines of movement, as epistemological clues. The core argument is that whilst spiritual traditions point to a better, essentially static, condition to be achieved through following their guidelines, the injunctions in their guidelines are do's and don'ts that give no sense of the dynamics of the experiential reality that their practice is claimed to enable. As a result they appear essentially static and moralistic, and disconnected from the patterns of movement that people find meaningful -- setting up, through misapplication of those guidelines, a somewhat antiquated moral barrier that prevents interpretation of those guidelines in ways that would be highly valued by those who seek a richer and more dynamic reality. The dynamic is likely to be essential to the viability of the emergent pattern.
Whilst such guidelines may well be vital to what might be understood as "attitude control" and coordination, the latter can be usefully understood as prerequisites to any process of shifting attitude into subtler perceptions -- described metaphorically through somewhat misleading terms such as "ascent" or "escape". The distinction between attitude control and ascent for an individual may then be compared with the various highly elaborated challenges of launching any vehicle into planetary orbit -- into orbit around the globe (Entering Alternative Realities -- Astronautics vs Noonautics: isomorphism between launching aerospace vehicles and launching vehicles of awareness, 2002; Noonautics: four modes of travelling and navigating the knowledge "universe"? 2006). Hence the cognitive challenge of "launching" a credible global strategy. But it is curious that this tends to take the form of launching a single "pillar" -- only too reminiscent of rocket, missile and projectile technology (Missiles, Missives, Missions and Memetic Warfare: navigation of strategic interfaces in multidimensional knowledge space, 2001).
Especially interesting in relation to model building of any kind is the extent to which elements of "attitude control", typical of spiritual disciplines (such as the "humility" highlighted above by Melko), are in fact vital to the construction and comprehension of any more subtle meta-model -- to the epistemology which it necessitates.
The role of a dynamic is also central to the study of Arthur Young (Geometry of Meaning, 1976), notably in relation to learning/action cycles, and tentatively adapted elsewhere (Characteristics of phases in 12-phase learning-action cycle, 1998; Typology of 12 complementary strategies essential to sustainable development, 1998). Of particular interest is the possibility that seemingly incommensurable cultural symbols should be reconciled through movement (Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation, 2008).
The cognitive challenges have also been variously highlighted through the following themes:
Fuller's "geometry of thinking" may be understood as implying a progressive engagement with globality which he indeed understood to be fundamental to the global management of resources of "Spaceship Earth" -- a term which he invented (Operating Manual for Spaceship Earth, 1969). But it might be said that it is precisely the instrumental "rationality" of his understanding, and those inspired by it, which has failed to reflect the psychosocial processes which undermine such rationality in response to the dramatic challenge of world crises. The failure of rationality associated with the global financial crisis of 2008, and its aftermath, is but one indication.
And yet the elements of cognitive geometry are implicit, if not evident, in the metaphorical geometry through which the attempt is made to develop global strategies:
Such examples are developed in more detail elsewhere (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009; Geometry of Organizations, Policies and Programmes, 1992).
Of particular interest is the continuing reliance on the simplest forms of geometry in strategic construction, notably the "pillar". Where Hammurabi (ca. 1728-1686 BC) and Ashoka (304-232 BC) both used one to articulate their historic principles, the "four pillars" of Barack Obama were announced in 2009 as: non-proliferation and disarmament; the promotion of peace and security; the preservation of our planet; and a global economy that advances opportunity for all people. Other strategic concerns may be articulated in terms of "axes" or "poles" at the global level, with a focus on "stakes" at other levels of society. There is a degree of irony to the fact that T. E. Lawrence (1888-1935) considered a larger number of "pillars" to be appropriate (Seven Pillars of Wisdom, 1922) with respect to the Middle East arena -- which continues to be strategically problematic.
Many intergovernmental institutions articulate their basic strategies in terms of sets of pillars (Coherent Value Frameworks: pillar-ization, polarization and polyhedral frames of reference, 2008). There is a metaphorical irony to the fact that "pillars" on their own do not constitute any form of shelter -- even if "stakes" may be configured to form a protective stockade. Curiously it is sacred architecture that tends to configure pillars to enable arches to be constructed to sustain vaults -- thus creating a form of cognitive shelter.
In a secular society, with respect to the earlier comment on "uprightness", it is curious that claims are "staked" for a protective set of "rights" which are merely presented as a list (Universal Declaration of Human Rights, 1948). The facility with which they may be configured as a system is not considered (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008).
of Human Rights
Selected polyhedral representations [see others]
(made with Stella Polyhedron Navigator)
|30 Articles displayed on
1 face-type of a rhombicosidodecahedron
into geodesic form
|Experimental representation of
Whilst pillars are of course basic to any design, such geometric elements are of the simplest kind in relation to the range of configurations explored by Fuller. A tensegrity may be fruitfully understood as a configuration of interlinked pillars in three dimensions. An effort was made to configure as a tensegrity a set of strategic dilemmas (as paradoxical forms of pillar) emerging from considerations at the 1992 United Nations Conference on Environment and Development (Configuring Strategic Dilemmas in Intersectoral Dialogue, 1992; Spherical Representation of Icosidodecahedral Net of Strategies, 1992).
|Spherically symmetrical tensegrity structures configuring interlinked strategc "pillars"|
Principles by Balancing Configurations of Functions: a tensegrity organization
Groupware Configurations of Challenge and Harmony: an alternative approach to "alternative organization" (1979)
Clearly of deep concern is the extent to which reliance is currently placed on elements so simplistic that they are incapable of encompassing the strategic challenge, for lack of requisite complexity. This is notably indicated by the challenge of "polarization" and "sides" in any current political discourse from which global strategies are expected to emerge. It is also indicative of the inability to elaborate sustainable configurations of points or parties -- requiring more complex geometry -- capable of holding higher orders of consensus. The question is how to move cognitively beyond "points" and "linearity", as discussed elsewhere (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009).
In the light of Fuller's geometrical argument, this suggests the merit of exploring the role of polyhedra in more effective strategic engagement with globality (Towards Polyhedral Global Governance complexifying oversimplistic strategic metaphors, 2008; Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008; Polyhedral Empowerment of Networks through Symmetry: psychosocial implications for organization and global governance, 2008).
But, despite Young's argument for learning cycles, little effort has been made to integrate cognitive cycles with the great circle geometry so fundamental to systemic viability in Fuller's terms (Emergence of Cyclical psychosocial Identity: sustainability as "psyclically" defined, 2007; Representation of Interlocking Elements for a Sustainable Global System: configuring strategic dilemmas in intersectoral dialogue, 1992; Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998).
At issue is the nature of "cognitive geometry" (cf Harada Toshinobu, et al. Extension of "Cognitive Geometry" and the application to primitive, Bulletin of Japanese Society for Science of Design, 2000; Rashi Glazer and Kent Nakamot, Cognitive Geometry: an analysis of structure underlying representations of similarity, Marketing Science, 1991). This may be explored through the extension beyond linearity through Cognitive Circlets (2009) and Cognitive Crowns (2009).
Use is as yet seldom made of available software to support non-linear integrative comprehension (Polyhedral Pattern Language Software: facilitation of emergence, representation and transformation of psychosocial organization, 2009). In theory web technology should provide communication protocols that are highly appropriate to enabling the emergence of such forms. This has been recognized in the case of the Team Syntegrity Initiative as described by J. Truss, et al. (The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003), where it is noted that:
TSI has not had the resources to simultaneously develop electronic applications, though many exciting Syntegrity-based designs are awaiting the right opportunity and resources. Our task for the future ranges from providing the ability for groups to conduct some or all of the Syntegration process online, to applying the geometric design principles underlying the architecture of the Syntegrity forms, and the cybernetic principles underlying the protocols, to support the evolution of and connections among groups and communities.
Insight into cyclic geometry may in fact be most readily understood through music (Carol L. Krumhansl, The Geometry of Musical Structure: a brief introduction and history, Computers in Entertainment, 2005). Hence the potential merit of musical representation of strategic challenges (Conversion of Global Hot Air Emissions to Music, 2009). This may relate to transforming "vicious" social cycles and feedback loops (Vicious cycles and loops, 1995; Dysfunctional Cycles and Spirals: web resources on "breaking the cycle", 2002).
Given the efforts of Stafford Beer to design and implement a cybernetically informed management system at corporate level and at the national level (in Project Cybersyn), it is curious that so little of that research continues with respect to the challenge of global governance. In contrast, however, it might be considered amazing the amount of effort and investment devoted over deacdes to the design of the Large Hadron Collider in pursuit of the so-called "God Particle" -- faith-based science at its best (Robin McKie, Second chance for Large Hadron Collider to deliver universe's secrets, The Observer, 1st November 2009). At an estimated cost of 4 billion euros, it is funded by and built in collaboration with over 10,000 scientists and engineers from over 100 countries as well as hundreds of universities and laboratories. Projet Cybersyn was of course destroyed at the instigation of Nobel Peace Laureate Henry Kissinger, due to inability to tolerate any alternative models or experiments in governance in the western hemisphere. The contrast is all the more astonishing in comparing the geometry of the two images below -- and what they imply for the underlying cognitive geometry.
|Logic diagram of a cybernetic
factory by Stafford Beer
[from Andrew Pickering,
The Science of the Unknowable:
Stafford Beer's cybernetic informatics, 2006, Fig 4]
|Cross-section of the 27km circular tunnel of the
Large Hadron Collider
|The diagram which merits display here derives from a document by Beer (1962). Reproduction of the diagram is however indicated as subject to the permission of John Wiley and Sons and Allenna Leonard. Similarly, although the CERN device has been developed entirely with taxpayer funds, that image is also subject to copyright. Such antiquated constraints on the rapid dissemination of ideas for non-commercial purposes is indicative of a more fundamental problem in interrelating any insights of potential value to global governance (as discussed in the Conclusion). Given their common focus on the integrative power of rings, such dysfunctional possessiveness may be well-caricatured by the role of the Gollum in The Lord of the Rings.|
The cognitive challenge, and its implication for more appropriate psychosocial organization, might be said to be a legitimate preoccupation of mainstream behavioural approaches -- to the extent that the psychological dimension is considered to be of any significance. Such approaches might however be seen as insensitive, if not hostile, to the degree of subjectivity implied by existential and experiential approaches. It is questionable whether such dimensions were considered meaningful by Fuller.
The global strategic challenge, and the question of a collective political will to change, would seem however to lie beyond the realm of behavioural psychology, notably as it is deployed to "motivate" people in conventional marketing of products and services, including the promotion of global social change strategies. The missing factor is a degree of existential engagement, whatever that may be held to be (Psychology of Sustainability Embodying: cyclic environmental processes, 2002).
The "geometry of thinking" might be understood as one approach to a "pattern language". But, as with other languages, a vital question is how people experience their identity as expressed and carried by any language. The exercise mentioned above (Patterns of N-foldness, 1984) presents various pattern languages. That of Christopher Alexander (A Pattern Language, 1977) has constituted an exemplar for other approaches to pattern language and experimental adaptations of it (5-fold Pattern Language, 1984).
As a metaphor, however, the point might be made that the visible stars have long been configured into patterns of named constellations with which legendary meanings have been associated. Such interpretations are however meaningless from an astronomical or astrophysical perspective. The patterns of constellations might be said to be as meaningless as the process of "joining up the dots" which led the intelligence communities to assert the existence of weapons of mass destruction in Iraq.
Crucially Alexander moves beyond the patterns he identifies, framing their significance in a new way, in an interpretive volume (The Timeless Way of Building, 1979):
There is a central quality which is the root criterion of life and spirit in a man, a town, a building, or a wilderness. This quality is objective and precise, but it cannot be named,
The search, which we make for this quality, in our own lives, is the central search of any person, and the crux of any individual person's story. It is the search for those moments and situations when we are most alive.
In order to define this quality in buildings and in towns, we must begin by understanding that every place is given its character by certain patterns of events that keep on happening there.
These patterns of events are always interlocked with certain geometric patterns in the space. Indeed, as we shall see, each building and each town is ultimately made out of these patterns in the space, and out of nothing else: they are the atoms and the molecules from which a building or a town is made.
The specific patterns out of which a building or a town is made may be alive or dead. To the extent they are alive, they let our inner forces loose, and set us free; but when they are dead, they keep us locked in inner conflict.
The more living patterns there are in a place -- a room, a building, or a town -- the more it comes to life as an entirety, the more it glows, the more it has that self-maintaining fire which is the quality without a name.
As he notes, the patterns are always interlocked with certain geometric patterns. Beyond human construction, Alexander has also explored these patterns in nature (Christopher Alexander, The Nature of Order, 2004). Arguably it is through this existential engagement with the geometric patterns that any meaning associated with globality emerges, on whatever scale. It is specifically this factor that is absent, or ineffectually present, in current efforts to engage people with global strategies, however urgent they are claimed to be -- as with climate change.
Although it is not the case, the term "pattern" is readily assumed to imply some configuration that can be meaningfully represented in two dimensions. Of far greater interest is the possibility of patterns in three or more dimensions, as suggested by the elaboration of Fuller's "geometry of thinking". A more appropriate term is then topology, especially because of the psychological significance of "topos" as place, as notably articulated by Frances Yates (The Art of Memory, 1966) with respect to the mnemotechnical method of loci. This is also fundamental to the philosophy of Martin Heidegger as described by J. E. Malpas (Heidegger's topology: being, place, world, 2006):
Almost all of Heidegger's language... is "iridescent" in the sense of constantly shining and showing different facets. The attempt to delineate the topology of thinking and the topology of being will always carry a certain "iridescence" of this sort. It is an iridescence that may also be compared to the "backwards or forwards" relatedness that is found in hermeneutical thinking and that is also tied to the topological. It is the reflection of the iridescent, the multiple, shifting character of place as such. (p. 37)
The configuration of any pattern in three dimensions then offers a new kind of template for identity and the dynamics of its engagement with encompassing cognitive reality, and any embodiment of it (Topology of Valuing: psychodynamics of collective engagement with polyhedral value configurations, 2008). It is through the potential of such topological configurations that as yet unexplored possibilities are to be found (In Quest of a Strategic Pattern Language: a new architecture of values, 2008). Being more generic, topology includes forms that are more paradoxical and less intuitively obvious, as already explored by some schools of psychoanalysis (Jacques Lacan, R. D. Laing). The simpler examples of great potential significance are the Möbius strip and the Klein bottle, as discussed elsewhere with respect to self-reflexivity (Intercourse with Globality through Enacting a Klein bottle: cognitive implication in a polysensorial "lens", 2009). Jacques Lacan and his followers, informed by topology and knot theory, focused on the significance of the manner in which the surface of the Klein bottle appears to be intersected by a portion of itself.
Again the argument here is that, whether in Fuller's work or those of other initiatives of comprehensive ambition, the challenge is one of "decoding" what is clearly there. But the challenge, as with a Rosetta stone, is that the code is effectively "lost" because it lies "beneath" or "within" the geometry that is so visible. In Maruyama's terms the existential significance has been "subunderstood". A degree of cognitive closure has set in through the manner in which geometry is conventionally apprehended. However, in principle at least, Fuller's work is not about closure but about degrees of openness -- which the satisfaction with premature cognitive closure has obscured.
Such openness is evident in the tensegrity structures and principles which underlie the viability of geodesic domes. The question is whether those principles are of relevance to the design of a new order of psychosocial organizations and strategies capable of configuring a higher degree of incommensurability (Consciously Self-reflexive Global Initiatives: Renaissance zones, complex adaptive systems, and third order organizations, 2007).
A quite distinct approach to cognitive geometry may be found in the patterns in sacred architecture, notably as assiduously explored by Keith Critchlow (Order in Space: a design source book, 1969; Islamic Patterns, 1976; Islamic Art and Architecture: system of geometric design, 1999). Of relevance is the contrast between the Western attitude (and its concentration on the external look of a building) and the traditional Islamic concern with the feel of space within. As argued by Geeti Sett (Interpretation of an exhibition entitled Kham Space and the Art of Space, New Delhi, 1986):
The geometry of architecture is clearly based on an elaborate symbolism. In Islam, the dome, the minaret, the arch, the quindi vaults and pendentives do not stand for themselves, as invented forms. based at times on engineering feats; but for another, supra-reality which cannot be depicted in any other way but pure geometry. The dome for instance, is associated with the Spirit which pervades all beings, as indeed the vault of the sky embraces its enclosed space. The dome unifies space and encompasses it. The arch expresses the human soul, repeated ad infinitum soaring and aspiring towards the heavens. The iwan, the arched doorway or corridor, is viewed as the locus of the soul, moving between the room, seen as the body, and the garden or courtyard which is taken as the spirit.
The nature of what tends to be lost in conventional abstraction -- in the geometry on its own -- has been highlighted by Steven M. Rosen (Topologies of the Flesh: a multidimensional exploration of the lifeworld, 2006; Dimensions of Apeiron: a topological phenomenology of space, time, and individuation, Value Inquiry Book Series, 2004). He notes the manner in which the richness of psychosocial engagement with the world has been completely undermined by formal discourse -- an "eclipse of the lifeworld" in his terms. Ironically, in a period of sensitivity to the challenges of "resources" and "energy", this view is echoed by other authors, notably arguing for the desirable potential of a participatory encounter with reality:
It is curious that there is widespread recognition of the function of aerial arrays. especially radio telescopes, to resolve distinct signals into a coherent image or message. The metaphor has been developed to detect cognitively "distant" or "larger" systems by Joël de Rosnay (The Macroscope: a book on the systems approach, 1979) who subsequently prefaced a sequel by Luc de Brabandère (Le Latéroscope: systemes et creativite, 1989). The latter noted that, as the basic tool of creativity, the latéroscope is necessarily impossible to construct. Such endeavours raise the question of the nature of the cognitive instrument by which meaning can be reflected. The suggestion is that like the telescope it has a particular geometry, but that its mirroring capacity has a "magical" quality to it, perhaps as in many mirrors of legend. The instrumental nature of the geometry is necessarily complemented and reframed by metaphor, most probably an array of complementary metaphors.
Metaphor: There is now a considerable body of literature regarding the cognitive role of metaphor, notably with respect to policy formation. The question is what are the productive metaphors, or sets of complementary metaphors, that might be a key to a new geometry of global governance (Metaphors as Transdisciplinary Vehicles of the Future, 1991). More challengingly, with respect to any enterprise as noted above by Bateson: We are our own metaphor.
With respect to comprehension through personification, Kenneth Boulding (Ecodynamics; a new theory of societal evolution, 1978) helpfully cautions against rejecting such metaphors in the following terms:
Our consciousness of the unity of the self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of a group, organization, department, discipline, or science. If personification is only a metaphor, let us not despise metaphors -- we might be one ourselves. (p.345)
More of a challenge is the extent (whether individually or collectively) to which cognitively one may become (in any integrative sense) a caricature of oneself, a mockery of oneself, or a shadow of one's former self -- or that a civilization may turn into a parody of what it believes it coherently represents. This is typically echoed in political satire and more concretely in the burning of an effigy by opponents.
Magic: The last of the three laws of Arthur Clarke suggests that: Any sufficiently advanced technology is indistinguishable from magic. However the "technologies" of the future are as likely to be cognitive as what is now conventionally recognized as technology -- as already implied by Robert Romanyshyn (Technology as Symptom and Dream, 1989). What might be the nature of the cognitive technology that would constitute a "magical" breakthrough in global governance -- given the optimistic dependence on future human ingenuity to circumvent foreseen and unforeseen crises (Thomas Homer-Dixon, The Ingenuity Gap: how can we solve the problems of the future? 2000).
"We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough."
To that Freeman Dyson added: "When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!" (Innovation in Physics, Scientific American, 199, 3, September 1958)
Decades later string theory, as the amazing new "craziness", is now the great hope for a unifying theory of the material universe. It endeavours to reconcile quantum mechanics with relativity by arguing that subatomic particles are strings vibrating through space and time, differing merely in the ways in which they vibrate -- through 10 or 11 dimensions (as in M-theory). This possibility has progressed so far beyond experimental provability that the only guide to the adequacy of a solution is its unifying beauty and elegance.
What might this imply for a form of global governance appropriate to the challenge of the 21st century? Pointers towards understanding this challenge are discussed elsewhere (Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007; In Quest of Mnemonic Catalysts for comprehension of complex psychosocial dynamics, 2007).
The extreme forms of symmetry (represented above), detected by mathematicians and claimed to be of fundamental significance, emerge from an extraordinary appreciation of unusual connectivity, which they themselves have nicknamed as "moonshine" (Marcus du Sautoy, Finding Moonshine: a mathematician's journey through symmetry, 2008). The question might be the challenging nature of the "moonshine" that would be of value to future understandings of global governance (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).
Geometry itself has been framed as "magic" (Siobhan Roberts, The King of Infinite Space: Donald Coxeter and the Magic of Geometry, 2006).There are indeed many references to the "magic" of Fuller, most typically from an architectural perspective (R J Evans, Geodesic Magic: There's No Place Like Dome, Architecture, July 2009). The thinking of Fuller has been related to the "magic mirror" of M. C. Escher (Victor Acevedo, Space Time with M. C. Escher and R. Buckminster Fuller, 2003). In describing the "magic" of the operation of the open source community, Eric S. Raymond (The Magic Cauldron, 1999) acknowledges the relationship with "ephemeralisation" (doing more with less) through the knowledge of synergy promoted by Fuller.
"Magic" is of cognitive significance in another respect, namely through the "magic numbers" that are the focus of extensive literature. Their relation to the geometry of thinking is discussed in Fuller's Synergetics (995.00: Vector Models of Magic Numbers, 1997). Of particular interest are the patterns of association of numbers characteristic of magic squares, notably recognized as fundamental to sacred geometry in various traditions (9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes, 2003). Fascination with such patterns is evident in the widespread interest in sudoku. Such "magic" raises the question of the relevance of such connectivity to governance (Governance through Patterning Language: creative cognitive engagement contrasted with abdication of responsibility, 2006).
Of particular relevance to such tenuous connectivity and coherence are various understandings of correspondence (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). Curiously some of these recall the cognitive implications of traditional understandings of the cognitive role of the image, whether in magic (Magic, Miracles and Image-building: Poetry-making and Policy-making, 1993), or as highlighted by Kenneth Boulding (The Image: knowledge in life and society, 1956). The role of such correspondences in relation to traditional cognitive engagement with the environment has been extensively documented by Darrell A. Posey (Cultural and Spiritual Values of Biodiversity: a complementary contribution to Global Biodiversity Assessment, 1999).
Mirror: The paradoxical nature of the cognitive challenge has been extensively explored by Douglas Hofstadter (Gödel, Escher, Bach: an Eternal Golden Braid, 1979; Metamagical Themas, 1985; I Am a Strange Loop, 2007).
A mirror has been used metaphorically in many cultures to facilitate comprehension of that challenge. Use of magical mirrors for scrying and "speculation" has been traced back to Euclid as the first geometer. It was the subject of a papal bull (Super Illius Specula, 1326) threatening excommunication of those confining demons in mirrors so as to foretell the future. Brian Dillon (Disc World, The Guardian, 24 October 2009) comments on the fascination for artist and magus of the black mirror as being actually and allegorically a portal between arcane practices and mainstream painting and sculpture. The use of mirrors, notably in the form of a jade bi disc, has been associated with the representation of cosmological principles in Chinese culture (Robert Dickter, The Ya Shape and: Bronze Mirrors, The Ming Tang, The Jade Bi Disc, The Mayan Calendar In: Number, Time and Archetype).
Of particular interest to any global strategy is the extent to which the patterns of order in nature (Alexander, The Nature of Order, 2004) themselves constitute a mirror (Stepping into, or through, the Mirror: embodying alternative scenario patterns, 2008; Self-reflective Embodiment of Transdisciplinary Integration (SETI): the universal criterion of species maturity? 2008). The "cognitively transformative" interaction with a mirror might be usefully understood through a conflation, "con-fusion" or "con-volution" of the implication of a simple mirror reflection, a torus, a Mobius strip and a Klein bottle (Engaging with Globality through Knowing Thyself, 2009). Rather than any form of "revolution", the new paradigm so widely sought may in fact require a paradoxical degree of cognitive "convolution". As with looking into any mirror, what is then seen may be terrifying -- readily to be condemned from a conventional papal perspective, as with the non-geocentric model of the universe (Thinking in Terror: refocusing the interreligious challenge from "Thinking after Terror", 2005).
The cognitive implications of the hole in the bi disc (with its two sides to be understood as cognitively twisted together) is reminiscent of the Eygyptian all-seeing Eye of Horus, as well the distorted projection of the two-sided hole into the traditional two-dimensional symbol of the Tao, as discussed previously (Snoring of The Other a politically relevant psycho-spiritual metaphor? 2006).
Curiously the hypothesized dodecahedral form of the space-time universe has an inherent mirroring property as described by Jean-Pierre Luminet (A Cosmic Hall of Mirrors, Physics World, September 2005; A Finite Dodecahedral Universe, Nature, 9 October 2003; Mirror, Mirror up above, Unesco Courier, May 2001). Potentially significant for any "cognitive universe", light leaving that dodecahedron through a given face immediately re-enters through the opposite face, such that any observer whose line-of-sight intercepts one face has the illusion of seeing a slightly rotated copy of their own dodecahedron. Luminet asks whether we could be living deep in a cosmic mirage, where rays of light multiply and distort our perceptions of space. Instead of being flat and infinite, might space not in fact be folded up -- and our sense of the universe's vastness just an illusion? How might this apply to any "universe of knowledge"?
A prudent approach suggests that the pattern of understanding held to be most appropriate to the unification of the material universe should be considered of greater probability with regard to comprehension of the unification of the intangible universe. Adapting the above statement with respect to string theory, the future unifying theory of psychosocial relations might endeavour to reconcile the cognitive engagement with locality and with globality by arguing that the disparate cognitive senses of individual identity are strings vibrating through cultural space and comprehension time, differing merely in the ways in which they vibrate -- through 10 or 11 dimensions.
It would be an amazing irony to discover a correspondence (as understood above) between the cognitive capacity to envisage a unifying physical theory (through the variety of string theories) and the capacity to imagine various representations of individual variety and integrity (with their different mappings, whether traditional or modern). The ultimate provocation to conventional thinking would be any recognition that the traditional horoscope constituted a crude intuitive 2-dimensional mapping of some form of 12-dimensional integrity intimately associated with comprehension of number theory -- however this may be understood in the future (Representation, Comprehension and Communication of Sets: the role of number, 1978).
Any cognitive unification would also require integration of the sense of identity with intellectual property and the competitive derivation of energy from it -- of continuing fundamental concern to the institutionalization of synergetics, syntegrity and analogous endeavours (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). A pattern corresponding to Wolfgang Pauli's Exclusion Principle is suggested by the challenge of epistemological antipathy. Also of interest is the extent to which the reflective, refractive and mirroring properties of the facet configuration of precious stones -- typically associated with "magic" -- offers a mirror indicative of the geometry of possible unification (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002).
However, just as the guide to the adequacy of a string theory is to be found primarily in its aesthetic elegance, that of any emergent theory of psychosocial relations in "10 or 11 dimensions" may also depend on its aesthetics and symmetry. It is in this sense that music, poetry and song offer a form of cognitive mirror, as argued by Gregory Bateson (repeating the quote above):
One reason why poetry is important for finding out about the world is because in poetry a set of relationships get mapped onto a level of diversity in us that we don't ordinarily have access to. We bring it out in poetry. We can give to each other in poetry the access to a set of relationships in the other person and in the world that we're not usually conscious of in ourselves. So we need poetry as knowledge about the world and about ourselves, because of this mapping from complexity to complexity.
Given the widespread engagement with aesthetic resonances (and an explicit recognition of "vibrations" and "vibes"), the challenge is how to engage that understanding to reframe global governance and the appreciation of the global problematique as previously discussed (A Singable Earth Charter, EU Constitution or Global Ethic? 2006; Poetry-making and Policy-making: arranging a marriage between Beauty and the Beast, 1993).
This has the potential of reframing current strategic challenges (Poetic Engagement with Afghanistan, Caucasus and Iran: an unexplored strategic opportunity? 2009; Conversion of Global Hot Air Emissions to Music: aesthetic transformation and instrumentalization of vaporware, 2009).
There is a curious mirroring, potentially of great significance, between the worldwide (popular) fascination with fooball and the dramatic global challenges of the future with which the world is believed (by an elite minority) to be faced. This is evident in the tragic complementarity between:
The cognitive entanglement between a "micro ball" (within a "macro context") and the "micro treatment" (of that macro global context) may be explored in terms of geometric commonalities:
In a period of intense concern with regard to climate change, a further geometric association may be added, namely the discovery in 1985 of buckminsterfullerene (C-60 or buckyball). This is an unusual spherical molecule composed entirely of carbon -- resembling a geodesic dome in appearance, and hence the name. It is the smallest fullerene molecule in which no two pentagons share an edge -- the most common fullerene in terms of natural occurrence, as it can often be found in soot. Most ironically, the chemists researching it had to be informed by mathematicians that what they were looking for was indeed commonly represented by the stitching of leather patches on a soccer ball.
As noted above, one of the shapes hypothesized for the universe is a Poincaré dodecahedral space, a positively curved space, colloquially described as "soccer ball shaped" (Jean-Pierre Luminet, A Finite Dodecahedral Universe, Nature, 2003; John Whitfield, Universe could be football-shaped, Nature, 2003). What might this imply for any "cognitive universe"? Perhaps, with apologies to William Blake (Auguries of Innocence):
To see the universe in a football game and heaven
in a ball ?
Hold infinity in awareness now and eternity in a mall ?
|What might be the cognitive analogues of these forms of organization?|
(and soccer ball)
(reproduced from Wikipedia;
click for rotating version)
|Dymaxion map or Fuller map
unfolding to an icosahedron
animation created by Chris Rywalt
C60 with isosurface of ground state
(reproduced from Wikipedia)
Geometrically the buckyball and the football are more complex than those structures discussed above (such as the icosahedron), where there is a modicum of attention to their psychosocial significance. As previously explored in relation to dialogue (Understanding Sustainable Dialogue: the Secret within Bucky's Ball? 1996).
Maybe the "ball" that gets kicked around between positions in any dialogue -- the topic -- needs to be understood as more complex. It is not just a "point". It is only this complexity that accounts for its resilience and ability to bounce back. Who would want a dialogue with a "ball" that did not bounce, but just stuck where it landed? Would a dialogue then be possible? What is the difference between games with a bouncing and a non-bouncing "ball"? C-60 is remarkable for its resilience. Like a manufactured football, it is a satisfactory approximation to a perfect sphere.
When we observe a soccer game, the ball moves unambiguously all over the field -- shifted onwards, or blocked and deflected, by players at different positions. Runs and plays can be watched and remembered as a whole -- but even then with some difficulty. For the game as a whole, the moves of the ball -- and the corresponding positions -- could only be remembered by those with photographic memory, or the mnemonic skills of a particular form of autism. The ball is explicit, whilst the context -- the whole within which it is set -- is implicit. We all have difficulty in understanding what is going on in a dialogue. How come it took people so long to understand the significance of C-60, having been exposed so long to footballs? What is so well known to us in the psychosocial realm that is of similar significance?
In subsequent consideration of engagement with globality (Engaging
with Globality through Cognitive Circlets, 2009) reference was made
to how astounding it was that greater sophistication is applied
to the analysis of patterns of interaction in various sports -- passing
patterns -- than is applied to the patterns of dialogue at vital strategic
gatherings (Mark Weston, Passing
Patterns, 2006; Athalie Redwood-Brown, Passing
patterns before and after goal scoring in FA Premier League Soccer, International
Journal of Performance Analysis in Sport, 2008; Association for Soccer
Education and Teaching, Passing
Patterns and Small Sided Games, 2008; Alan Reifman, Network
Analysis of Basketball Passing Patterns II, 2006; ).
The approach has been adapted to message passing in complex organizational
networks. With respect to global dialogue, the situation is all the more
curious given the widespread metaphoric use of "ball" in strategic dialogue
-- as in the "ball
is in their court".
Given the constant pressure for higher quality performance in such contexts, how is the quality of dialogue regarding the current crisis to be evaluated? The movement of a ball, or a set of clubs, has long been tracked and visualized to improve performance (Daniel A. Reed, Matthew J. Gardner and Evgenia Smirni, Performance Visualization: 2-D, 3-D, and Beyond). How is it that no effort is made to do so for strategic dialogue -- especially widely publicized panel sessions? (Future Generation through Global Conversation: in quest of collective well-being through conversation in the present moment, 1997).
The question here is whether there is a degree of credible "moonshine" connectivity possible between the universal engagement with football, its well-recognized geometric form, and its potential as a patterning template for new forms of global psychosocial dialogue and organization -- beyond current conventions? Might the panels of stitching of a football be fruitfully understood as indicative of the desirable configurations of the panels in a global conference (Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998)? Might the viability of sustainable global governance depend on the credibility of isomorphism with the familiar patterns of of a football?
Is the very dynamic of the game essential to the viability of such an emergent structure (Playfully Changing the Prevailing Climate of Opinion: climate change as focal metaphor of effective global governance, 2005)? Perhaps as a form of resonance hybrid (Configuration of Modes as a Resonance Hybrid, 1986; Patterns of Alternation: cycles of dissonance and resonance, 1995)?
More challenging is the paradoxically mirrored cognitive relationship between the (micro) football and the (macro) globe. Is there a form of cognitive inversion associated with that mirroring, perhaps usefully to be captured by the Mobius strip as previously discussed (Psychosocial Energy from Polarization within a Cyclic Pattern of Enantiodromia, 2007)? Might football be as important as climate change to global governance?
There is the possibility that buckminsterfullerene may indeed be of requisite cognitive complexity such that the panels of any corresponding football could be used to explore in a learning environment the configuration of global issues or those of climate change -- labelling each panel with one topic. Footballs could even be distributed with such topic maps as a guide to multi-panel conferences. This would be something tangible for delegates to being home from conferences to their family and friends. However it is also possible that such a configuration might be fundamental to any viable global arrangement transcending conventional patterns of agreement and disagreement. In this sense a football would be a vital means of rendering comprehensible its underlying complexity. As remarked by Chris Lucas: An interesting analogy in terms of a labeled football, is that if any panel is eliminated the ball goes flat, if any are seriously changed in size the distortion of the ball makes it impossible to play with.
The explicit or implicit antipathy between those undertaking any consideration of the "geometry of thinking", or any cognitive Theory of Everything, necessitates the search for frameworks of greater complexity to encompass such divergent perspectives. Astrophysics may offer pointers (Towards an Astrophysics of the Knowledge Universe: from astronautics to noonautics? 2006). Are the cognitive wormholes -- through which each is able to "wriggle" from one "universal framework" to another -- configured in some special way, perhaps forming a polyhedral cognitive metaverse. The notion of a metaverse has been explored from an artificial intelligence perspective by Ben Goertzel and also in science fiction. Given the efficiencies of a polyhedral organization of computer (memory) systems, what might then be the appropriate geometry of any future global brain? How might its emergence be comprehended, given the challenge of Atkin?
Ass argued above, however, a "cognitive inverse" may also be implied through mirroring. Hence the cognitive value of the Stella Polyhedron Navigator software developed by Robert Webb, notably Stella4D which enables exploration of 4D polytopes -- with 4D models projected into 3D, offering visualizations of their 3D cross-sections and nets in real-time. As in Stella4D, Chris Lucas also uses the partial mirroring of stereoscopic images to enable a sense of four dimensions (Four Dimensional Strange Attractors in Stereogram form, 2002)
Both in the case of Fuller's innovations and those of Beer, it is much to be regretted the extent to which these remain subject to constraints of intellectual copyright -- inconsistent with their purported global significance in a period of global crisis. Such constraints might be understood as an appropriate metaphor of the flaws inhibiting effective global initiatives (Future Coping Strategies: beyond the constraints of proprietary metaphors, 1993).
It is perhaps to be expected that every comprehensive initiative should have a central flaw -- fundamental in some aesthetic schemes to its appropriateness. As discussed with respect to lipostrategies the challenge is how to incorporate this constraint into a tragic poetic epic, a civilizational magnum opus, as a commemorative mnemonic for the future (Lipoproblems: Developing a Strategy Omitting a Key Problem -- the systemic challenge of climate change and resource issues, 2009). This might offer a framing corresponding to that of the mysterious wound of the Fisher King in the epic poem Parzival (cf Richard Sanderson, Wounded Masculinity: Parsifal and The Fisher King Wound) which inspired the opera Parisfal by Richard Wagner.
In the case of Buckminster Fuller, despite his uniquely global engagement, one aspect of this tragic flaw is the manner in which his inventions of potentially fundamental significance are subject to copyright -- inhibiting their development by others, contrary to the spirit through which resources are shared in an open source community. As with others discovering geometric applications of potential value to remedial global strategies, this might be held to be the collective equivalent of "shooting oneself in the foot", or shooting an own goal in a football metaphor -- or friendly fire in military terms.
As with the Fisher King, Fuller's "wound" may be essentially a topological "subunderstanding" resulting from failure to appropriately embody the four-dimensionality exemplified by the paradox of the Klein bottle -- a perspective on knots notably explored in Lacanaian psychoanalysis. This has been a theme of the recent work of Melanie Purcell (Towards A New E.R.A.: Epistemological Resolution Analysis By Through and From Klein-Bottle Wholeness and Transdisciplinary Education) who explicitly recognizes the inadequacy of platonic geometry and the need for higher-dimensional topologies. In this sense the seeming flaw in in any comprhensive "geometry of thinking" is a consequence of inappropriate self-reference (even "self" destructive) which is corrected in an appropriate "topology of thinking".
Curiously the dissemination of most key insights relevant to the challenges of global civilization is already severely constrained by copyright. Is the pattern such that global civilization will be held to ransom by a future inventor of the ultimate remedial technology for its ills? Is distribution of the swine flu vaccine in 2009 an indication of that model? Will some geo-engineering technology follow that pattern? This mindset may be the key to the question posed by Jared M. Diamond (Collapse: how societies choose to fail or succeed, 2005).
A similar phenomenon is evident in the probability that the most legally binding constraints to emerge from the United Nations Climate Change Conference (Copenhagen, 2009) may well be those relating to wider use of any documents presented to it to clarify the global challenge -- and, perhaps most significantly, that governing the copyright of the symbol of the Copenhagen event. In contrast to other such UN symbols, this is a valuable effort to reflect that challenge in spherical geometry (reminiscent of Fuller's own endeavours).
representation of section headings of this document (indicative)
(made with Stella Polyhedron Navigator)
In both images, the 18 section headings are on the square faces of a rhombicuboctahedron
set within the framework of its dual, a deltoidal icositetrahedron (with the section
headings on its corresponding vertices). The great circles of both structures are individually coloured.
(click each image for enlarged versions; gif animation of rotation not included)
|The attribution of section headings to faces and vertices is of course arbitrary, suggesting contiguity adjustments (as with Rubiki's Cube). It is however indicative of a possible future approach to the introduction of greater cognitive coherence into the organization of a linear text. The software used permits the addition of many other geometrical features (symmetry axes, reflection planes) as well various possibilities of morphing the configuration into its dual or unfolding the polyhedron into a net. In interactive use, or by generation of animations, a more dynamic overview is necessarily obtained.|
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