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Beyond dispute in 5-dimensional space: Pentagramma Mirificum ?
Correlating a Requisite Diversity of Metaphorical Patterns (Part #9)
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The cybenetician Stafford Beer is known for his application of such insights to new patterns of dialogue (Beyond Dispute: the invention of team syntegrity, 1994). These processes were organized in terms of the 3-dimensional structure of the icosahedron, notably characterized, by 15 great circles. Given the questions raised by "dis-" in relation to "ease", a degree of closure to the pattern of correspondences of this argument can be explored through recalling the etymology of "dispute" -- from the Latin dis-"separately" with putare"to count or consider".
Wu Xing: A speculative closure to the pattern of 5 correspondences tentatively correlated in this discussion could be consistent with the arguments of A. C. Graham (Yin-Yang and the Nature of Correlative Thinking, 1986). Given the above-mentioned conventional recognition that a region's climate is generated by a climate system of five components (atmosphere, hydrosphere, cryosphere, lithosphere, and biosphere), these can be associated through a circular pattern fundamental to Chinese thinking, namely the dynamics of 5-phase Wu Xing.
The trigrams of BaGua are related to the five elements of Wu Xing, used by feng shui practitioners and in traditional Chinese medicine. The elements correspond with the trigrams: water and fire directly; earth corresponds with two (earth, mountain), wood with two (wind, thunder), and metal with two (heaven, lake). Hence the implied relationship to the pattern of I Ching hexagrams -- suggesting the possibility of a "collapsed" variant of the latter.
Curiously this pattern is very similar to a diagram fundamental to the Pythagoreans. The former continues to be valued with respect to understanding of health, the latter is at the origin of the understanding of hygiene, as may be variously discussed (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012; Memorable dynamics of living and dying: Hygeia and Wu Xing, 2014).
The pattern is now being explored from a mathematical perspective in China (Ziqing Zhang and Yingshan Zhang, Mathematical Reasoning of Economic Intervening Principle Based on "Yin Yang Wu Xing" Theory in Traditional Chinese Economics. Modern Economy, 2013; Yingshan Zhang and W. Shao. Image Mathematics: mathematical intervening principle based on "Yin Yang Wu Xing" theory in traditional Chinese mathematics, Applied Mathematics, 2012; Guang-hong Ding and Tao WanG, Mathematical analysis of Yin-Yang Wu-Xing Model in TCM, Journal of Acupuncture and Tuina Science, 2008; Zhaoxue Chen, Researches on Mathematical Relationship of Five Elements of Containing Notes and Fibonacci Sequence Modulo 5. The Scientific World Journal, 2014).
| Correspondence between 5-fold patterns from contrasting cultures | |
| Chinese 5-phase Wu Xing cycle (Image adapted from Wikipedia) | Hugieia Pentagram of Pythagoreans (Image adapted from Wikipedia) |
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The pattern can be speculatively used to order current understanding of a climate system. It is closely associated with the requisite attitudes identified in the organization of the famed Japanese strategic study The Book of Five Rings.
| Speculative ordering of current understanding of a climate system | |
| Speculative use of Wu Xing cycle to order climate | Book of Five Rings arrayed according to Wu Xing |
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The original use of rings in this depiction, and with respect to strategic attitudes, invites reflection on its relevance to the adequacy of comprehension of any system of climate. The question acquires greater focus through the symbolic value associated worldwide with the Olympic Games
| Symbol of Olympic Games |
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| Author: Pierre de Coubertin (public domain image via Wikimedia Commons) |
"Sparsity" of the "pattern that connects": The juxtaposition of the above patterns, with the suggestion that they are indicative of an underlying pattern, raises the question as to how disparate and "disconnected" can the elements of the "pattern that connects" appear to be in correlative thinking, as suggested by research on structured sparsity (Junzhou Huang, Tong Zhang, et al, Learning with Structured Sparsity, Journal of Machine Learning Research, 2011).
This returns to the indications of small world theory of six degrees of separation, cited in the introduction. Whereas connection to any "other" may indeed be possible through six intermediaries -- as with the problems of society -- is this also the case amongst seemingly disparate categories characteristic of requisite variety in cybernetic terms? How unrelated may distant intermediaries be credibly asserted to be? A corollary to that theory might then call for recognition that there are 5 (or 6) degrees of "cognitive separation" from an experiential sense of globality, namely its comprehension -- despite the expressed belief that "everything is connected to everything".
The challenge to comprehension of the coherence of global patterns -- aided by "path-ology" -- may lie in the confusing "subliminal" intuitions regarding the elusive nature of that complexity, so dimly apprehended. This challenge is clarified to a degree by the mathematics of Ron Atkin (Multidimensional Man: can man live in 3-dimensional space? 1981) and by the arguments of Magoroh Maruyama (Peripheral Vision: polyocular vision or subunderstanding? Organization Studies, 2004).
Is weather a valuable key to engaging with patterns of a higher order (Engaging with Insight of a Higher Order: reconciling complexity and simplexity through memorable metaphor, 2014)?
Embedding in higher dimensional space: In commemoration of the tragic death of Nobel Laureate John Nash at the time of writing, further speculation is justified in the light of his work on the mathematics of isometric imbedding, This resulted in the discovery that any surface can be embedded into 5-dimensional space (Daniel Mathews, Every world in a grain of sand: John Nash's astonishing geometry, The Conversation, 27 May 2015; Ben Andrews, Notes on the Isometric Embedding Problem and the Nash-Moser Implicit Function). For John Conway: Nash's result is one of the most important pieces of mathematical analysis in the 20th century.
Might this have implications for the "superficial" cognitive possibilities by which the above 5-fold speculations are indicated? Of relevance to this argument is the considerable psychological unease from which Nash suffered for many decades -- possibly not unrelated to his remarkable innovations in game theory and his later quest for more collaborative games.
Of particular relevance is the comment of the physicist Gerard 't Hooft with regard to a holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. Such "curvature" is clarified in a discussion of Is the universe 5 dimensional space-time or 4? (Carlos Romero, et al, The Embedding of General Relativity in Five Dimensions. General Relativity and Gravitation, 1996). How might this be recognized in efforts to comprehend the coherence of globality?
Pentagramma Mirificum: Potentially relevant to such considerations is the unusual 5-fold structure known as the Pentagramma Mirificum, so named by the mathematician Carl Friedrich Gauss -- after its discovery by John Napier in exploring the hypergeometric origin of spherical action (deriving what are termed Napier's rules for right spherical triangles). Gauss later showed how the spherical pentagramma, and its projection, are both artifacts of an elliptical function, which itself is an artifact of a superseding hypergeometry. The argument relating to this pattern is developed separately in further detail with animations (Annex 5: Global Psychosocial Implication in the Pentagramma Mirificum, 2015)
Related details and epistemological implications of non-Euclidean geometry are articulated by various authors (Joel Silverberg, Napier's Rules of Circular Parts, 2008; Fragmentary Notes from Carl F. Gauss' Work on the Pentagramma; Bruce Director, From Plato's Theaetetus to Gauss's Pentagramma Mirificum: a fight for truth, EIR, 7 October 2005; Pentagramma Mirificum: investigations in geometry, ScienceLarouchePAC.com). The latter includes many helpful schematics and animations. As noted by Bruce Director ( Riemann for Anti-Dummies: Part 65 : On the 375th Anniversary of Kepler, LYM Canada, 16 November 2005)
Gauss was intrigued by the relationship of Napier's spherical pentagram to his elliptical transcendentals. He realized that Napier's pentagramma mirificum established that a spherical surface had an intrinsic five-fold periodicity.
Recognizing metaphorical self-closure: The point to be stressed is that this unusual form is generated by 5 great circles. It derives from the most fundamental characteristic of spherical geometry, the "great circle-pole" relation, making of it a reflection of spherical geometry itself. It thereby makes evident a 5-fold periodicity as being intrinsic to the sphere and to globality.
| Pentagramma Mirificum -- a non-regular spherical pentagon | ||
| As sketched by John Napier | Gauss's sketch of Napier's | Adaptation to Wu Xing pattern |
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Napier's diagram indicates the unusual self-closure of a chain of 5 spherical right triangles to form a non-regular spherical pentagon. The spherical pentagon is self-polar, which means that each vertex is the pole of the opposite side. The form offers an insight into globality in non-Euclidean terms, which otherwise remain even more elusive.
There is a degree of irony to the sense in which a sequence of perspectives, each upheld as "upright" in contrast to that preceding it, suggests that the progression brings one back to the point of departure -- but seen from a "humbler" angle. This recalls the insight of T. S. Eliot:
| We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know it for the first time. (Little Gidding, 1942) |
Comprehension of globality through 5-fold patterns of metaphor? Exploration of the possible relationship between the pattern of Wu Xing, Hygeia and the Pentagramma Mirificum can be usefully related to that of the non-sequiturs of so-called Knight's Move thinking. As a move in chess, with its correspondence in go, it offers a pattern for metaphor as being orthogonal to conventional linearity. The Wikipedia entry on orthogonality offers a valuable summary of the challenges this constitutes to communication in various domains. For example: In communications, multiple-access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. Metaphor enables this conventional constraint on connectivity to be transcended -- notably that between information silos. Ironically it might be said, following the argument of the introduction with respect to birds, metaphors enable arguments to "fly". The point is carefully argued by Douglas Hofstadter and Emmanuel Sander (Surfaces and Essences: analogy as the fuel and fire of thinking, 2013).
Appropriately Knight's move thinking is both valued as such for strategic creativity and deprecated as a thought disorder -- a pathology of disconnected thinking, notably categorized in DSM (Knight's move thinking: appreciated or deprecated, 2012). Consequently the Knight's move can be considered as a valuable indication of use of metaphor in reframing linearity in complex systems. As a feature of spherical geometry, the self-closing sequence of right angles is the defining characteristic of Napier's pentagon -- the Pentagramma Mirificum. Aspects of the argument can be explore in terms of the relation between "planarity" and "globality" (Adhering to God's Plan in a Global Society: serious problems framed by the Pope from a transfinite perspective, 2014).
The experience of the abnormal is necessarily a matter of surprise -- as extensively explored by Nassim Nicholas Taleb (The Black Swan: the impact of the highly improbable, 2007). The surprising nature of the abnormal -- in contrast with linear "business as usual" -- can be creatively envisaged through metaphor as an exercise in cognitive preparedness. Alternatively, any disaster (consequent on failure to do so) may also be framed through metaphor -- typically weather metaphors (as indicated above), for lack of any more meaningful framework. Strategically, the orthogonal transformation of direction is indicative of the need to see the process "in another way" -- an indication of the creative challenge of "new thinking".
Related arguments were developed in separate sections (Insights from Knight's move thinking; Alternative representations: Naturomimicry: sourcing nature for strategic metaphors; Stratagems and ploys characteristic of Knight's move thinking). The Knight is part of the emblem for the US Psyops as a traditional symbol of "special operations" -- signifying the ability to influence all types of warfare. A challenging comparison may be made between the Knight's move, BaGua and the Swastika, illustrated by separate animations (Knight's move, Swastika and BaGua? 2012).
There is a case for recognizing that the 10 Napier's Rules for navigation in spherical geometry are indicative of the distinctive forms of metaphor required for the navigation of globality (Napier's Nifty Rules, ThatsMaths, 12 August 2012). Ironically these are associated with what are termed Napier's Analogies (What is the analogy in Napier's Analogy? Quora).
Cycle of self-closure: A knight's graph is a representation of all legal moves of the Knight over the board. A Knight's tour is the mathematical problem of determining the path the Knight may follow in order to visit each square on the board only (see animation in Wikipedia article). Further insights into the contrast between Predictability and pattern-breaking with respect to the Knight's move are presented separately (Implication of Toroidal Transformation of the Crown of Thorns: design challenge to enable integrative comprehension of global dynamics, 2011). Of potential relevance in this respect, in the light of development of the work of Nash, is progress in the visualization of Flat tori in three-dimensional space and convex integration -- the focus of the Hevea Project.
The animation on the left below derives from the argument regarding the Knight's moves. It was notably used to address the challenges of the "blame game" in public discourse -- and the use of weasel words (namely the "slithers", "flips" and "twists" between possible interpretations). In the animation on the right, both forms of the Swastika: left-facing (green) and right-facing (red) are engendered (note the switch in colour and direction -- to the "other" Swastika -- following each "move"). The animations do not necessarily take account of additional patterns of directionality or arrangement. In the case of the BaGua mirror, these are clarified by the screen shots which follow the animations.
| Experimental animations indicative of a cyclic pattern of metaphors (each use of metaphor within an animation is associated with an orthogonal direction) | ||
| 8 of the Knight's moves across neighbouring cells of a chess board | Wu Xing cycle (one direction) superimposed on one BaGua mirror arrangement | Knight's moves superimposed on one BaGua mirror arrangement |
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These animations together highlight the role of orthogonality, as indicative of the metaphorical connectivity required to achieve self-closure -- in a global context. Orthogonality is thus indicative of a creative leap of association to a new framework -- engendered at each of the 5 right-angled points of the Wu Xing cycle. There are 4 possible metaphorical transformations at each such right angle, as indicated by the following screen shots -- each of the 5 points constituting a 4-fold nexus. This makes a total of 20 such transformations, suggesting a further correspondence meriting exploration (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).
The 5 circles on the traditional depiction could be considered a reminder that the Wu Xing pattern is a spherical pentagram -- not planar. Of related significance is the fact that on a sphere the points of the Pentagramma Mirificum are not visible from each other, being effectively "over the horizon" in terms of any "flat earth" perspective from each -- suggesting a fundamental challenge for the comprehension of globality and the implementation of any "global plan"..
| Indication of 4 distinct Wu Xing patterns in terms of the Knight's Move transition at each of the 5 points (colours of no significance; left-most corresponds to the central animation above) | |||
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The formal contrasts and commonalities between the animations recall arguments regarding the value of exploring the relationship between the 5- and 6-pointed stars that are held to be so profoundly symbolic in the conflicts in the Middle East (Middle East Peace Potential through Dynamics in Spherical Geometry: engendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks, 2012). The possibility that belief systems of any significance are necessarily embedded in a 5-dimensional cognitive space then reinforces the arguments for investment in mathematical theology (Mathematical Theology: Future Science of Confidence in Belief: self-reflexive global reframing to enable faith-based governance, 2011).
Integrative inversion: Missing from such 2-dimensional animations is the sense implied by the Pentagramma Mirificum framed by the spherical geometry of globality. There is therefore the implication that only through recognizing the operation of such connectivity on a 3-dimensional surface that the alternative directionalities (indicated by the screen shots above) can be integrated into a single global pattern.
Of relevance in this sense is the fact that the Pentagramma Mirificum has an inverted variant on the reverse side of the sphere -- both being connected by the 5 great circles around the sphere (namely the extension of the 5 curves in the middle animation above). This geometry is remarkably illustrated by the interactive animations of the LaRouche group (Full Circle; The Pentagramma Solid). Less evident are the cognitive, epistemological and experiential significance. Rather than being understood as circles in terms of spherical geometry, the 5 great circles are more appropriately understood as 5 cyclic processes through which the two variants of the Pentagramma Mirificum / Wu Xing are globally integrated.
The nature of the sphere onto which the complementary forms of the Pentagramma Mirificum may be mapped is usefully indicated by exploration of the possibility of mapping the 64 I Ching hexagram conditions onto a sphere (József Drasny, The Reconstruction of the Yi-globe: the spherical arrangement, 2011) as discussed separately (Correspondence between blastosphere and spheroidal I Ching? 2010).
Requisite global connectivity requiring 5-dimensionality? It is through such devices that it may be appropriate to explore the nature of the connectivity characteristic of 5-dimensional space -- and the challenge to its comprehension. Self-closure is presumably a characteristic of the coherence of a "pattern that connects". Indeed if Napier's insights proved vital to navigation around a sphere, why should such insights not be of some relevance to cognitive navigation in a (virtual) global context -- if not a necessity?
It could be argued that this characteristic is helpful with respect to the ordering of cybernetics through the greater degrees of self-reflexivity, implying increasing degrees of cognitive closure. The following schema exploits the Pentagramma Mirificum to offer a tentative holding pattern for seemingly disparate cognitive modalities. It derives in part from the 15 elements of that pattern (separately coloured) which compose the Knight's Move style transformations (3 segments per move), with 4 types of move passing through each pentagram point. Not to be forgotten is that on the other side of the globe there is a second such pattern indicative of a totality of 30 elements of the pattern as a whole, variously associated with the 5 great circles.
| Holding pattern for disparate cognitive modes of global integration? | |
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Of particular interest is the manner in which 5, 10, 15, 20 and 30 can be transformed between each other, most comprehensibly through mappings onto polyhedra and their duals, as separately illustrated (Global strategic significance of 20-fold configurations, 2015; Geometrical configuration of Alexander's 15 transformations, 2010). The work of Beer inspired an effort to map the 1992 Earth Summit issues in that way (Spherical Representation of Icosidodecahedral Net of Strategies: configuring strategic dilemmas in intersectoral dialogue, 1992)
Given the references to chess, go and the Olympic games, to what extent is the complexity of global governance intuitively understood through play (at least to some degree), as separately argued with respect to "Playing" with interrelated metaphors (in Playfully Changing the Prevailing Climate of Opinion; climate change as focal metaphor of effective global governance, 2005; Enacting Transformative Integral Thinking through Playful Elegance, 2010)? It is in this sense that there is a delightful irony to the strategy of the "Pentagon" in seeking to achieve global hegemony through monopolar comprehension.
Poetics of cognitive weather patterns: This argument has emphasized the value of weather in facilitating comprehension of complexity through its multiple manifestations -- beyond those promoted by conventional models. The self-reflexive cognitive mirroring offered by "weather" (and its relation to "whether") is succinctly framed in terms of poetry by biologist/anthropologist Gregory Bateson in explaining why "we are our own metaphor" to a conference on the effects of conscious purpose on human adaptation:
Curiously it could be said there is a long tradition of representing humans as "5-dimensional", as exemplified by the Virtruvian (hu)man of Leonardo da Vinci. -- although little is made of the cognitive dimensionality corresponding to that visual metaphor. As remarked by Kenneth Boulding: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 are 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. (Cited by Mary Catherine Bateson, Our Own Metaphor, 1991, pp. 288-9)
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. (Ecodynamics; a new theory of societal evolution, 1978, p. 345)
| Vitruvian animation? |
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