Eliciting the pattern that connects via bull?

Zen of Facticity: Bull, Ox or Otherwise? (Part #11)

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The argument above addresses the challenge of clarifying subtle distinctions, rendering them memorable, and configuring them usefully. There is however the fundamental issue of the transitions between such distinctions, the pathways these may imply, and their complementarity under a variety of conditions.

This can be framed as the "pattern that connects", as highlighted by Gregory Bateson with respect to Steps to an Ecology of Mind (1972), and widely discussed (The Pattern That Connects, Expanding Your World; The Pattern That Connects, Global Vision Foundation; Søren Brier, Bateson and Peirce on the Pattern that Connects and the Sacred, 2008). It can be variously discussed (Hyperspace Clues to the Psychology of the Pattern that Connects, 2003; Walking Elven Pathways: enactivating the pattern that connects, 2006).

Muck? Given the cognitive equivalent of "muck", so characteristic of the present times, it is appropriate to recall that the so-called industrial revolution in England can be recognized as having been framed by the slogan Where there's muck there's brass (Liam Brunt, "Where There's Muck There's Brass": the market for manure in the Industrial Revolution, 2000). Ironically the slogan currently features in advocacy for new approaches to the environment.

Of interest, however, are the implications of "muck" -- understood here as "bull" -- for the next cognitive revolution and paradigm shift. It could well be that the challenge "lies" in the "bull" in which one believes (if not a "sacred cow") -- especially including a belief in nothing (Emerging Significance of Nothing, 2012). There is a sense in which if one does not understand how one is part of the problem, one cannot understand the nature of the solution required. There is then good cause for epistemological panic (Epistemological Panic in the face of Nonduality: Does nothing matter? 2010).

"Magic" squares? As commonly framed, there is no lack of "bull" in the argument above, especially through the manner in which it touches on symbolism and numerology. This is highlighted in the case of "magic squares", despite the considerable importance associated with them from a mathematical perspective, most notably with respect to number theory. Thus for Ezra Brown (Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves, College Mathematics Journal / Mathematical Association of America, September 2001):

I came across a little gem of a result that ties many mathematical threads together, threads that originate in several different areas of mathematics. The result is that every elliptic curve has nine points of inflection which can be arranged, in a very natural way, to form a 3 x 3 magic square.

[Note generalizations of this argument (Edray Herber Coins, Semi-Magic Squares and Elliptic Curves. arxiv.org, 2009; B L Kaul, et al., Generalization of Magic Square (Numerical Logic) 3x3 and its Multiples (3x3) x (3x3), International Journal of Intelligent Systems and Applications, 2013) and summary of implications for communication (Elliptic curve cryptography; Meenu Sahni and D.B. Ojha, Magic Square and Cryptography, Journal of Global Research in Computer Science, 2017; Nitin Pandey, et al., Secure Communication with Magic Square, Journal of Global Research in Computer Science, 2012)].

Correspondences: Of relevance, as noted above, is then the question of the cognitive implication of mathematical insight (George Lakoff and Rafael Nuñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). How is number theory to be distinguished from the "muck" with which numerology is variously associated?

The challenge of dubious correspondences has been highlighted with respect to what mathematicians themselves have caricatured as moonshine theory, through which the high order symmetries of the so-called monster group were discovered. As discussed separately, how then to consider the possibility of analogous correspondences enabling discoveries of significance to a paradigm shift (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007; Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

Obscurantism? In navigating facticity, how is it that the order so valued by mathematicians has been so systematically dissociated from the significance valued by individuals and in their governance -- as discussed separately (Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008)?

Can mathematicians be accused of indulging in deliberate obscurantism, as is evident from answers to questions raised in various fora (How do I study and conduct research in the areas of elliptic curves and modular forms? Quora, 2014; What is so special about elliptic curves? Physics Forums, 2013). Self-reflexively, are elliptic curves themselves relevant to the challenge of comprehension, in the light of some responses offered: Elliptic curves are interesting because they are the simplest algebraic structure that is not yet completely understood; The problem I fear is that's it's not going to be easy to understand if we're not already familiar with the subject.

Given the early slogan of the industrial revolution, what indeed is the "manure" from which "brass" can be derived? Appropriately "brass" was then a reference to money -- and is therefore relevant to the fundamental issues of confidence which are now proving so critical with respect to facticity.

Connectivity: enneagram vs magic square: If the issue is one of comprehensible connectivity, one exercise deriving from the above argument is to confront the connectivity of the enneagram with that of the magic square.

Enneagram	Enneagram rotated	3x3 Magic square

Numbers in corresponding positions in the enneagram sum to 7, 8, 9, 10 or 11, as indicated in the image on the left (below). If the corresponding numbers in the columns, rows or diagonals from the magic square are summed, they give the totals 9, 10, and 12, as indicated in the image on the right -- based on their relative positions in the enneagram. The areas in that diagram are then necessarily indicative of the 15 to which the three numbers total in each case (as a property of the magic square indicated above). Note that these results would have been visually identical had the enneagram been rotated in the other direction.

Enneagram with totals indicated	Rotated enneagram with indication of totals (from magic square rows, columns and diagonals)

Such patterns are indicative of the connectivity within which the transitions between the Zen insight phases could be understood to be embedded. Especially intriguing as an indication of comprehensibility are the various symmetries in the image on the right (above) which are more apparent in the images which follow.

The enneagram is otherwise valued for mapping cognitive styles (A. G. E. Blake, The Intelligent Enneagram, 1996). As a management cybernetician concerned with integration processes, Stafford Beer has however noted the manner in which a 3D variant of it "hangs within the icosahedron" (Beyond Dispute: the invention of team syntegrity, 1994). This is discussed and represented separately (Correspondences between Traditional Constellations and Pattern Languages, 2014). A representation of the image above could be developed in 3D. The relevance of Knight's move thinking (from chess) is discussed separately with animations (Knight's move thinking: appreciated or deprecated, 2012; Knight's move thinking in relation to magic squares, 2015).

Points of inflection and a 3x3 magic square: The above-mentioned exploration of magic squares by Ezra Brown suggests an unexpected convergence of threads relevant to this argument. He frames a Magic Square Theorem:

Every elliptic curve has nine points of inflection, and these points form an affine plane of order 3. That is, each point of inflection lies on exactly four lines, each of which contains two other points of inflection -- making 12 lines in all -- and each pair of points of inflection determines a unique line (Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves, 2001)

Five "plus or minus four"? Noting the resemblance to the argument of Viktor Prasolov and Yuri Solvyev (Elliptic Functions and Elliptic Integrals, American Mathematical Society, 1997), Ezra Brown concludes by showing how the 9 inflection points can be arranged as a 3x3 magic square (left half of table below). He argues that this result ties together threads from finite geometry, recreational mathematics, combinatorics, calculus, algebra, and number theory.

Magic theorem (Brown, 2001)			3x3 Magic square
B	-A	-D	2 (=5-3)	7 (5+2)	6 (=5+1)
C	O	-C	9 (=5+4)	5	1 (=5-4)
D	A	-B	4 (=5-1)	3 (=5-2)	8 (=5+3)

From these correspondences, in Brown's pattern on the left (above:): A=2, B=3, C=4 and D=1. This suggests that "5, plus or minus 4" might be usefully explored as an alternative to 7 in the classic paper cited above (George Miller, The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review. 63, 1956). Given the use of the torus as a framework for the Zen images, it is appropriate to note that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane.

Indication of relation between enneagram mapping and magic square
Enneagram mapping	Magic square	Indications on base 5
	diagonals: 2-5-8, 4-5-6 rows: 2-7-6, 9-5-1, 4-3-8 columns: 2-9-4, 7-5-3, 6-1-8 Knight's moves: 1-2, 2-3, 9-9, 8-7

Memorability: foldability and degeneracy? With respect to learning or insight, why 5 rather than 7 ? Intuitively interrelation of 5 points is already a challenge, as with comprehension of the subtleties of orders of feedback and self-reference greater than 3 -- with 5 at the limit of current cybernetic considerations. The sense in which 5-fold comprehension "decays" to 5-1, 5-2, 5-3 or 5-4 comprehension can be readily recognized, with dysfunctional polarization the most common of psychosocial processes, and with the concerns regarding the unitary assertion of fact being the focus of the above argument. Potentially relevant is the coherence associated with the five points determining a conic in projective geometry.

Could this "entropic decay" in cognitive capacity be compared with mathematical use of "degenerate" -- in contrast with the complexity implied by its opposite? Similarly intriguing is the sense in which the icosahedral great circles are understood as fundamental to the foldability of polyhedral structures, as explored in a presentation to the Interdisciplinary Conference of the International Society of the Arts, Mathematics, and Architecture, by C. J. Fearnley (Exploring Foldable Great Circle Geometries, Synergetics Collaborative, 2009). It also features in a study within the Cluster of Excellence of the Image Knowledge Gestaltung: An Interdisciplinary Laboratory by Michael Friedman and Joachim Krausse (Folding and Geometry: Buckminster Fuller's provocation of thinking, 2016) in which the authors conclude:

... folding suggests an interlacing of thought and contemplation together with a materialized geometry and partially overlapping events. This enables Fuller to tie together his conception of geometry as a material mathematical science -- a point is a place where two lines pass through but not at the same time -- and the scenario as a form of thinking. Folding, weaving and interlacing engender this form of thinking.

To what extent might "foldability" be relevant to the coherent organization of memory with respect to different levels of complexity?

Given the above-cited approach of Lakoff and Nuñez (2011), it might also be asked whether the frequent use by mathematicians of "embedding" is indicative of the poorly extent to which human subjective experience is embedded in the objective reality of mathematics -- matched by the extent to which the latter "informs" subjective experience.

Points of inflection in the learning process? Curiously the literature focuses on the challenge of learning about the importance of elliptic curves but not on their implication for the learning process itself -- with the exception of some hints to that effect (Apostolos K. Doxiades and Barry Mazur, Circles Disturbed: the interplay of mathematics and narrative, 2012; Roberto Festa, et al. Cognitive Structures in Scientific Inquiry, 2005). The symmetries of the image on the right above are however suggestive of correspondences in the comprehension of 4 (=5-1) and 6 (=5+2), as well as of 3 (=5-2) and 8 (=5+3), for example.

Such an image also provocatively recalls the form of the Tree-of-Life / Sephirot pattern much-studied in the Judaic Kabbalah (readily framed by science as yet another form of "bullshit"). Kabbalah (but not the Sephirot) has evoked various comments on the relation between it and "ox-herding" (Kabbalistic Meditation on Ancient Ox Herding Photos; Ox-herding Pictures of Zen Buddhism, Donmeh West, 2004; Michael Eigen, A Felt Sense: more explorations of psychoanalysis and Kabbalah, 2014). As might be expected, there is a long association with magic squares (Tasha Lindsay, Kabbalah and Magic Square, The Magic Square Blog, 15 February 2016).

The relationship established by Brown to elliptic curves is valuable because the kind of learning associated with the Zen pattern of distinctions is essentially non-linear and usefully understood in terms of elliptic curves and points of inflection -- from concave to convex -- potentially to be considered in terms of the insideness (subjectivity) or outsideness (objectivity) of perspective (Transformation of worldview from "inside-outside" to "outside-inside", 2013).

Stabilizing facticity: encycling the serpent and the snake lemma: The points of inflection are explained otherwise in Wikipedia by use of the following animation (on the left) with its suggestively snake-like dynamic.

9 Points of inflection in an elliptic curve animation	"Encycling" the serpent/snake	Benzene snake-dream (August Kekulé)
	Traditional Ouroboros symbol
From Wikipedia, produced by Thomas Cooper; created with a GNU Octave script; Public Domain, Link	Zen image #8	Screen shots of animation by Michael Verderese

The serpentine movement of the animation ironically suggests -- however naively -- that it is indeed snakes that have the most instinctive understanding of an "elliptic dynamic" which they embody in their movement. The challenge of self-reflexive learning might then be usefully framed by the snake "biting its tail", as with the Ouroboros symbol on the right above -- echoed by one of the images in the Zen pattern. The tail-biting image is also renowned for the insight it offered into the discovery of the structure of the benzene molecule fundamental to biological tissue, as illustrated in the screen shots above from an animation by Michael Verderese (An HTML-based Representation of Kekulé's Benzene Dream, ChemDoodle, 2011), and variously discussed (Malcolm W. Browne, The Benzene Ring: dream analysis, The New York Times, 16 August 1988).

Of relevance to such exploration is the literature on "continuous elliptic curves" and "cyclic elliptic curves", and the associated literature on the "snake lemma" or the "serpent lemma" (Snake Lemma, Wolfram MathWorld; Snake Lemma, nLab, 30 October 2015; 2015; Jonathan Wise, The Snake Lemma, Stanford Department of Mathematics, 17 February 2011). The latter introduces the mathematical argument with the biblical phrase: the serpent was more subtil than any beast of the *field which the Lord God had made (Genesis 3:1).

The challenge is exemplified otherwise by the plasma snake effects in toroidal fusion reactors (Taming Plasma Fusion Snakes, Berkeley Lab Computing Sciences, 2014). Again, it is of course extremely ironic that the extensive current interest on comprehension of elliptic curves is related to cryptography, readily to be recognized as a device for the systematic prevention of learning (Elliptic Curve and Finite Field Cipher Visualization Tools, 2012; ECvisual: A Visualization Tool for Elliptic Curve Based Ciphers, 2012). There is seemingly no trace of their relevance to enabling learning of other kinds.

So framed it becomes clearer that "facts" only acquire the stability -- that science and convention would claim they have -- under very particular conditions. Otherwise, like some isotopes, "facts" may only be meta-stable, retaining their coherence only relatively briefly -- before "decaying" in some way (Samuel Arbesman, The Half-Life of Facts: why everything we know has an expiration date, 2012). As yet to be fully clarified, this transformation may come to be indicated by some form of "post-truth table", as separately discussed (Towards articulation of a "post-truth table"? 2016). The alleged "facts" justifying military action in Iraq and Syria merit evaluation in such terms (Truth Test on Syria: Religious oath -- Polygraph -- Ouija board? Systemic mapping of the pattern of affirmations and denials, 2013).

Given the problematic association of facticity with problem perception, the challenge of "encycling" the snake can be explored in relation to that of coherence in governance (Encycling Problematic Wickedness for Potential Humanity, 2014).

Encycling a paradoxical snake in a Möbius strip: As widely noted, an elliptic curve is a challenge to comprehension. Elliptic curves are of the following topological types: a torus, an annulus, a Möbius strip, and a Klein bottle (North-Holland Mathematics Studies, 54, 1981). A Möbius strip is a challenge to comprehension in mathematical terms -- although easy to visualize through constructing one (Understanding the Equation of a Möbius Strip, Mathematics Stock Exchange, 2014). Any challenge to learning, with the requisite "cognitive twist" of a paradigm shift, might then be understood as usefully embodied into a Möbius strip in some way -- beyond the relatively simplicity of framing the Zen pattern within a torus (as indicated above).

"Re-cognition" of the nature of the challenge is usefully indicated by the manner in which depiction of wisdom in iconography may use some form of halo or annulus, even taking the form of a torus. Such depiction may include use of the infinity symbol, readily implying the paradoxical complexity of a Möbius strip. The cognitive peculiarity of the relationship between "global experience", the "toroidal journey" of eternal return, and the "paradoxical twist" linking alternative world views can be explored in terms of the corpus callosum nexus of the human brain (Corpus Callosum of the Global Brain? 2014; Engendering Viable Global Futures through Hemispheric Integration, 2014). Any dynamic integration of "facts" and modalities, as implied by the Zen "herding" metaphor, can then be contrasted with the implication of "herding cats".

Better still would be to explore any way of combining a torus mapping with that of a Möbius strip. One approach is suggested by a double twisted Möbius strip fitted into the form of a torus; otherwise known as a bifilar torus knot, this is discussed and depicted separately (Möbius and Torus Relationship, The Life Field Transformer).

Experimental association of Möbius strip with torus
Test error A	Test error B	Animation of progression

Presentation of the test errors above recalls the range of topological distortions of the Möbius strip through introduction of extra twists -- with their potential cognitive implications.

DNA as a biomimetic template: DNA as a coiled coil (a double helix, or supercoiling) is the fundamental schema of biological communication (J. L. Campos, et al., DNA coiled coils, PNAS, 102, 2005). It can be explored in terms of elliptic curves (P.Vijayakumar, et al., DNA Computing based Elliptic Curve Cryptography, International Journal of Computer Applications, 36, 2011; Enhanced Level of Security using DNA Computing Technique with Hyperelliptic Curve Cryptography, International Journal on Network Security, 4, 2013). As might be expected, related exploration also involves consideration of the Möbius strip, as indicated by Dongran Han, et al, (Folding and Cutting DNA into Reconfigurable Topological Nanostructures, Nature Nanotechnology, 5, 2010):

Here, we show that DNA origami can be used to assemble a Möbius strip, a topological ribbon-like structure that has only one side. In addition, we show that the DNA Möbius strip can be reconfigured through strand displacement to create topological objects such as supercoiled ring and catenane structures.

Verbal explanation of any "twist" is essentially unhelpful to forms of comprehension dependent on linearity. The following images and animations can therefore be understood as experiments in the facilitation of comprehension implying "paradigm twists". The challenges to their construction using 3D visualization facilities are also significant in that respect -- notably through the failures to engender results which combine both comprehensibility and appeal. The question is on what forms can new insight be meaningfully "hung" -- and in what forms can it be fruitfully embodied (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007).

Torus knotting and helicoidal animation: Mnemonic clues to to the nature of the cognitive challenge may also be offered by topological complexification of the basic torus as suggested below -- and as previously developed (Configuration of a Toroidal Helix, 2016).

Potential use of a continuous toroidal knot with 9 windings as a container for the Zen or BaGua pattern (Interactive variants: X3D or VRML)
Options for disk dynamics -- with cognitive implications
disk movement along Möbius strip
disk movement through knot

Rather than the Zen images, the contrasting subtleties of 9 insight modalities can be encoded by colour, as shown below. The representation can be readily varied by modifying parameters of: colour, cycle period, ball size, number of balls, period of visibility, radius of helicoidal winding. Any given combination of choices may well range from appealing to alienating -- and meaningless. This suggests that such an animation is best explored like an interactive musical instrument on which a variety of "cognitive melodies" can be explored (possibly enhanced with associated tonal encoding typical of sonification). Those recognized as appealing are then evocative of more integrative insight. Those which are variously alienating are then suggestive of possible cognitive traps. Such an instrument lends itself to depiction of both habitual modes of busyness and occasional flashes of insight.

Animation in 3D of movement of 9 balls around a continuous toroidal knot (.x3d source)
Screen shot of full rendering (.mp4 version)	Screen shot of wireframe rendering (.mp4 version)

A related design metaphor that can be used as a suggestive container for the dynamics of the Zen pattern is indicated below. There the images circulate through a hierarchical winding. Again a wide variety of dynamics can be associated with the images as an aid to "re-cognition". As noted earlier, a tenth image could be associated in some way with the pattern as a whole (perhaps as suggested by that on the right). The pattern could be rendered in an 8-fold form using the distinctions of the BaGua trigrams (as above).

Contrasting movements of 9 Zen images circulating through a helical winding in 3D (screen shots of animations)
Rotating vertically	Vertical flipping	Horizontal flipping	Occupation of centre
.mp4 version	.mp4 version	.mp4 version	.mp4 version

Any depiction of circulation through a helical winding is a provocative stimulus to consideration of the implications of the dynamic reframing by Nikola Tesla of electromagnetic polarity (Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014). In a period in which binary thinking is the primary dynamic enabling violence, there is therefore a case for exploring analogues of greater subtlety in the psychosocial domain, as discussed there (Potential implications of alternation and rotation in psychosocial fields, 2014).

Correspondence with the coaction cardioid? There is a further possibility that the learning process associated with the pattern of the magic square could be fruitfully explored in the light of the work on the coaction cardioid, within a similar pattern, by Edward Haskell (Full Circle: The Moral Force of Unified Science, 1972), and further developed by Timothy Wilken (UnCommon Science, 2001) into the following generic form:

Coaction cardioid (Haskell / Cassidy) Geometric representation of 8-fold conditions [see also articulations by Wilken, pp. 157-161)	Correspondence: coaction cardioid / magic square (indicative combination of Wilken/Haskell)

This is discussed separately (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005). Especially intriguing is the manner in which that cardioid could be interpreted in the light of the patterns of subjectivity and objectivity associated with conditions of the learning process.

Reality and unreality? With respect to any sense of "reality", Brown shows that only two of the nine points of inflection on an elliptic curve are defined by real coordinates, the others have at least one nonreal coordinate -- meaning that they cannot be readily seen. An insight of this kind would seem to merit careful consideration in terms of the subtlety of the learning process indicated by the Zen pattern. This "unreality" is seemingly acceptable to science -- perhaps as "good shit" in the widely circulated schema of such perspectives (Shit Happens: close-to-complete ideology and religion shit list).

Of potential relevance is the relation of such curves to the work of René Thom, basic to catastrophe theory -- with learning readily understood as a succession of cognitive catastrophes (Structural Stability and Morphogenesis: an outline of a general theory of models, 1972; Esquisse d'une Sémiophysique: physique aristotélicienne et théorie des catastrophes, 1989).

This association has been developed by Wolfgang Wildgen (Catastrophe Theory and Semiophysics: with an application to "movie physics", Language and Semiotic Studies, 2015) and by Jean Petitot (Morphogenesis of Meaning, 2003; Cognitive Morphodynamics: dynamical morphological models of constituency and syntax, 2011).

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