Misleading triadic approximations of ring dynamics

Symbolizing Collective Remembering Otherwise (Part #4)

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It is appropriate to note the enthusiasm of the "head modality" for triangular representations of the most fundamental cognitive dynamics. This can be variously summarized (Triangulation of Incommensurable Concepts for Global Configuration, 2011). The latter notes the focus on triadic logic?, triadic dialectics, triadic strategic applications, triadic conceptualization, and triadic education and learning.

As illustrated below, the arrangement also echoes that of various much-cited patterns of significance to the study of cognition and associated experience, as separately discussed (Visualization in 3D of a trinity of connotations as a cognitive pill, 2017). The relevant controversies, preumably predictable, are usefully reviewed by Maurizio Meloni (A Triangle of Thoughts: Girard, Freud, Lacan, Psychomedia, 14, 2002). That of Jacques Lacan is a version of the triangulated Oedipus complex (mother-child-father) combining Freud's theory with structural linguistics, developed from the theories of Saussure, Levi-Strauss and Jakobson (John Phillips, Lacan and Language).

The image of Ron Atkin is used to illustrate the challenge of integration beyond any binary clash using the mathematics of q-analysis (Multidimensional Man; can man live in 3-dimensional space?, 1981), as separately summarized (Comprehension: Social organization determined by incommunicability of insights). Using the geometry of a triangle, Atkin illustrates the challenge of comprehension in relation to experience "within" -- especially with regard to the perspective necessary to comprehend the geometry of the triangle as a whole -- namely the trinity. The perceptual significance of this approach is well-illustrated by visual sensitivity to colours resulting from the three primary hues (red, green and blue). These are represented on a simple triangle (below). Here the vertices (O-simplexes) represent the primary hues, the sides are twofold combinations (1-simplexes), and the combination of the three hues makes the central white (2-simplex).

Semiotic triangle of meaning (Charles Ogden)	Triangulated Oedipus complex (Jacques Lacan)	Triangle of comprehension (Ron Atkin)	Triadic depiction indicative of omitted/implicit curvature

As illustrated by the schematic above-right, there is a sense in which such representation is a radical form of reductionsm in which the "cognitive curvature" with which comprehensions is associated is conflated with the "cognitive linearity" of a highly questionable form of simplification, itself usefully illustrated compared with those which follow. Typically an "explanation" of each line is offered separately, notably using a curve as a form of "meta-explanation" to offer comprehensible meaning to the relations between the points of that triangle -- considered to be especially fundamental. That meta-relationship may be especially challenging as in the case of the Christian Trinity.

Roerich Pax Cultura	Borromean rings and knots	Phenomenological epoché (Francisco Varela)	Phenomenological epoché (remapped on Borromean rings)

Especially striking is an unusial depiction of the Christian Trinity (below-left) from long before the formal recognition of the Borromean rings. Such depiction could be explored further in the light of the recent declaration by the UN Secretary-General (UN must reform to defend enlightenment values, secretary-general says, The Guardian, 10 May 2017) pointing to the three separate pillars of the UN: peace and security, human rights and sustainable development. He asked whether these could continue to be addressed separately, calling for them to be combined into a single program -- one central to any strategy of of reform.

In an earlier argument the suggestion was made that these could indeed be understood as "value pills", anticipating discussion of their necessary three-dimensionality (Psychosocial Transformation by "Pill Pushing"? Model-making, strategic advocacy and the myth of the "red pill", 2017). It could then be argued that this implies a "multi-pill" complex as being fundamental to the UN, if it is to be recognized as the "big pill" through which the ills of global civilization are to be remedied. Speculatively, since these are otherwise in no way comprehended as integrated, adaptations (below) of the depiction of the Christian Trinity are presented below (Cognitive Cycles Vital to Sustainable Self-Governance, 2009).

Depiction of Christian Trinity	Interwoven metrics of relevance to global governance ?	Three-fold "pill" of enlightenment values?	Triskelion (traditional Celtic knot pattern)
Illustration from a 13th-century French manuscript, as reproduced in from Wikipedia)	Adaptations of the image on the left		See variants in Wikipedia

Of relevance to the development of this argument, the traditional Celtic image (above) takes the complexification of the triadic even further. Anticipating further discusssion regarding three-dimensionality, this was the catalyst for a more general discussion (Cognitive Implications in 3D of Triadic Symbols Valued in 2D: representations of the triskelion in virtual reality and implications for quantum consciousness, 2017). A 3D variant of the set of Borromean rings (above) is itself significant in this respect -- especially given its value as the logo of the International Mathematical Union. Highly appropriate to the geometric imagery of the argument above is its logo (on left below). Similarly relevant, with respect to the pattern of interlocking circles (on right below).

Logo of the International Mathematical Olympiad (reproduced from Wikipedia; see IMO animation)	Logo of International Mathematical Union (see Wolfram Mathematica animation)

Rather than a two-dimensional emphasis on the cognitive merits of the triangle -- as a potentially misleading approximation to a 2D ring -- its geometry can be used to explore triangulation as an approximation to a 3D sphere, most notably through spherically regular polyhedra (Interrelating disparate threefold cognitive patterns as a polyhedron, 2017; Triangulation of Incommensurable Concepts for Global Configuration, 2011).

To the extent that dynamics imply forms of curvature, anticipating discussion below of the curves of the heart symbol (and their dynamics), another approach is to use the 2D triangle as an encoding template for significance that is typically only implied by the linearity of its form -- if at all. Can the triangle -- favoured by the "head modality" -- be considered an approximation to the heart symbol -- as favoured by the "heart modlity"? The question is relevant to reframing the triadic forms presented above.

The nature of that potentially misleading approximation to the ring -- or to the heart symbol -- is evident from the image below-left. When its inversion is superimposed, as also illustrated below, it is suggestive of a need for comprehension which is primarily implicit -- and of potential significance to any interpretation of the Star of David. A speculative exploration of the dynamics inherent in that symbol in 2D is offered by its use as a template for encoding the 64 transformation conditions of the I Ching, as discussed separately -- from which the animations are reproduced (Mapping of I Ching hexagram coding onto Star of David, 2008; Star of David as reinforcing dangerous cognitive reductionism? 2017).

Triangle as approximation to heart symbol?	Superimposition of inversion of image on left	Superimposition of inversion (another variant)	Hexagram codes of I Ching mapped onto Star of David	Hebrew alphabet mapped onto Star of David

Reference to the Star of David featured in am earlier discussion of its polyhedral complexification and its embedding within a spherical configuration of cycles -- of some relevance to aspects of the argument below with respect to dynamics in 3D (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017).

Reference to both the heart and depictions of the Christian Trinity also recalls the great importance attached to their association with the Sacred Heart -- one of the most widely practiced and well-known Roman Catholic devotions.

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