Contrasting orientations indicative of complementary cognitive modalities

Reimagining Guernica to Engage the Antitheses of a Cancel Culture (Part #9)

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Global interrelationship of distinctive orientations: The earlier argument offered interactive models in 3D of the "tennis ball curve" (and the baseball curve). Views of the same curve from different perspectives are presented below with quarter portions distinctively coloured. These 4-fold perspectives all suggest how the pattern could lend itself to "simplification" to a 2-fold pattern -- a reduction convenient from a binary perspective. With respect to a reimagined Guernica -- as a crisis to be confronted -- a key question is in how many distinctive ways should it lend itself to being perceived, as discussed more generally (Interrelating Multiple Ways of Looking at a Crisis, 2021).

Mutually orthogonal views of the tennis ball / baseball curve in 3D
Adapted from images of Robert Ferréol (Bicylindrical Curve, 2018) Interactive variant in 3D (x3d)

Both the simplicity and complexity of the "tennis ball curve" can be understood through the following screen shots -- with and without the ball around which it curves. Despite its relative complexity, or because of it, its elusive elegance can be readily appreciated.

Screen shots of 3D model of tennis-ball/baseball curve
One perspective		Another perspective
without ball	with ball	without ball	with ball
Interactive variant in 3D (x3d)

Rather than its recognition as a 4-fold or 2-fold pattern, the curve can be recognized as suggestive of an 8-fold pattern indicated by its passage through 8 octants as shown in the screen shots below, as discussed separately (Non-linear pathways curving  between octants, 2006). The octants are then to be understood as indicative of distinctive cognitive modalities vital to the dynamic integrity of the whole.

Tennis-ball/Baseball curve indicative of transformative movement between octants

Requisite variety of cognitive modalities? The excessive reliance on binary thinking can be usefully challenged -- as above. It is however intriguing to note that recognition is accorded to modalities ranging from 3-fold to 9-fold, partially in accord with the argument of George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review. 1956; Nelson Cowan, George Miller's Magical Number of Immediate Memory in Retrospect: observations on the faltering progression of science, Psychological Review, 122, 2015, 3), as discussed separately (Comprehension of Numbers Challenging Global Civilization, 2014).

Given the variety of proposals, as yet missing is any clarity on when a particular framing is appropriate and how the modalities framed by one proposal are to be understood as related to another (Systems of Categories Distinguishing Cultural Biases, 1993; Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980).

The geometry of the tennis ball curve, with its smoothly changing orientation, is especially valuable in allowing various degrees of segmentation according to preferences for 2, 4 and 8 modalities. A number of sports recognize distinctive postures or orientations which suggest a cognitive analogue potentially most evident in dialogue (The Eight Basic Shots In Tennis, Tennis World, 2019; The Eight Fencing Parries, YouTube, 3 September 2008; Breaking Balance in Eight Directions, AikiWeb Aikido Forums). There is of course the challenge of how the extensive 6-fold articulation by Edward de Bono might be related to it (Six Frames For Thinking About Information, 2008).

Dynamic connection with complexity: Guernica is inherently complex with the need for requisite complexity seemingly calling for recognition. However such complexity readily engenders a dysfunctional cognitive disconnect. The means of enabling connection with complexity therefore merits particular consideration. The Mereon Trefoil (animated above) could be considered an exemplar of requisite complexity posing such a challenge to comprehenion. One approach to its comprehension is through the manner by which it ican be engendered, as illustrated by the remarkable 3D animation of Sergey Bederov (below right).

The animation shows how the Mereon Trefoil is generated from a torus by rotating a circle of 3 spheres moving around the torus, then rendering visible the winding path they trace in the course of the combined movements. For clarity the animation is also reversed. Bederov's technique can also be applied using 2 spheres moving around a torus oriented vertically (below centre). In this case it generates a more classical trefoil. With the same method, using 1 sphere, another form can be generated in 3D (below left). The animations are unfortunately jerky due to constraints on video recording and web display. (Smoother interactive 3D variants can be explored using the X3D variants as indicated, and potentially via an X3DOM context)

Animations indicating the generation of relatively complex forms around s torus Adapted from models developed by Sergey Bederov of Cortona3D
One-sphere	Two-sphere	Three-sphere
Classic twisted form (screen shot)	Classic trefoil generated around a torus	Mereon trefoil generated around a torus
	Interactive x3D	Interactive x3d

The generation of the Mereon trefoil through the helical winding together of 3 spheres is especially relevant to fruitfiul comprehension of the complexity of the Triple Helix model of innovation, namely the set of interactions between academia, industry and government, to foster economic and social development. By contrast, the trefoil generated with 2 spheres offers a means of exploring the challenge of binary thinking -- exemplified by the reconciliation of the "headless hearts" and the "heartless heads", or between the "two cultures".

The form generated with 1 sphere is of interest in a period when a new logo is promoted for Facebook, rebranded as Meta (Kari Paul, 'Live in the future': Zuckerberg unveils company overhaul amid shift to metaverse, The Guardian, 16 February 2022; Gray Beltran, When a Logo Doesn't Risk It All: Meta's Brand Is Designed for Unknown Worlds, The New York Times, 10 November 2021).

Curiously the logo bears a strong resemblance to the seam curve of the tennis ball and baseball, as animated above and presented separately (Interactive display of tennis-ball / baseball curve in 3D -- with timed transparency of ball). As a hypotrochoid, the curve invites extensive commentary (Re-membering the Globe from a Flatland Perspective, 2020). This refers to the an Interactive display of generalized baseball and tennis-ball seam curves in 3D also developed by Sergey Bderov.

Necessarily of relevance to the argument, the Bederov approach can be extended to a greater number of spheres, and using different parameters, aas discuss separately.

Dialogue container? A reimagined Guernica, responsive to the antithetic dynamics of a cancel culture. would offer insight into a variety of orientations. This is otherwise readily reduced to binary resolution in dialogue, namely the eradication of one and the triumph of the other -- following a vote or a verdict. Dialogue of that form is only too evident in "parliamentary discourse" -- marked by insult, shouting and even violence. On a larger scale it is institutionalized in the "speciation" into multiple political parties with their characteristic orientations and antagonisms.

Missing is the sense in which the emergence of such variety is predictable as a systemic response to any proposal of a strategy (analogous to a "serve" in tennis). Curiously the number of political parties, with their distinctive modalities, is typically limited in practice to "seven, plus or minus two" as exemplified by the European Parliament (Experimental Visualization of Dynamics of the European Parliament in 3D, 2019).

Arguably this could be consistent with the distinctions made by Douglas Walton (Eight Types of Dialogue: How can logic best be applied to arguments? Logic Journal of the IGPL, July 1997). However other approaches would distinguish 3, 4, or 5, etc modes of dialogue with little effort to reconcile such patterns. One exercise resulted in a Typology of 12 complementary dialogue modes essential to sustainable dialogue (1998) adapted from distinctions made by Arthur Young (Geometry of Meaning, 1978).

The unfruitful collapse of binary dialogue can be contrasted with the trivial case of the Longest tennis volley rally (Guinness World Records) of 30,576 volleys, raising the question of examples of the longest binary exchanges in dialogue and their fruitfulness. More intriguing would be examples of dialogue engaging the interest of a maximum number of distinctive participants (and of an audience) for the longest period of time -- in contrast to a filibuster. How would the variety of participant perspectives contribute to the sustainability of the dialogue and prevent it collapsing unfruitfully? (Sustaining the Quest for Sustainable Answers , 2003).

Colour coding of contrasting perspectives? Edward de Bono uses six colours to distinguish the different thinking modalities mentioned above (Six Thinking Hats: an essential approach to business management, 1985). There are many examples of colour wheels of different complexity, of which the RGB images on the left below are indicative. There is however little consideration of how these might be suggestive in some way of a wider spectrum of cognitive modes than those used by Edward de Bono.

Colour coding also tends to be characteristic of the self-identification of political parties.  This raises the question of how cross-party consensus could be appropriately colour coded -- as in the case of mainstream discourse regarding universal vaccination and the exclusion of any sense of requisite variety in response to complexity..

One exploration of such a possibility is to take advantage of the 64 cognitive modalities articulated by the classical Chinese I Ching, as discussed separately (Transformation Metaphors -- derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle , 1997). These were notably presented in the circular form of Shao Yung -- influential in the thinking of Gottfried Leibniz in 1703 regarding binary coding (Anne D. Birdwhistell, Transition to Neo-Confucianism: Shao Yung on Knowledge and Symbols of Reality, 1989).

The pattern is notably consistent with that of a so-called compass rose (Ways of looking distinguished in terms of "compass" orientation, 2021). It is however suggestive of "directionalities" in cognitive space (which it traditionally purports to encompass). The clusters of 8 hexagrams around the circle could be recognized as cognitive articulations of the 8 trigrams in the Fuxi arrangement of the BaGua.

Use of circles of colour as indicative of distinctive cognitive modalities
RGB colour wheel	RGB colour wheel	Colour coding of circle of I Ching hexagrams
		Coding sequence (out-to-in)	Animation with inverse
DanPMK at English Wikipedia, CC BY-SA 3.0,	LászlÃó Németh, CC0, via Wikimedia Commons

Reference to a 64-fold pattern usefully recalls the 8 queens puzzle on an 8x8 chess board. The question is how the queens can be positioned such that they do not attack each other -- given that they are each capable of moving "powerfully" in any direction on the board. There are 92 solutions but only 12 do not involve simple reflection and rotation. The hexagram circle (above right) is organized into 8 complementary clusters. The RGB wheel on the left offers a 12-fold configuration.

The puzzle can be understood as indicative of the challenge of configuring an array of disparate cognitive modalities to highlight their complementarity, as discussed separately (Global Coherence by Interrelating Disparate Strategic Patterns Dynamically, 2019; Dynamics of N-fold Integration of Disparate Cognitive Modalities, 2021). The 8-fold puzzle is understood as the simpler variant of a greater puzzle only recently (partially) solved. As a clue to "peaceful" global organization, in the case of an n-by-n board, there are approximately (0.143n)ⁿ ways to place n queens so that none can attack each other. (Ben Turner, Mathematician cracks 150-year-old chess problem, Live Science, 4 February 2022).

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