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8 March 2021 | Draft

Framing Dynamic Transcendence of Simplistic Cognitive Polarization

Implication of circular configuration of targets, dartboards, mandalas, codons and hexagrams

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Introduction
Cultivating the pretence of binary simplicity
Objectively understood configurations indicative of fundamental cognitive implication
Purportedly objective configurations with potentially subjective implications
Subjectively understood configurations indicative of fundamental cognitive implication
Configuring personality types and gender orientation as the archetypal challenge
Interactively understood configurations indicative of fundamental cognitive implication
Configuring polarities for cognitive engagement of higher order
Exercises in centro-symmetric encoding using hexagrams
Container for polarization -- an analogue to geodesic domes?
Systematic configuration of ambiguity and liminality?
References

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Introduction

History may well find it remarkable the extent to which the present period cultivates the pretence of the adequacy of binary distinctions. Arguably these have even been embodied in the technologies on which civilization is now so dependent. In the case of the computer, this is most evident in the on/off distinction of logic gates -- ironically at a time when progress in quantum computing is revealing other practical possibilities. It is also obvious in the nature of many competitive games in which the triumph of the winner over the loser is the objective. It is clearly fundamental to democratic processes and the emergence of a governing party or coalition

There is little interest in more complex possibilities, despite the poisonous nature of public discourse on many matters (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016). A major difficulty is that typically there is limited consensus on the appropriateness of the outcome of binary decision-making, as is only too evident in democratic elections -- exemplified by that of the US presidential election. Each side effectively defines itself as exemplifying the "good" in contrast to the other then appropriately framed as "evil". This process is captured by the proverb "one man's meat is the other man's poison" (Edward de Bono, I am Right and you are Wrong: from rock logic to water logic, 1992).

The "dilemma" is how to ensure a viable society in which the co-existence of the "good" and the "evil" is so evident -- especially when it is "them" who are "evil", and "us" who are "good", if only by implication. There is little acknowledgement of the wisdom of the insight of Pogo: We Have Met the Enemy and He Is Us (1970).

It is of course the case that the tradition of binary distinctions dates back over millennia -- especially reinforced by religions allowing for little subtlety and humility in the matter. In a civilization faced with crises of every kind, it can usefully be questioned whether the dependency on this mode of thinking is "fit for purpose". Despite vain and desperate efforts at "Getting to Yes", there is an extensive literature on the case for transcending such dualism (Edward de Bono, Po: Beyond Yes and No, 1990). Ironically the world is now witness to the highly controversial emergence, in formally recognized practice, of categories beyond male or female -- exemplified by the initialism LGBT, variously extended to LGBTIQ and beyond

The concern here is more general and focused, to whatever extent this is deemed more fundamental. The question is how might polarized discourse be framed such as to enable and allow for emergence of a richer pattern of modalities. Curiously polarized discourse does not encourage visualization of such possibilities since it only provides for two possibilities -- thereby rendering trivial any visualization, with few exceptions. There are however many familiar clues to more complex patterns. It is these which are highlighted here in quest of a more generic pattern.

Emphasis is given here to a wide variety of circular patterns. These include circular configuration of targets, dartboards, logos, mandalas, genetic codons and hexagrams. Related clues are to be found in the design of some game boards. Some are presented by their advocates because of their relevance to comprehension and learning in contrast with tabular arrays.

As yet to be clarified is the extent to which such arrays reinforce the silo thinking reflected so widely in urban grid layouts of skyscrapers -- in a period in which joined-up thinking in increasingly perceived as vital, especially to communities. More provocatively it might be asked whether the "magic" of magic squares has been lost, despite having been deemed so important to governance by Benjamin Franklin, as a Founding Father of the US (Salvation Enabled by Systemic Comprehension -- via aesthetics of magic squares? 2015). Can the new challenge be speculatively presented, as argued separately (Reframing the Square Wheels of Global Governance: transcending vain hopes of squaring the circle in global decision-making, 2017). Time for magic circles?

Given the possibility of richer patterns, of interest is how they might apply to problematic polarities variously conflated, such as: good/evil, guilt/innocence, right/wrong, winner/loser, positive/negative, relevant/irrelevant, profit/loss, secret/open, or beauty/ugliness? How are richer patterns to be thought about and discussed?

Cultivating the pretence of binary simplicity

In a world torn by polarization, it is strange to note that there is little sense of how many polarities might be usefully recognized, how they are variously conflated, or how they might be fruitfully configured -- nor is there any collective motivation to explore such matters, and the strategic dilemmas to which it gives rise. One exercise to that end distinguished 230 value polarities, tentatively clustered into 45 value types.

Another approach can be recognized in the many questionnaires with a yes/no response designed to determine the "purity" with respect to knowledge or attitudes in a variety of domains (John DuBois, Purity Tests!, Amory). Generally, for each "yes" answer the respondent loses a purity point, with the result scaled to indicate what percentage of purity is still retained.

How many strategic dilemmas are there -- in a period in which there is also concern with trilemmas and quadrilemmas?

Fundamental polarities? The following selection variously illustrates how pervasive is the dynamic of pretence currently undermining global civilization and the emergence of more fruitful patterns of order.

• Good / Evil: Every effort is made to lay claim to "good" in absolute terms, or at least to "better", if not to "best". This process is a feature of every kind of performance, from sports, to media, to academia -- formally recognized with awards and prizes. Most striking are the claims with regard to spirituality and holiness, epitomized by the process of beatification and recognition of sainthood.
  
  Corresponding claims are widely made with regard to the "evil" of others, whether absolute or not (Framing by others of claimants of evil as evil, 2016). Some are readily held to be "monsters".
  
  Obvious difficulties are evident in the case of those who are purportedly good in the extreme, such as religious leaders and clergy, who are discovered to be engaged in sexual or financial abuse. More striking is perhaps the case of political leaders and presidents, elected for their qualities, who are later indicted for corruption of various kinds. Similarly there are those widely recognized to be evil -- gang lords and war lords -- but whose goodness and kindness is much appreciated by some with direct experience of them.
  
  
• Guilt / Innocence: The system of justice is valued for the establishment of "innocence" and the determination of "guilt" in absolute terms -- whatever the claims for mitigating circumstances.
  
  Obvious difficulties arise from cases of miscarriage of justice -- in some cases many years after the judgment. More problematic is the degree to which many who are incarcerated claim (or feel) themselves to be innocent, and may be so held by others -- giving rise to questionable processes of appeal by those with resources. Many who are determined to be innocent by the system of justice (purportedly for "lack of proof") are held to be guilty by those who have experienced their actions or have been informed of them. There is much concern about the impunity of those with influence, and with the manner in which only token punishments are accorded to the guilty (a "slap on the wrist") -- despite the gravity of actions in each case.
  
  
• Right / Wrong: Rather than of concern to the system of justice, more generally there is a preoccupation with whether someone (or some group) is "right" or "wrong". This is notably a striking feature of assertions in politics and academia, as well as of religion. In the latter case "wrong" may be associated with sin -- evoking stringent the provisions in that regard (even justification for execution).
  
  The obvious difficulty is the extent to which right/wrong is readily conflated with guilt/innocence and/or good/evil. There may be some understanding of "partially right" or "partially wrong", as in assessments of performance. Rather than being absolute, some provision is made by religions for the distinction of sins of different orders -- especially in the case of clergy. Sin other wise considered to be grave may be questionable framed as minor -- and therefore excusable.
  
  
• Positive / Negative: Very extensive emphasis is placed on the need to be "positive", most notably in public relations and in team leadership roles. Any emphasis readily held to be "negative" is condemned as unproductive, if not dangerous.
  
  The degree of misguided emphasis on the positive has been clarified by Barbara Ehrenreich (Smile Or Die: How Positive Thinking Fooled America and the World, 2010), and can be otherwise argued (Being Positive Avoiding Negativity, 2005). As indicated in the latter, a major consequence of overvaluing the positive and excluding the negative is the vital requirement for negative feedback in order to determine why anything is not working as intended -- including nuclear power stations. More generally this difficulty detracts from the need for critical thinking to enable remedial action and creative responses to crises. The problematic nature of this deficiency is currently highlighted by the quality of testing for disease which may result in "false positive errors" or "false negative errors".
  
  

  A further issue is the valuable challenge to any assumption that such logos are a focus or container for the "positive" and desirable only -- thereby designing out the "negative" and undesirable. Arguably the more fundamental challenge is to find ways of using the implied cognitive dynamics of such configurations so as to contain both positive and negative in a more fruitfully integrative manner.

  A valuable metaphor in that respect is the design of electrical circuits. These require two wires in order to carry positive and negative currents, whether directly or by alternation. Stripping out one wire to eliminate the negative is neither feasible nor desirable. The argument has been developed otherwise by Barbara Ehrenreich (Bright-sided: how the relentless promotion of positive thinking has undermined America;  Smile Or Die: how positive thinking fooled America and the World, 2009). The argument clearly has implications for the emerging trend towards suppressing all communications on the internet which use any form of negative expression.
  

• Success / Failure: There is an obvious preoccupation with achieving success and desperately avoiding failure. This may well be framed in terms of winning, and being a "winner", in contrast with losing, and being framed as a "loser". The pattern is most evident in the driving dynamic in competitive sports, in business and in military initiatives. Success is celebrated and rewarded; failure is deplored and even condemned.
  
  A notable difficulty is evident in that the success of one party is typically achieved at the price of the failure of another. This may result in a situation which there are relatively few who succeed remarkably and many who fail -- thereby losing both the esteem of others and self-esteem. Particular attention is curiously given to learning from failure, notably in business (Amy C. Edmondson, Strategies for Learning from Failure, Harvard Business Review, April 2011).
  
  
• Profit / Loss: As an especially evident aspect of the success/failure dynamic, there is an obvious driving concern with the desirability of "profit" -- possibly even at any cost. This is matched by an abhorrence of "loss", especially as evaluated in economic and financial terms.
  
  Difficulties are evident in that profitability for one party may well be matched by a loss for another -- exacerbating socioeconomic inequalities. Over time it may become evident that desirable levels of profitability are not sustainable and that there is a need to adapt to cycles of profit and loss. 
  
  
• Appropriate / Inappropriate: With respect to performance in many settings there is a concern with what is "appropriate", however this may be understood (Comprehension of Appropriateness, 1986). In some settings this may be equated with being "useful", notably in group undertakings.
  
  Clearly there is a corresponding preoccupation with what is deemed inappropriate or useless -- most obviously in the assessment of people in a team. In the development of any group, it is considered desirable to limit inappropriate behaviour and to eliminate those who are useless -- possibly caricatured as being a "waste of space".
  
  
• Relevant / Irrelevant: This can be understood as a more general form of the appropriate/inappropriate distinction. Consideration of what is "relevant" is most evident in design and construction, especially creative innovation. The focus is then on what merits inclusion and on what can (or should) be excluded from consideration.
  
  Integrated systems can be understood as interrelating the relevant within a boundary from which the irrelevant is excluded -- whether as potentially threatening or calling for intervention and exploitation of some kind. The mandates of organizations and disciplines may be understood as defining explicitly what is relevant -- whilst implicitly defining what is deemed irrelevant.
  
  An obvious consequence is the accumulation in the wider context of what is deemed irrelevant -- epitomized by waste and problematic waste disposal. A further difficulty arises from the failure of many systems from which the irrelevant has been mistakenly excluded (Variety of System Failures Engendered by Negligent Distinctions, 2016). System design may be deliberately or inadvertently negligent to a dangerous degree (Reintegration of a Remaindered World: cognitive recycling of objects of systemic neglect, 2011).
  
  
• Open / Closed: Value is widely attached to systems which are described as "open" in contrast to those that are "closed". Open may be understood as "inclusive" compared to "exclusive". The contrast may be equated with "transparent" rather than "opaque".
  
  The difficulty is that the quality of openness is vulnerable to what comes to be perceived as forms of degradation. Value is then attached to the closed, the exclusive and the less transparent (Orrin E. Klapp, Opening and Closing: strategies of information adaptation in society, 1978). This has become most evident in the pressures of migration on countries valuing openness, leading to problematic interactions with migrants and a preoccupation with degrees of closure. Locally this may encourage the emergence of gated communities.
  
  
• Subjective / Objective: Considerable importance is attached to objectivity, notably by science and management. Value is otherwise attached to subjectivity with respect to the appreciation of quality and in interpersonal relations.
  
  Difficulties necessarily arise when decision-making is confronted by a problematic reconciliation of subjective and objective considerations, especially in relation to value judgement. In such contexts subjectivity may well be deprecated and condemned -- but objectivity may similarly be called into question
  
  
• Beauty  / Ugliness: There is considerable emphasis on "beauty" and the quest for achieving appreciation as beautiful by some criteria. This may be epitomized by cosmetic surgery or by the acquisition of what are deemed to enhance beauty, namely cosmetics, clothing, and other status symbols. This extends beyond individual aspiration to cultivation of the elegance of locations. Beauty and the qualities associated with elegance are widely celebrated, reinforced to a considerable degree by marketing.
  
  There is a corresponding deprecation of personal ugliness -- in private discourse, if not in public. Such deprecation has been extended to whole countries by Donald Trump as leader of the free world in a much-cited notorious reference (Trump derides protections for immigrants from 'shithole' countries, The Washington Post, 12 January 2018)
  
  One difficulty is that beauty is very often a matter of perception, as widely recognized in the adage: Beauty is in the eyes of the beholder. Potentially far more problematic is the effect of aging on beauty -- whereby the formerly beautiful are susceptible to recognition as ugly.
  
  
• Masculine / Feminine: The rigid distinction between "male" and "female" continues to be designed into official identity profiles, grammatical construction, clothing and toilet facilities.
  
  The difficulty, as noted above, arises from the highly controversial emergence in formally recognized practice of categories beyond male or female, exemplified by the initialism LGBT, variously extended to LGBTIQ and beyond.

A fundamental question with respect to this set of polarities -- as dilemmas challenging psychosocial dynamics -- is how to explore the commonality they imply. Obviously challenging is that any effort to do so is vulnerable to the "charge" associated with one or more polarities. It could be asserted that (necessarily?) there is currently no acceptable method for engaging with the chaotic dynamics to which they variously give rise.

The configurative mappings below are an indication of one possibility of evoking coherent reflection on the above set of 12 polarities. As one example, the central animation has the polar extremes mapped onto 24 "opposing" vertices of a truncated octahedron. The animation on the left has the polar extremes mapped onto the 24 "opposing" triangular faces of the tetrakis hexahedron (dual of the truncated octahedron).

Indicative configurations of selected major polarities
Circular configuration	Mapped onto truncated octahedron	Mapped onto tetrakis hexahedron

Animations prepared using Stella Polyhedron Navigator

Part of the difficulty lies with the degree to which each polarity is identified by labels which are conventionally defined. The definitions can however be recognized as simplistic suggesting that each is overdefined for the convenience of conventional frameworks, whereas the experiential reality in practice (as mentioned) is far more complex and subtle. Each term is more fruitfully recognized as a memeplex -- cognitively "encrypted" -- as discussed separately (Cognitive Encryption enabling Collapse of Civilization: drowned by the undertow of pseudophilia, 2021).  As such, cognitive engagement with such misplaced concreteness and reification is better undertaken through indirection and "unsaying" (Misplaced concreteness as a form of encryption, 2021).

It is argued that a polarity, or paradox, is a situation in which opposing forces within a system, pull at each other to keep things balanced, with each "pole" unable to exist without the other -- a natural tension that is an attribute of the system (Larry Clark, Navigating Complexity: managing polarities, Harvard Business Review, 17 December 2018).

Controversies and dichotomies: There is indeed an extensive literature on "controversies" and how they might be modelled (List of Wikipedia controversies; Myungha Jang, et al, Modeling Controversy within Populations, Center for Intelligent Information Retrieval, 2016; KazimierzZielinski, et al, Computing Controversy: formal model and algorithms for detecting controversy on Wikipedia and in search queries, Information Processing and Management, 54, 2018, 1). As noted by the latter, controversy is a complex concept that has been attracting attention of scholars from diverse fields. Many researchers have studied controversy but, surprisingly, no formal computational model for detecting content controversy has yet been proposed or widely accepted -- and there is seemingly little justification for doing so (Brian Martin, The Controversy Manual, Irene Publishing, 2014).

Somewhat ironically, it is significant that even where formal models have been proposed -- as in the case of a "world model" -- this remains  a "mixed controversy", involving different views on facts, theories, principles and values which evolved over three decadesbut has not ended (Magne Myrtveit, The World Model Controversy, The Systems Dynamics Group, 2005).

Arguably controversies arise when participants hold reverse roles on one or more polarities -- each claiming to be right, with the other held to be wrong, for example. In particular cases, one approach is that of Polarity Mapping, also known as Polarity Thinking. This is a four-quadrant visual framework for tackling problems in team collaboration, and other challeges. It is used for paradoxical or costly challenges -- problems that can be exacerbated by focusing on one component more than the other (Marjan J B Govaerts, et al, Managing Tensions in Assessment: moving beyond either-or thinking, Medical Education, 53, 2018, 1; Laurie Levknecht, Using 'Polarity Thinking' to Achieve Sustainable Positive Outcomes, 8 January 2013).

As these approaches imply, typical of the preoccupation with "controversy" and its modelling is the focus on a particular issue or selection of issues rather than on the set of fundamental polarities which trigger the controversy (such as those above), as explored in terms of axes of bias by W. T. Jones (The Romantic Syndrome: toward a new method in cultural anthropology and the history of ideas, 1961).

A somewhat related body of literature focuses variously on the modelling of dichotomies (Georgios Koutsopoulos, et al, Modeling the Dichotomies of Organizational Change: a state-based capability typology, The Practice of Enterprise Modelling, 2019; Knud Thomsen, It Is Time to Dissolve Old Dichotomies in Order to Grasp the Whole Picture of Cognition, Explainable AI and Fuzzy Logic Systems, 2018; Vivian L. Vignoles, et a;, Beyond the 'East-West' Dichotomy: Global Variation in Cultural Models of Selfhood, Journal of Experimental Psychology: General, 145, 2016, 8)

Objectively understood configurations indicative of fundamental cognitive implication

It is useful to explore configurations which are suggestive of richer patterns of articulation of binary distinctions -- whether explicit or by implication. Those of a purportedly objective nature might include the following.

Targets: One mode of resolving the difficulties of polarization is through decision-making -- most obviously executive decision-making, with the potential implication of eliminating the perspective of an opponent through their "execution", whereby they are "taken out". There are of course many dubious examples of the latter, most obviously in the specific targetting of the leadership of groups deemed to be a threat (Osama bin Laden, and the like) and of environmental journalists (for example). Those who could be considered to be targetted in this way notably feature on the "most wanted lists" of law enforcement agencies by whom bounties may be offered.

Of interest with respect to decision-making is the focus offered by a "target" of a physical nature (a shooting target, or "bull's eye"), as with the targetting of intercontinental missiles. A metaphorically equivalent can be recognized in the case of strategic objectives, most obviously a target market and any sense of a financial target.

Potentially problematic are implications resulting from the the uncritical use of military metaphors derived from the focus on physical targets (Enhancing Sustainable Development Strategies through Avoidance of Military Metaphors, 1998). The latter included discussion of Use of "targets" and "bullets" and Mutual targeting in a democratic process. The use of "bullets" is of course a feature of strategic presentations characterized by an array of "bullet points" whereby a target is defined or implied.

There is considerably  irony to the fact that the participants in a polarized situation (and in controversies) frame their opponents as "targets" for campaigns, whereby hopefully the other party will be defeated.

It is somewhat remarkable that goals in the quest for a strategic solution should be framed by what may be recognized as a hunting metaphor -- with all that that may imply cognitively in terms of ensuring some analogue to death in the environment. Are strategic solutions to be understood as targets to be "taken out" -- as in the log-jam metaphor? Alternatively, what is the significance of creating a fatal hole in a living system -- given the mysterious nature of holes as variously explored (Terrence Deacon, 2011; Roberto Casati and Achille  Varzi, 1994). Understood as a metaphor borrowed from sport, what has to be inserted where to satisfactorily transform the environment?

As a device for focusing individual or collective attention, a subtle contrast can however be recognized between a target (below left) and the cross-hairs of a reticle (below right). The latter is a pattern of fine lines or markings built into the eyepiece of a sighting device, such as a telescopic sight in a telescope, a microscope, or the screen of an oscilloscope, to provide measurement references during visual examination.

Target and reticle
Shot grouping on target	Reticle cross-hairs
Binksternet, CC BY-SA 4.0, via Wikimedia Commons	Jellocube27 at English Wikipedia, Public domain, via Wikimedia Commons

Missing from any such configuration, although implied by their use, is the cognitive reality experienced in causing some form of death through successful targetting. Reference is frequently made to this with respect to the actual experience of warfare or of being able to kill a person or an animal. The experience is given particular significance in hunting as a rite of passage ("being blooded") and in the proven capacity to kill as requirement of some urban gangs for membership (as an evocation of respect). Related practices have been evident in academic fencing in which duelling scars (German: Schmisse) were seen as a "badge of honour", a mark of membership of an upper class, and "good husband material". They have been perceived as a way of showing courage in being able to stand and take a blow, as opposed to inflicting the wound.

Given the existence of an extensive array of missiles around the globe, variously targetted on the infrastructure of distant countries, there is a case for noting the use by the United Nations Sustainable Development Goals (SDGs) of an explicit set of 169 targets. Metaphorically understood these may be understood as an array in a multidimensional strategic space through which an analogue to the military strategy of full-spectrum dominance is to be achieved by the UN with respect to global challenges. The targets are only too evident with respect to climate change: reduction of emissions, reduction of global warming.

The articulation of the 169 global targets does not seemingly offer any sense of the coherence of that pattern in systemic terms -- if pattern there is -- and has attracted specific criticism (The 169 Commandments: the proposed sustainable development goals would be worse than useless, The Economist, 28 March 2015). Rather than accept that criticism, and in the spirit of the argument developed here, there is a case for tentatively envisaging how those targets might be coherently configured in a multidimensional space. The approach follows from that indicated separately (Identifying Polyhedra Enabling Memorable Strategic Mapping: visualization of organization and strategic coherence through 3D modelling, 2020).

As with the absence of any suitable polyhedra for mapping the 17 SDGs beyond their tabular arrangement, none was apparent for the 169 targets. Relatively simple mapping possibilities are however available for the 16 primary goals (Functional dynamics of a 16-fold configuration of strategic goals, 2019). The exercise here therefore focused on a polyhedron with 168 vertices (of 8 types) with which targets might then be associated, namely the dual of the so-called "12 Gishi Slice 25" in 4D -- of which a 3D projection can be visualized in 3D as shown below. The Gishi "regiment" is one of 16 regular polychora (or regular 4-polytopes).

The requisite complexity of the multidimensional structure is notably indicated by the manner in which visible vertices are indicated with three conflated numbers of the 168 (below left). The coherence can be variously indicated by colouring options for the edges (parallel, great circle, same type, etc).

Exploratory configuration of 168 targets of the UN Sustainable Development Goals (presented as a 3D projection of a 4D structure -- a polychoron)
Animation	Animation with alternative colouring option
Animations prepared using Stella Polyhedron Navigator

The 4 nested "layers of coherence" can be further indicated by selectively highlighting faces of different types to reveal nested structures as shown below -- the outermost being an icosahedron.

Potential coherence of 168 targets of the UN Sustainable Development Goals (highlighting face patterns of 3D projection of a 4D structure)
Prepared using Stella Polyhedron Navigator

Dartboard: There is an intriguing contrast between a target and a dartboard in that the former is widely adapted for metaphorical purposes whereas the latter is a feature of a widely known physical game involving use of darts (as a substitute for "bullets"). As a metaphor, it is only the seemingly unrelated notion of a "dashboard" that is used for navigation purposes (most notably in vehicles and on web sites) without taking the circular form of a target.

Points can be scored by hitting specific marked areas of the board, though unlike in sports such as archery, these areas are distributed all across the board and do not follow a principle of points increasing towards the centre of the board.

Dartboard and score configuration
Typical dartboard	Scores for each region of a dartboard (not to scale, shaded by value)
Tijmen Stam, CC BY-SA 3.0, via Wikimedia Commons	Cmglee, CC BY-SA 4.0, via Wikimedia Commons

Mathematical studies have considered the design of an "optimal" dartboard among the many possible alternatives. It is somewhat curious to note that these share a focus on factorial 19, given that this number is so fundamental to the design of a Chinese go board.

Magic circles: There is a fair degree of familiarity with magic squares, especially in recreational mathematics. Far less evident is the importance attached to them by Benjamin Franklin, one of the Founding Fathers of the US -- also renowned in Freemasonry, for which such squares are of symbolic significance (Magic Squares, Masonic Dictionary; Paul Pasles, Franklin Squares 2006; Benjamin Franklin's Numbers: an unsung mathematical odyssey, Princeton University Press, 2007). Of particular relevance to this argument is how any sense of "magic" relates both to comprehensibility and to memorability, and more especially in the case of magic circles -- or those of higher dimensionality.

Franklin called his 16x16 magic square the most magically magical of any magic square ever made by a magician -- with which many mathematicians and mystics would now be held to agree, as separately discussed (Magic square integrity and implications for the US Constitution, 2015). The following animations are reproduced from the latter, although how these may have influenced the elaboration and operation of that Constitution remains far less evident.

Franklin's 16x16 magic squares: animations of movement of selected bent diagonals
Vertical movement	Combined movement	Horizontal movement

Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). These are the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles (below left) was constructed from 33 natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the centre.

Examples of magic circles and magic sphere (from commentary in Wikipedia)
Magic circle (Yang Hui)	Magic circle (Ding Yidong)	Magic circle from square	Andrews's magic sphere
numbers on each circle and diameter sum to 138 (ignoring the middle 9)	numbers on each circle (solid colour) sum to 200 and numbers on each diameter (dashed grey) sum to 325	derived from 4x4 most-perfect magic square, containing numbers 1 to 32, with each circle and diameter totalling 132	numbers 1 to 62 arranged along intersections of 5 circles of latitude (dashed grey) and 6 circles of longitude (coloured solid), each circle has 12 numbers totalling 378
Cmglee, CC BY-SA 4.0, via Wikimedia Commons	Cmglee, CC BY-SA 4.0, via Wikimedia Commons	Cmglee, CC BY-SA 4.0, via Wikimedia Commons	Cmglee, CC BY-SA 4.0, via Wikimedia Commons

The argument can be developed in the light of the geometric magic square as indicative of a new way of thinking about the UN Sustainable Development Goals, as an extension of an earlier argument in relation to the Rubik Cube (Interplay of Sustainable Development Goals through Rubik Cube Variations Engaging otherwise with what people find meaningful, 2017). 

Flow diagrams: There are numerous accessible circular flow diagrams (although typically subject to copyright), whether of systemic significance or otherwise. These include representations of:

• circular economy (The Circular Economy in Detail, Ellen MaArthiur Foundation, 2017)
• systems development life cycle
• circular flow diagrams of governance economics
• circular sequencing of human genes by pathways
• circular data visualization

It is appropriate to note that circular flow diagrams may be primarily indicative of flow around an empty centre and with little systemic interaction across the circle.

Classificatory configurations: Flow diagrams are to be contrasted with approaches to circular classification in which categories are nested to various degrees in a circular configuration -- but with little explicit indication of system links between the categories configured. Typically subject to copyright, they include:

• Bloom's taxonomy, namely the many visualizations of a set of three hierarchical models used to classify educational learning objectives into levels of complexity and specificity. The three lists cover the learning objectives in cognitive, affective and sensory domains. (Tips for Writing Instructional Objectives - Bloom's Taxonomy Job Aids,  Social Learning Systems, 25 July 2009)
• Functional classification of proteins based on protein interaction data (What-When-How.com) as a feature of proteomics
• circular periodic tables of the chemical elements exist in many accessible variants, many of which feature a spiral organization

Examples of circular/spiral configurations of periodic table of chemical elements
TROPE ring of periodic elements	Harrison Spiral Periodic Table
Alexander Braun, CC BY-SA 3.0, via Wikimedia Commons	Robert W Harrison, CC BY-SA 3.0, via Wikimedia Commons	Cbuckley at English Wikipedia, CC BY 3.0, via Wikimedia Commons

Systems biology makes use of circular cladograms and dendrograms and phylogenetic trees. A related diagrammatic approach to cognitive biases is that of the Cognitive Bias Codex. as discussed separately with animations (Global configuration of cognitive biases: towards mapping G7 susceptibility, 2018).

Phylogenetic tree	Cognitive bias codex
Highly resolved, automatically generated tree of life, based on completely sequenced genomes
Ivica Letunic: Iletunic. Retraced by Mariana Ruiz Villarreal: LadyofHats, Public domain, via Wikimedia Commons	By Jm3 [CC BY-SA 4.0], from Wikimedia Commons

Genetic code: There is of course intense focus on the genetic code as fundamental to the organization of life. More commonly these are represented in tabular form (DNA and RNA codon tables). RNA and DNA are polymers made of long chains of nucleotides, the basic building block of nucleic acids.. A nucleotide consists of a sugar molecule (either ribose in RNA or deoxyribose in DNA) attached to a phosphate group and a nitrogen-containing base. The bases used in DNA are adenine (A), cytosine (C), guanine (G), and thymine (T). In RNA, the base uracil (U) takes the place of thymine. Nucleotides contain either a purine or a pyrimidine base.

There are however numerous circular configurations of the 64 genetic codons accessible on the web ("circular genetic code" or "codon circle"), with explicit indication of the manner of their nested articulation, as indicated below. The DNA variant below centre is reproduced from an explanatory discussion by Vijini Mallawaarachchi (Starting off in Bioinformatics: turning DNA sequences into protein sequences, Towards Data Science, 13 August 2017).

Examples of circular configurations of genetic codon tables
Standard RNA codon table	DNA variant of codon table	In preparation
Mouagip, Public domain, via Wikimedia Commons	Reproduced from Genetic Code Chart for DNA

The associated circularity is itself a feature of studies including:

• Jerome Lacan and Christian Michel: Analysis of a circular code model (Journal of Theoretical Biology, 213, 2001)
• Christian Michel: Circular Codes in Genes (Université Louis Pasteur de Strasbourg, 2007)
• Christian Michel: A 2006 review of circular codes in genes (Computers and Mathematics with Applications, 55, 2008, 5)
• Anton A. Komar: The "periodic table" of the genetic code: a new way to look at the code and the decoding process (Translation, 4, 2016, 2)

Whilst such diagrams, whether tabular or circular, are recognized as a key to the organization of life, it is somewhat curious that the dynamics of whatever constitutes "life" are only evident in them by implication (if at all), if only with respect to mutation. The genetic code was once believed to be universal, with a codon coding for the same amino acid regardless of the organism or source. It is now accepted that it might vary more frequently than was thought (Variations in the genetic code of life, New Scientist, 1 May 2006). While there is much commonality, different parts of the tree of life use slightly different genetic codes (List of genetic codes).

It is now agreed that the genetic code evolves, resulting in discrepancies in how a codon is translated depending on the genetic source. Arguably the quest for alien life on Mars and elsewhere could be understood as a desperate need to clarity such matters, given the importance attached to that quest and the resources expended.

Purportedly objective configurations with potentially subjective implications

The following cluster could be considered intermediary between an objective and subjective focus, notably in the light of development of circular mind maps.

Traditional knowledge systems: Especially provocative from the objective perspective of science is the degree of credence accorded worldwide to circular configurations, most notably the zodiac (for purposes of astrology) -- arguably far exceeding the degree of credibility accorded to the systems sciences. The inclusion here of such insights is justified by according significance to "what real people think" rather than to "what people ought to think" from some alternative perspective -- however justified.

Of great interest therefore is the careful articulation by Arthur Young of the measure formulae of physics in relation to the circular framing of the zodiacal configuration (Geometry of Meaning, 1976). The luopan or geomantic compass is a Chinese magnetic compass, also known as a Feng Shui compass. It is used by a Feng Shui practitioner to determine the precise direction of a structure, place or item.

Traditional knowledge systems
Zodiac ( with associated "measure formulae")	Chinese Wu Xing and BaGua configurations	Geomantic compass (Luopan)
Following the attributions of Arthur Young (1976)	Reproduced from Wikipedia	Pcae18 at the English Wikipedia, CC BY-SA 3.0, via Wikimedia Commons

System diagrams: Many system diagrams are obviously to be understood from an objective perspective, although the more abstract they are -- and the more they take account of the psychosocial context -- the more likely they are to include a subjective dimension. From the perspective of cybernetics, this is increasingly obvious to the extent that consideration is given to feedback processes, notably third and higher orders -- to the extent that these are recognized (Consciously Self-reflexive Global Initiatives: Renaissance zones, complex adaptive systems, and third order organizations, 2007).

An intermediary condition is notably indicated by the work of Yingxu Wang (Towards the abstract system theory of system science for cognitive and intelligent systems. Complex and Intelligent Systems, 1, 2015; On Cognitive Foundations and Mathematical Theories of Knowledge Science, International Journal of Cognitive Informatics and Natural Intelligence, 10, 2016, 2). Indications of this from those publications are presented in the schematic image below-left.

A truly remarkable analysis of why things ought to work in theory is provided in a magnum opus by Fritjof Capra and Pier Luigi Luisi as The Systems View of Life: a unifying vision (2014). Curiously the cover of the book has an evocative illustration suggestive of the possibility of an integrative visual systemic overview absent from the study itself. In a critical review, following this lead, the schematic (below centre)t was produced in the light of the elements featured on the contents page  (Asystemic organization of unifying vision? 2014).

The argument can be taken further through the circular framework (below right) elaborated by Arthur Young in the light of his preoccupation with cycles of learning in relation to a generic understanding of navigational control (Geometry of Thinking, 1976). His inspiration derived from the innovative insight he acquired in developing the original Bell helicopter with its unusual piloting challenges. He subsequently generalized this in philosophical terms to include the speculative possibility of designing a "psychopter". It could be argued that the quest of global governance for sustainability bears comparison with the challenge of getting the psychosocial system "off the ground" and enabling it to "fly" -- as with a "psychopter" (Clues to 'Ascent' and 'Escape' -- in Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002).

Young's approach can be usefully compared to another, possibly to be understood as controversial to a lesser degree, namely sociophysics (or social physics). There are many flavours to sociophysics, the term having been appropriated by authors with contrasting perspectives (if not mutually antagonistic) -- most being irrelevant to this comparison, or to the argument with respect to discourse at a round table. The framework of sociophysics is usefully summarized by Frank Schweitzer (Sociophysics, Physics Today, 71, 2018). A commentary with respect to Young's presentation features in the work of Paris Arnopoulos (Sociophysics: Cosmos and Chaos in Nature and Culture, 1993). The comparison is discussed separately (Associating significance with a dodecahedron, 2018).

The image (below right) is tentatively augmented with the 15 fundamental concepts of Paris Arnopoulos (Braiding the Triadic Codex and Triple Helix: the sociophysics of nature-culture-nurture and academy-industry-polity, 2000) [text].). Note that 3 of the latter's concepts have been set below, arguably because they do not lend themselves to direct experience in the same manner as the other 12.

Examples of circular configurations of systems
Hyperstructural model of abstract systems	Systemic view of life?	"Rosetta stone of meaning" of Arthur Young supplemented by fundamental concepts of sociophysics
Reproduced from Yingxu Wang (2015)	Configuration of contents of Capra and Luisi (2014)	Using both Arthur Young (1976) and Paris Arnopoulos (2000)

Cycles: Wikipedia provides a very extensive List of cycles. Unfortunately there is little sense of how these might be interrelated -- whether fruitfully or problematically. Of fundamental significance to life are naturally the metabolic cycles which are indeed variously interrelated in metabolic pathway maps (see Full map of metabolic pathways). WikiPathways is a database of biological pathways, depicting many cycles, maintained by and for the scientific community. The image below left is one example of a metabolic cycle within such a map.

With respect to global cycles, the image below centre is a possible configuration of the separate images presented in Biogeochemical Cycles (OpenStax, 20118), as  modified Matthew R. Fisher.

Examples of micro and macro-systemic cycles -- and systemic boundaries
Citric acid cycle (Krebs cycle)	Interrelated biogeochemical cycles	Nine planetary boundaries
Narayanese, WikiUserPedia, YassineMrabet, TotoBaggins, CC BY-SA 3.0, via Wikimedia Commons	Reproduced from Biogeochemical Cycles by OpenStax (2018)	Reproduced from Planetary Boundaries: exploring the safe operating space for humanity, (2009)

Strategic configurations: The configuration indicated below centre derives from an effort to indicate one strategically relevant set of cyclic relationships relating to refugees (Evaluating the Grossness of Gross Domestic Product: Refugees Per Kiloton (RPK) as a missing indicator? 2016)

Examples of circular configurations of systems of strategic significance
Oxfam Doughnut	Systemic relationships framing Refugees per Kiloton	Global "underground map"
Reproduced from Kate Raworth (A Safe and Just Space for Humanity: can we live within the doughnut? (2012).	RPK indicated above as the systemic cycle in red Relationships in blue are associated with unconstrained growth	Reproduced from Mapping the Global Underground, (2010)

Geometrical configuration of music: The organization of tones as experienced in music has progressed far beyond what might be readily associated with scales and musical notation as conventionally represented, as discussed separately (Tone-of-voice insights from music and the organization of the Tonnetz, 2020). This suggests that there are new ways of comprehending the coherence of music which may be of value to comprehension and use of tones-of-voice -- and to the articulation of strategy.

The research of particular interest is notably associated with the manner in which the human brain appears to organize the patterns of tones that is appreciated, as clarified by the work of Dmitri Tymoczko relating music and geometry (A Geometry of Music: harmony and counterpoint in the extended common practice, 2011; The Geometry of Musical Chords, Science, 313, 2006). The manner in which such organization of tones in musical tuning and harmony has long been been explored in terms of the Tonnetz (a tone-network), namely a conceptual lattice diagram representing tonal space. Tymoczko has developed his understanding through its generalization (The Generalized Tonnetz, Journal of Music Theory, 56, 2012. 1).

Selection of graphical representation of the Tonnetz
Circle of fifths showing major and minor keys	Toroidal view of the neo-Riemannian Tonnetz	37 Tonal values in hexagonal lattice array
Just plain Bill, CC BY-SA 3.0, via Wikimedia Commons	Adaptation of Davidwbulger, Public domain, via Wikimedia Commons	Reproduced from McClain (Meditations Through the Quran: tonal images in an oral culture, 1981, p. 95)

Subjectively understood configurations indicative of fundamental cognitive implication

In this cluster far greater emphasis is placed on subjective engagement with the configuration -- potentially to a lesser degree as in the case of logos but to a far greater degree in the case mandala-type structures where the configurations may be held to mirror psychic structure.

Centro-symmetric logos: The widespread use of centro-symmetric logos as indicative of a fundamental preoccupation with strategic integration is of relevance to this argument.

Typically it is however unclear how the structure of such logos relates explicitly to the cognitive articulation of the initiative -- other than by implication of an aspiration especially valuable in public relations and marketing. They notably serve a vital symbolic purpose in expressing the coherence of the initiative -- as a rallying focus for adherents. The highly controversial role of the swastika merits particular attention in this regard (Swastika as Dynamic Pattern Underlying Psychosocial Power Processes, 2012).

Mandalas / Yantras: As a notable variant of centro-symmetric logos, both mandalas and yantras are explicitly intended as an effort to articulate zones of reflection as a focus for meditative attention.

As a geometric configuration of symbols in various spiritual traditions, mandalas may be employed for focusing attention of practitioners and adepts, as a spiritual guidance tool, for establishing a sacred space and as an aid to meditation and trance induction. As a mystical diagram, yantras are also used as an aid in meditation. They are appreciated for their aesthetic and symmetric qualities.

These manifest ions of Eastern cultures are matched by a variety of "rose windows" of Western Christian inspiration, most notably found in Gothic cathedrals and churches. The most common symbolism of the imagery is a reference to the Last Judgement.

Examples of mandala-style configurations
Tibetan mandala of the Naropa tradition	Shri Yantra of the Hindu tradition	Rose window of Strasbourg cathedral
Anonymous, improved by Poke2001, Public domain, via Wikimedia Commons	N.Manytchkine, CC BY-SA 3.0, via Wikimedia Commons	Clostridium, Public domain, via Wikimedia Commons

**** (Eliciting Insight from Mandala-style Logos in 3D: interactive engagement with mandalas and yantras in virtual reality, 2020)

Configuring personality types and gender orientation as the archetypal challenge

Irrespective of the traditional interest in the circular configuration of the zodiac and its appreciation in terms of the astrological typing, many other approaches to personality typing have been developed and legitimated within the psychological disciplines. Some notably date from the work Carl Jung on a 4-fold articulation (Four functions: sensation, intuition, thinking, feeling), subsequently extended to an 8-fold configuration of types. Several modern theories are closely associated with Jung's set of types. The range of initiatives is usefully summarized by Stefan Boros (Introduction to Typology, Open Journal of Jungian Typology, 1 September 2017)

Seemingly unrelated approaches include that of the Big Five system and the Enneagram of Personality. An embedding of the enneagram (seemingly a 2D figure) has been discovered within an icosahedron in 3D, as illustrated below right and discussed separately (Representation of Creative Processes through Dynamics in Three Dimensions: global insight from spherical reframing of mandalas, the zodiac and the enneagram, 2014).

Several 16-fold articulations exist, notably the Sixteen Personality Factor Questionnaire (16PF) and the MBTI system. As discussed below, also of relevance are proposals for 32-fold and 64-fold articulations (Erik Thor, Personality Test, December 2020).

Perhaps the most comprehensive articulation is that deriving from integral theory, as articulated by Ken Wilber. This is centred on the 4-fold AQAL pattern ("All Quadrants All Levels"), interrelating levels or stages, lines, streams, or intelligences, states, and types. The pattern of insights is so complex that it has proved difficult to represent it visually as a comprehensible whole -- a challenge all the greater in that the AQAL system has been integrated with an understanding of spiral dynamics, itself subject to various copyright constraints.

Curiously the complexity suggests the possibility of a "gyroscopic" representation derived from the classical configuration of the most complex chakra of Hindu tradition (Global Insight from Crown Chakra Dynamics in 3D? Strategic viability through interrelating 1,000 perspectives in virtual reality, 2020). The latter included the animation below centre.

Personality typing configurations
AQAL Map of integral theory	Animation of "gyroscopic" movement of 1,000 petalled chakra	9-fold enneagram embedded within an icosahedron
Reproduced from Open Source Integral	interactive web -- mp4 -- x3d

Somewhat ironically, just as the astrological approach is deprecated by professional psychologists, the articulations of the different personality type theories tend to be deprecated within those professions by their respective advocates. They may well defend their systems by copyright and vigorous litigation. For that reason it is problematic to reproduce here any of the many circular configurations through which these may be presented.

In a world bedevilled by conflicting relations between personality types, it is curious to note the inability of the advocates of contrasting systems either to reconcile their perspectives or to enable reproduction of their visualizations. This is all the more curious in that it is the deprecated traditional systems which circulate freely without copyright constraint or threat of litigation.

The irony is all the greater in that the originators of such systems are seldom explicit about the type by which their own system defines them, and how this compares with that of the proponents of alternative types which they may vigorously deprecate. This is somewhat curious in that Jung's original effort was conceived as a means of reconciling the insights of his peers through recognition of their contrasting types. 

More problematic is the degree to which a motivation for such typing is effectively to "pigeon-hole" individuals permanently, with little expectation that they may shift between the associated cognitive functions during their daily life or over time -- nor that their experiential challenge may be to integrate the functions in the kind of individuation process envisaged by Jung. There is little explicit concern with how contrasting types may fruitfully communicate with each other -- beyond the indulgence in deprecation. Nor is there any strong preoccupation with the complementary nature of contrasting types. The challenge has been framed to some degree by a number of authors, as separately discussed (Systems of Categories Distinguishing Cultural Biases, 1993), noting the particular perspective of the philosopher W. T. Jones (The Romantic Syndrome: toward a new method in cultural anthropology and the history of ideas, 1961).

Especially valuable to this argument is the reconciliation between 5-fold patterns and the extensions of the binary pattern, as articulated by Ulrich Schimmack (32 Personality Types, Replicability Index, 9 September 2019):

The dominant framework in personality psychology is the Big Five model that conceptualizes personality traits as five independent continuous dimensions. If we were to create personality types by splitting each dimension at the median, it would create 32 personality types, where individuals are either above or below the median on neuroticism, extraversion, openness, agreeableness, and conscientiousness. if these five dimensions were perfectly independent of each other, we would see that individuals are equally likely to be assigned to one of the 32 types....

Personality psychologists are frustrated that they have discovered the Big Five factors and created a scientific model of personality, but in applied settings the Myers-Briggs Type Indicator (MBTI) dominates personality assessment. One possible reason is that the MBTI provides simple information about personality by classifying individuals into 16 types. These 16 types are defined by being high or low on four dimensions.

Gender identity: Whereas psychological typing focuses on 4-fold and greater articulations, implicit is the more fundamental distinction made between the styles of thinking that are held to be characterized by the 2-fold male-female pattern. As noted above, the rigid distinction between "male" and "female" continues to be designed into official identity profiles, grammatical construction, clothing and toilet facilities -- despite the increasingly successful challenge of the LGBT communities.

A primary concern is with the increasing levels of harassment and rape between those distinctively identified, as discussed more generally (Global Civilization through Interweaving Polyamory and Polyanimosity? Loving/Hating the world otherwise through contractual bonding with any significant other, 2018). Psychosocial variants are evident in the conflicts between those of distinctive personality types, whether individually or as collectivities. Reproduced from that discussion, the associated ambiguities are suggested by the following "centro-symmetric" imagery.

Schematizing the representation of harassment?	Basic gender-orientation symbols Configured centro-symmetrically as animations

Given the confusion regarding the extended distinction of gender-orientation symbols, it is as yet unclear what such a corresponding set might usefully signify, and how they might be reduced to a coherent pattern of 8, for example, as discussed separately (Encompassing the "attraction-harassment" dynamic with a notation of requisite ambiguity? 2017). A set of 12 with associated computer codes is available (Gender Symbols Female and Male Signs, Alt-Codes). A larger set is presented by Anunnaki Ray (All the Gender Symbols), with indications of an even more comprehensive set as shown below left

Of interest with respect to a 3D interpretation is the sense in which the linear form of the exclamation mark -- adapted for use as "Cupid's arrow", is better understood as a circle in the third dimension -- seen here edge on (below right). The alternative directionality of the arrow can then be understood as suggesting the two distinctive directions of rotation of that circle. Below right is an experimental array of visual anagrams derived through progressive geometrical transformation, discussed separately (Indicating transformations between forms of polyamory and polyanimosity as "visual anagrams"? 2018).

Possible gender orientation symbols
Comprehensive set of gender-orientation symbols	Experimental array of visual anagrams
Reproduced from Cari-Rez-Lobo (Gender Symbols, Deviant Art), which includes a further clarification of each	Generated using transformation facilities of Adobe Illustrator (with improvements to the design to be envisaged)

Interactively understood configurations indicative of fundamental cognitive implication

It could be argued that the primarily static images above have been considered inadequate to a form of essentially subjective comprehension which requires dynamic engagement, consistent with the argument of Nassim Nicholas Taleb (Skin in the Game: hidden asymmetries in daily life, 2018). This is most obvious in the ritual construction and destruction of a sand mandala, as practiced in Tibetan Buddhism (The Symbolism behond the Creation and Destruction of the Buddhist Sand Mandala, Buddhist Art, 21 August 2015).

Labyrinth: Also of notably interest is any ritual associated with traversing a labyrinth -- especially one combining circular and spiral forms. Recognized as a form of labyrinth, the traditional Neolithic image of the triskelion (below right) invites exploration using visualization technology (Cognitive Implications in 3D of Triadic Symbols Valued in 2D: representations of the triskelion in virtual reality and implications for quantum consciousness, 2017).

Examples of labyrinths
Chartres Cathedral	Cretan labyrinth	Triskelion with coloured spheres moving between focal points (video)
Ssolbergj, CC BY-SA 3.0, via Wikimedia Commons	Reproduced from Wikipedia

Angelic arrays and magic circles: The argument above with respect to magic circles has invited further traditional reflection regarding their configuration, as indicated below right and discussed in Wikipedia (Yang Hui magic circles). The configuration merits comparison with that of angels and demons, as indicated below left and discussed separately (Hyperbolic reframing of the Demonique and Angelique of tradition, 2016) The allocation of sets to the star array of "tables" is based on the tabular form in which the 72 angels and demons are typically presented. The rows are presented "around the tables" in one schematic, and the columns are presented "around the tables" in the other. The sequence around the tables is clearly questionable, demanding further consideration.

Comparable arrangements from a traditional perspective?
Alternation between configurations of 72 angels and demons of tradition (animation)		Arrangements of magic circles (according to Yang Hui)
Animation of 8 sets of 9 (enlargements for detail: angels / demons)	Animation of 9 sets of 8 (enlargements for detail: angels / demons)	8 magic circles in a square (Yang Hui)	9 magic circles in a square (Yang Hui)
Reproduced from Hyperbolic reframing of the Demonique and Angelique of tradition (2016)		Yang Hui 13th century. ????????, Public domain, via Wikimedia Commons

Games: Especially as a consequence of the development of interactive web facilities, circular configurations feature in a number of games -- or game-like representations. These include

• Target shooting can take a game-like form -- most obviously in the game of darts (as mentioned above), potentially as a surrogate for hunting (or a prelude to it)
• Ward's® Genetic Code Wheel Game
• SanHe FengShui LuoPan turns iPad into a tool for any practitioner
• Interactive Metabolic Pathways Map (Merck), in valuable detail with numerous highlighted cycles, although not presented within a general cycle
• The New Harlequin Game (1800/50)
• Rubik game
• circular games
• periodic table

Configuring polarities for cognitive engagement of higher order

This argument was introduced by recognizing a selective set of polarities, These raise the question of how many polarized situations can be coherently handled simultaneously or through some pattern of alternation between them. The question could be framed in terms of the strategic dilemmas, paradoxes or contradictions which any form of governance is required to handle (Governance as "juggling" -- Juggling as "governance": dynamics of braiding incommensurable insights for sustainable governance, 2018). Traditionally this has been recognized in terms of a riddle or a Gordian knot (Global Governance as a Riddle, 2018).

Polarities might be configured coherently as an exercise, notably as lemmas of higher order, following from some of the clues presented above from other sources (Vital distinctions beyond dilemmas, 2015).

It could be assumed that any N-fold lemma is an effective block on coherent comprehension. Each can be recognized as a form of Gestalt experienced as a whole, namely a knot that cannot be disentangled. It is in this sense that greater coherence can be sought through articulation -- ironically by using configurations of polarities, triplicities or quadruplicities as clarified to different degrees by the following. It is also useful to recognize that these can be recognized as feedback loops of second or higher order.

It is especially interesting that the engagement with these complexes is partially undertaken through engagement in games, and especially competitive games. Curiously the complexity can be "collapsed" by the throw of a dice -- a form of decision-making from which a singular result is derived, whether the result is held to be a consequence of chance or supernatural intervention (Decision-making capacity versus Distinction-making capacity. 2015). In relation to the following table, it is surprising to note the 31 distinctive N-faced polyhedral dice offering from 1 to 144 possibilities (Dice by number of sides, Wikimedia), although the games requiring such dice are not indicated.

As is evident in the commentary on the following table, comprehension of complex patterns is primarily facilitated by the role of symmetry in relation to polarity and duality -- implied to a degree by the considerable appreciation of symmetry in the arts. Mathematics has developed an extensive literature on the relation between symmetry group theory, with the the polyhedral group being any of the symmetry groups of the Platonic solids, as can be discussed in mathematical terms (S. De Haro and J. Butterfield, On Symmetry and Duality, Synthese, 2019; Marcus du Sautoy, Finding Moonshine: a mathematician's journey through symmetry, 2008). With respect to any sense of coherence, a summary of relevance is offered by Wikipedia (List of finite spherical symmetry groups).

There is however a fundamental challenge in using the rich insights of mathematics to clarify complex configurations of polarities. There is a strangely irritating disconnect between the ability of mathematicians to explore very high orders of symmetry (exemplified by the Monster Group) and the degree to which articulations in mathematical language are comprehensible outside that realm of discourse, as discussed separately (Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008). The point is curiously made Marcus du Sautoy:

I can portray the geometry in four dimensions without having to concern myself at all with trying to visualize it... Although I can't see the hypercube, the mathematical language allows me to manipulate it and explore its symmetries. The numbers give me... a sixth sense -- the feeling that I really can see in four dimensions. (2008, p. 27).

The challenge of the disconnect in managing complexes of strategic polarities is seemingly of little interest to mathematicians, but is usefully summarized by the following schema from that argument. The unresolved question with respect to complex arrays of polarities is which of the variety of forms of symmetry enhance comprehension and memorability of coherence -- and to what degree? Of value in this respect is the study by Mihai Gabriel Constantin, et al. (Computational Understanding of Visual Interestingness Beyond Semantics: literature survey and analysis of covariates, 2018). This notes that among a variety of dimensions symmetry and comprehensibility were "mostly unexplored". The authors note:

Symmetry reflects the extent to which shape and position of visual parts match after translation, rotation or mirroring with reference to an axis or a point in space... Balance/Harmony -- Somehow related to symmetry, compositional balance "unifies the structural elements of a pictorial display into a cohesive organization or framework that helps determine the role of each element within a composition"...The memorability of an image is its intrinsic ability of being stored by our visual memory: it reflects the extent to which the image can be remembered by human mind.

Indication of relationship between dimensions discussed relating to engagement with symmetry (tentative)

This simplistic figure holds some dimensions of the more complex diagram that is the subject of Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation (2007) and of an animation in Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori (2006).

Recognizing symmetry as a harmonious balance -- possibly fundamental to sustainability -- the strategic role of games is usefully reviewed in relation to a time-management survival game (Symmetry) set in a post-apocalyptic world (Jonathan Moore, Symmetry Review: a harrowing and uneven survival experience, GameSkinny).

In the quest for possible configurations of polarities, the following table is consistent with an earlier exploration (Identifying Polyhedra Enabling Memorable Strategic Mapping: visualization of organization and strategic coherence through 3D modelling, 2020).

1x2 (polarity, dilemma)	2x2 [=4]	22 [=4]		2x3 [=6]	2x5 [=10]
1x3 (trilemma)	2x3 or 3x2 [=6]	23 [=8]	32 [=9]	2x32 [=18]	3x5 [=15]
			33 [=27]	2x33 [=54]
1x4 (quadrilemma; tetralemma)	2x4 or 4x2 [=8]	24 [=16]	34 [=81]	2x34 [=162]	4x5 [=20]
1x5 (quintilemma; pentalemma)	2x5 or 5x2 [=10]	25 [=32]	35 [=243]	2x35 [=486]	5x5 [=25]
1x6 (hexalemma)	2x6 or 6x2 [=12]	26 [=64]	36 [=729]	2x36 [=1458]	6x5 [=30]
1x7 (heptalemma)	2x7 or 7x2 [=14]	27 [=128]	37 [=2187]	2x37 [=4374]	7x5 [=35]

• #2: seemingly eluding greater clarity as archetypal incommensurables, as noted above especially with respect to dilemmas (Systemic Mapping of Strategic Dilemmas, 1992; Configuring Strategic Dilemmas in Intersectoral Dialogue: summary of analysis on the occasion of Earth Summit, 1992)
  • Visualization: rendered curiously comprehensible to a degree in 2-sided games and competitions through which one party emerges and triumphant as a consequence of the crushing defeat of the other (Reframing the Game of Strategic Dilemmas: a 12-fold interplay of possibilities of otherwise, 2009; Reframing Strategic Dilemmas through 12 Modes, 2009; Sustainability through the Dynamics of Strategic Dilemmas: in the light of the coherence and visual form of the Mandelbrot set, 2005). Given the role of dice, tossing a coin is commonly used to collapse the complexity.
    
    
• #3: essentially mysteriously problematic, as with any Middle Way, trilemma, trinity, or "eternal triangle" (Arielle Saiber and Aba Mbirika, The Three Giri of Paradiso XXXIII, Dante Studies, 131, 2013), except as substructures (Triangulation of Incommensurable Concepts for Global Configuration, 2011):
  • Stavros Sintos and Panayiotis Tsaparas: Using strong triadic closure to characterize ties in social networks (August 2014)
  • Marco Winkler: On the Role of Triadic Substructures in Complex Networks (Dissertation, 2015)
  • Paris Arnopoulos: Braiding the Triadic Codex and Triple Helix: the sociophysics of nature-culture-nurture and academy-industry-polity. (2000)
  • Visualization: triadic symbols in 2D; basic surface features of many 3D polyhedra. Rarely rendered explicit in 3-sided games, most notably 3-sided football. Decision-making with 3-sided dice
    
    
• #4: tantalizingly comprehensible, as evident in 4-fold games like bridge (and other card games), but evidently problematic, potentially as with any sense of being at "cross purposes", or with reconciling 4 contrasting cognitive functions (Four-dimensional requisite for a time-bound global civilization? 2015):
  • Andrew J. Mitchell: The Fourfold: reading the late Heidegger (Northwestern University Press, 2015)
  • John Sellars: The Fourfold Remedy: Epicurus and the Art of Happiness (Allen Lane, 2021)
  • André Unterberger: The Fourfold Way in Real Analysis: an alternative to the metaplectic representation (Birkhäuser, 2006)
  • Burton A. Weisbrod: The Health Care Quadrilemma: an essay on technological change, insurance, quality of care, and cost containment (Journal of Economic Literature, 29, 1991)
  • Arthur Schopenhauer: On the Fourfold Root of the Principle of Sufficient Reason (1813) 
  • As tetralemma, this features prominently in the classical logic of India regarding fourfold negation (Notes on the Tetralemma, Twelve Links, 2006)
  • Visualization: various forms of cross in 2D, subject to controversial interpretation; implied by the diagonals of a cube; tetrahedral structure. Potentially understood through games using a 4-fold design of playing cards (Distinguishing normality using playing card suits as a pattern language, 2015). Possible through the exploration of 4-sided games (Insights for democratic global governance from 4-team ball-games: 2-way football? 2016). Decision-making with 4-sided dice
    
    
• #5: framed as an intuitive reference to quintessence, seemingly rendered explicit in the Hygieia of the Pythagoreans and the Chinese Wu Xing pattern (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012), potentially of strategic relevance (Five-fold ordering of strategic engagement with time, 2015; Five-fold cognitive dynamics of relevance to governance? 2015):
  • István Hargittai (Ed): Fivefold Symmetry (World Scientific, 1992)
  • References to a quintilemma as a general argument for Christianity; evangelical references to fivefold ministry (Understanding 5-Fold Ministry; Do You Know Why Fivefold Ministry is Essential?). Although various references are made, the most specific are the The Pentalemma: an Erisian elaboration on the Catsukoti (The Open Source Religion Social Network, 2006) and the Need to recognize quintucrisis or this five headed monster (Global Sustainability Project of Rediluvism).
  • Requisite Muslim performance of salah -- five daily prayers
  • Helmut Braun: Das Quintilemma im Gesundheitswesen: ein Beitrag zur ökonomischen Theorie der Diffusionsprozesse medizin-technischer Grossgeräte zur Diagnose und Therapie (1995)
  • Visualization: various 5-pointed stars in 2D; surface features of many 3D polyhedra; design of a traditional pyramid. Decision-making with 5-sided dice
    
    
• #6: potentially the first level at which configuration of polarities can be presented coherently and independently. It is a primary feature of the benzene molecule as the resonance hybrid structure fundamental to organic life:
  • Edward de Bono: Six Frames For Thinking About Information, Vermilion, 2008; Six Thinking Hats, Little Brown, 1985)
  • Puja Ahuja: Sixfold Spiritual Intelligence Scale: calibration and standardisation (Educationia Confab, 2, 2013, 11)
  • Raymond Abellio: La Structure absolue: essai de phénoménologie génétique (Gallimard, 1965)
  • Rachel Roe-Dale and Rachel Seligman: Sixfold Symmetry Pattern in Art and Science (Tang Teaching Museum, 2018)
  • Reference to hexalemma in relation to Derrida and Madhyamika Buddhism: from linguistic deconstruction to criticism of onto-theologies (Chinese Buddhist Encyclopedia; Cai Zongqi, Derrida and Seng-Zhao: linguistic and philosophical deconstruction, Philosophy East and West, 1993). Scott Aikin describes the challenge of a "hexalemma" in terms of justification scepticism: foundationalism, contextualism (unjustified justifiers), externalism, circularism, coherentism, and infinitism (Epistemology and the Regress Problem, Routledge, 2011, p. 46)
  • Visualization: *** Decision-making with 6-sided dice (the most common form).
    
    
• #7: A heptalemma is allegedly significant in Jain Dharma (Saman Suttam)
  
  
• #8: An octalemma is discussed in relation to Strategic Octalemmas (2009) and by Edward Anderson (Problem of Time in Quantum Gravity, 2012). The Euthyphro "dilemma" is restated as an octalemma by Richard Goode (Nothing Is Permitted: an argument for moral eliminativism).

These possibilities are separately explored further (Transcending Psychosocial Polarization with Tensegrity: biomimetic clues to collective resilience and unshackling knowledge, 2021)

Exercises in centro-symmetric encoding using hexagrams

An underlying inspiration of this exploration is the merit in a global civilization of taking respectful advantage of both the emerging insights driven by science and traditional approaches to challenges otherwise conceived. The point has been remarkably made by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999) as discussed separately (Enhancing the Quality of Knowing through Integration of East-West metaphors, 2000).

Given its degree of articulation, and a degree of correspondence with other circular configurations presented above, the use of the Chinese trigram, hexagram and tetragram encodings merit further attention in the quest for higher degrees of coherence, as argued separately (Discovering Richer Patterns of Comprehension to Reframe Polarization: comprehension phases, 1998; Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007). The concern here is with identifying an appropriate pattern language to encompass such contrasting perspectives -- not with premature interpretations of whatever such a language may imply.

20-fold encoding? In that light the widely noted degree of correspondence between the Chinese articulation and that with respect to the array of genetic codons can be understood as more than a coincidence and potentially indicative of cognitive constraints on comprehensible order. Some relevant arguments include:

• Martin Schönberger: The I Ching and the Genetic Code: the hidden key to life (Aurora, 1992)
• Sergey Fedotov: Simple association of the genetic code and hexagrams of the Book of Changes (I-Ching) (Cardiometry, February 2016)
• Deyi Wang: The Human Genetic Code and I Ching's 64 Hexagrams (May 2019)
• Fernando Castro-Chavez:
  • The Digram I Ching Genetic Code Compresses the Genetic Code into 24 Compatible Main Codons, July 2019).
  • Defragged Binary I Ching Genetic Code Chromosomes: compared to Nirenberg's and transformed into rotating 2D circles and squares and into a 3D 100% symmetrical tetrahedron coupled to a functional one to discern start from non-start methionines through a stella octangula (Journal of Proteome Science and Computational Biology, 1, 2012 3)
• Sergey Petoukhov: The genetic code, 8-dimensional hypercomplex numbers and dyadic shifts (arxiv.org, 2011)
• Sergey Petoukhov and Matthew He: Matrix Genetics and Culture (IGI Global, 2010)
• Tsung-I Dow: The Identicalness between the DNA Replication Sequence and the Hexagram Progression Sequence in I-Ching (Presentation to the 9th International Conference on the Studies of I-Ching, University of Hawaii, 1992).
• Quincy Robinson and Paul Martyn-Smith: Re-integration: Nine Realms of Middle Earth (2015)
• Paul Martyn-Smith: Finding both the Binary Opposites in Cubes (Hexagrams) and their Reversals (2016)

Some 500 amino acids are currently known but only 20 appear in the genetic code as the the building blocks for life as it is known. More intriguing is the manner in which the 64 codons code for the 20 amino acids so fundamental to life -- evoking preoccupation with how they are to be memorized (Tips and Mnemonics for Memorizing Amino Acid Structures, Cambridge Coaching, 18 January 2017).

It could then be asked how, in systemic terms, the 64 hexagrams of the I Ching pattern "code for" a psychosocial analogue to those amino acids (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). Many clues to the possibility are evident (Requisite 20-fold Articulation of Operative Insights? Checklist of web resources on 20 strategies, rules, methods and insights, 2018).

Relation to binary computer coding: There is no lack of references in the history of mathematics and computing to the role of the encoding of the I Ching (Yi Jing) in providing inspiration to Gottfied Leibniz in 1701 with regard to the development of binary logic, as previously noted ((Framing Cognitive Space for Higher Order Coherence: toroidal interweaving from I Ching to supercomputers and back? 2019). These include Mary von Aue (Gottfried Wilhelm Leibniz how the 'I Ching' inspired his binary system, Inverse, 1 July 2018; Will Buckingham,  Forget Prophecy: the I Ching is an uncertainty machine, Aeon, 11 October 2013). The latter argues that using the I Ching is a weirdly useful way to open one's mind to life's unexpected twists -- surely a vital requirement in a surreal period of global chaos.

In contrast to the more widely known tabular configurations, the circle of Shao Yong (1011-1077) or the I Ching hexagram circle was an influential feature of the communication to Leibniz in 1701 (James A. Ryan, Leibniz' Binary System and Shao Yong's "Yijing", Philosophy East and West, 46, 1996, 1). Features of the configuration are discussed separately (Diagram of 384 Relationships between I Ching Hexagrams, 1983; Bagua and the sequence of 64 hexagrams, Shanghai Daily, 20 December 2015).

The question here is how to embody more fruitfully the psychosocial dynamics implied by the I Ching encoding patterns. How can the coherence of seemingly incommensurable contrasts be represented with the aid of new technologies? Previous experiments with this pattern offer access, from each of the 64 positions, to classical Chinese commentary adapted with respect to: Dialogue / Vision / Conference / Policy / Network / Community / Lifestyle

The possibility had been clarified in an earlier study in the light of the alternation between two orientations as shown below centre and right (Alternating between Complementary Conditions -- for sustainable dialogue, vision, conference, policy, network, community and lifestyle, 1983).

Map of transformations encoded by a circle of hexagrams and their relationships
Shao Yong circle of hexagrams as communicated to Leibniz (1703)	Global, 'heads-together' networking conditions ('top-in')	Local, 'back-to-back' networking conditions ('top-out')
By Unknown - Perkins, Franklin. Leibniz and China: a commerce of light. Cambridge UP, 2004. 117., Public Domain, Link	Reproduced from Alternating between Complementary Conditions (1983)

Reconciliation with tabular arrangements?  The Shao Yong circle above left includes the more familiar tabular arrangement and therefore implies a reconciliation. There are however four such 8x8 traditional presentations Fu Xi; Jing Fang; King Wen; Mawangdui, as discussed separately ((Strategic Patterns in terms of Knowing, Feeling and Action, 2008; Steve Marshall, Yijing Hexagram Sequences, 2005; Yijing Dao, Archive of Yijing scans from Chinese and other sources, 2006). That 8x8 patterning could be understood as a form of "periodic table" -- but with respect to intangible matters.

The four traditional tabular configurations can be provocatively combined into a single animation reproduced from Identifying the Root Cause Focus of Radical Identity: reframing the complex space of radicalisation dynamics (2015).

Animation combining Fu Xi; Jing Fang; King Wen; Mawangdui

Cognitive engagement? Of interest is how people have chosen to engage cognitively and strategically with the 8x8 pattern so fundamental to chess. Opening theory in chess recognizes 20 legal moves for each player in the starting positions (First Moves for White, Chess Strategy Online). Are the "most important tactical patterns" indeed to be understood as 20-fold (20 Motifs Chess Tactics Course, ChessFox)? Why is the typical dartboard configured in 20 sectors, as noted above? And why is the game of go played on a 19x19 board? The 1,000 petalled lotus chakra of Hindu tradition, as mentioned above, is composed of 20 rings of 50 petals.

In the quest for coherence, the argument above suggests that the comprehensibility of tabular animations is inadequate to the challenge of the times. This frames the question as to whether dynamic patterns can be presented otherwise, as explored below.

The image on the left below derived from separate discussions (Living with Incomprehension and Uncertainty: re-cognizing the varieties of non-comprehension and misunderstanding, Towards the Systematic Reframing of Incomprehension through Metaphor, Towards the Dynamic Art of Partial Comprehension, 2012)

The images below (centre and right) derived from a discussion of Changing Patterns using Transformation Pathways, 2015) which explored "camp-us" inspiration in the context an alien world view.

Experimental association of "colours" to the 64 hexagrams of the I Ching	Depiction of Rummer's transformations in relation to the circle of hexagrams (surrounded by a circle of codons)	Depiction of transformations based on Rummer in relation to the circle of hexagrams (surrounded by a circle of codons)

Of particular interest in the case of the images on the right is the extensive study by Anagarika Govinda (The Inner Structure of the I Ching; the Book of Transformations, 1981), as discussed with imagery (Circular representation: inner structure, 1983).

Rotation of configurations? Another approach is through rotation of the configuration, as illustrated below from an earlier exercise (Experimental Revolutionary Animations of a Chinese Pattern of Metaphors: based on rotations of a circular configuration of I Ching hexagrams, 2015)

Screen shots of experimental animations presented separately (in case of browser difficulties in presenting animations)
Inner circle of trigrams rotating relative to outer circle of trigrams [animation; mp4]	With rotation of outer ring [slower] [faster]	With counter-rotation of 2 outer rings [slower]	With counter-rotation of 4 outer rings [faster]

Container for polarization -- an analogue to geodesic domes?

It is too readily assumed that systemic configurations, especially those of symbolic significance, can be adequately represented and comprehended in 2D, as argued separately (Cognitive Implications in 3D of Triadic Symbols Valued in 2D, 2017). There is every possibility that polarization of the most fundamental kind can only be clarified and resolved in four dimensions or more (Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness, 2004; Comprehending the shapes of time through four-dimensional uniform polychora and Five-fold ordering of strategic engagement with time, 2015). Hence the justification for the tentative representation above of the UN's 168 SDG targets using a polychoron.

The fundamental polarities, as discussed above, noted the recognition of the role of tension. The associated dynamics tend to be completely lost in static representations in 2D. The contrast is evident in visual representation of the tensegrity structures whose organizing principle is fundamental to both the architecture of biological cells and to geodesic domes (Donald E Ingber, et al. Tensegrity, cellular biophysics, and the mechanics of living systems, Reports on Progress in Physics, 77, 2014, 4; Samantha Pires, 8 Incredible Structures Around the World That Use Tensegrity to Defy Gravity My Modern Met, 2 January 2021).

The simplest tensegrity structure (a T3-prism) is presented in the animation below left. Each of three compression members (green) is symmetric with the other two, and symmetric from end to end. Each end is connected to three cables (red), which provide tension and precisely define the position of that end. The green elements could each be understood as polarities separating incommensurable extremes. Missing in the common approach to polarities is the existence and nature of the red tension elements by which seemingly unrelated polarities can be configured into coherent structures.

A problem -- of which the world now has many -- can be understood as deriving from a conflictual polarity. Studied as "problem jostling" by management cybernetician Stafford Beer, he mapped distinct problems onto an icosahedron (Beyond Dispute: the invention of team syntegrity, 1994). Beer related the pattern of tension and compression to that of a tensegrity structure.

The tensegrity possibilities had also been on the occasion of the UN Earth Summit (1992) in Configuring Strategic Dilemmas in Intersectoral Dialogue (1992). The image on the right below shows the unfolded tensegrity of dilemmas from a table of inter-sectoral strategic dilemmas of sustainable development (see also Alternate B and commentary).

Indications of tensegrity configurations
Simplest tensegrity	Spherical tensegrity	Configuration of Earth Summit dilemmas
Cmglee, CC BY-SA 3.0, via Wikimedia Commons

Given the demonstrated possibility of geodesic domes -- despite their seeming improbability in architectural terms -- it would seem that there is indeed potentially the possibility of analogues in psychosocial terms as "containers of polarities" in a knowledge-based civilization dependent on knowledge architecture, as previously argued (From Networking to Tensegrity Organization, 1984; From Zoom Organization to Zome Configuration and Dynamics, 2020).

Other possibilities of configurations in 3D are explored separately, from which the animations below are reproduced (Framing Cognitive Space for Higher Order Coherence: toroidal interweaving from I Ching to supercomputers and back? 2019).

Drilled truncated cube with rotation of mutually orthogonal circles of 64 hexagrams
Single circle of hexagrams	Double circle of hexagrams	Triple circle of hexagrams
X3D model; prepared using X3D-Edit and Stella Polyhedron Navigator

Drilled truncated cube with animations of circles of hexagrams (augmented by transformation pathways)
Fast rotation of single circle	Circles associated with faces and verticals	Circles on faces of polyhedron
	video; X3D

Systematic configuration of ambiguity and liminality?

Rather than the oversimplistic focus on "black-or-white" decisions, especially as theoretical and ethical abstractions, the circular arrays above imply the possibility of nuanced insights of greater or lesser complexity. Ironically the implication of the argument is that a black-and-white coding system might be used to highlight such nuances to varying degrees. Given the manner in which that binary system is so fundamental to computer operation, the organization of computer memory, and logic, it calls for attentive re-examination, especially now that the simplicity of binary operations is challenged by quantum computing and the possibility of superposition.

The adequacy of the circular arrays of hexagrams which are the focus above usefully frame the need to articulate their central domains. Their structure is indeed implied, but it is not articulated. By contrast the circular arrays of genetic codons do indeed have concentric central rings, although not articulated to the degree implied by the hexagram coding.

As an exercise there is therefore a case for using the binary coding system to articulate the array from a central circle through rings with 2, 4, 8, 16, 32 to 64 distinctions (a total of 126 or 127). The hexagrams of the I Ching highlight a further challenge to comprehension deriving from the direction in which they are "read", whether "bottom-up" or "top-down" and the significance which that distinction may imply. This can be discussed more generally (Unquestioned Bias in Governance from Direction of Reading? Political implications of reading from left-to-right, right-to-left, or top-down, 2016).

Given that perspective it is useful to consider the following distinctive orientations of the augmented circular array of hexagrams -- highlighting the contrast with that traditionally presented.

Distinctive orientations of circular array of nested coding patterns
Original orientation	Rotated right	Inverted	Rotated left

Comprehension can be enhanced otherwise by variously incorporating the four distinctive perspectives above into animations. Those below offer a contrast between a clockwise rotation and one involving reflection with a contrasting perspective (vertically or horizontally). The animations reveal a degree of technical imperfection in the original, however these fortuitously give rise to small movements which enhance the sense of a living system. Advantage was also taken of a technical facility for fading between the perspectives -- offering subtle rather than abrupt transitions.

Animations between distinctive orientations of circular array of nested coding patterns
Clockwise rotation	Reflection of opposites

The problematic nature of polarization engenders incomprehension and uncertainty in the guise of offering comprehension and certainty. The challenge is to navigate such dynamics, as argued separately (Living with Incomprehension and Uncertainty: re-cognizing the varieties of non-comprehension and misunderstanding, 2012). This explores the following themes:

Omnipresence of non-comprehension and ignorance Nature of incomprehension Incomprehension versus Misunderstanding Incomprehension in personal experience Incomprehension in relation to governance Incomprehension and ignorance within systems of belief Incomprehensible constraints on comprehensive capacity Experiencing incomprehension personally

Incomprehension in the face of information overload Incomprehensible inadequacy of collective response Being misunderstood and incomprehension of the other Re-cognition, epistemology and metaphysics Towards the systematic reframing of incomprehension through metaphor  Towards the dynamic art of partial comprehension

The context in which people are now called upon to live and move and have their being -- if they venture beyond bubbles of comfort -- can be framed as the art of "betwixt and between" (Living as an Imaginal Bridge between Worlds: global implications of "betwixt and between" and liminality, 2011). This explores the following themes:

Bridges: iconic, archetypal and otherwise Beyond "Us" and "Them" Living "betwixt and between" Liminality of betwixt and between Contemporary references to "betwixt and between" "Reality" of "betwixt and between" Vestigial "pillar" of the middle way Avoiding constraining pillar-dependency

Dynamics of cognitive possibility: "possum"? Living  "as" an imaginal bridge, rather than "on" one Enacting, engendering and embodying Negative capability: a bridge to nowhere as a bridge to knowhere? Cyclic bridging between worlds of modality Enabling autopoiesis through poiesis Questing for a dynamic quality without a name

As highlighted there by Victor Turner (Passages, Margins, Poverty, 1974) in the liminal stage of transformation, the between stage, one's status becomes ambiguous; one is "neither here nor there", one is "betwixt and between all fixed points of classification", and thus the form and rules of both an earlier state and a state-to-come are suspended. For the moment, one is an outsider; one is on the margins, in an indeterminate state -- a liminal person betwixt and between the positions assigned and arrayed by law, custom, convention, and ceremonial. For Turner:

The attributes of…liminal personae are necessarily ambiguous, since this condition and these persons elude or slip through the network of classification that normally locate states and positions in cultural space. Liminal entities are neither here nor there; they are betwixt and between the positions assigned and arrayed by law, custom, convention and ceremonial. As such their ambiguous and indeterminate attributes are expressed by a rich variety of symbols in the many societies that ritualize social and cultural transitions. Thus, liminality is frequently likened to death, to being in the womb, to invisibility, to darkness, to bisexuality, to the wilderness, and to an eclipse of the sun or the moon.

Traditionally depicted and dressed in chequerboard, whether coloured or black-and-white, the Harlequin may prove to be a symbol of central significance to this period. Recognized as the chief actor and protagonist of pantomime, Harlequin is of high significance to sociology, namely as the character who performs the prime functions of critic and caricaturist (Amelia Defries, The Origins and Social Significance of Harlequin and of the Commedia Dell' Arte, The Sociological Review, October 1927). As Defries notes: As master of the revels, and with license accordingly, he is seen as staging, with pungency, a criticism of the life of his period. Potentially comparable with Loki of Norse mythology and Wagnerian drama, could the late leader of the free world be seen as having successfully performed this trickster role?

With leaders of every kind now being indicted for abuse, and rather than identifying misleadingly with any polar extreme, is it now more appropriate to recognize the extent to which all now have a tendency to be of "chequerboard" character, or be perceived as such? As celebrated operatically in Arlecchino, this embodies ambiguity and hyperbole in order to place others momentarily in a position of slight doubt -- confronted with the illusory dynamics of light and shadow. Progressives may especially be seen in this light, as argued by J. D. Alt (Who will play the Harlequin? New Economic Perspectives, 9 January 2017).

At the centre of the above configuration, by implication traditionally is the yin-yang Tao symbol indicative of the paradoxical ambiguity of polarization. One exercise explored the possibility of representing that symbol in 3D, of which one result is reproduced as an animation below (Exploring Representation of the Tao in 3D: virtual reality clues to reconciling radical differences, global and otherwise? 2019). This is provocatively framed by images of the Harlequin -- so central to the art of Picasso, now recognized as pre-figuring the 20th century (Harlequin, 1901 by Picasso; Pablo Picasso's Harlequin; Aaron Wasserman, Self-Identity and Picasso's Harlequin, 2005).

Maurice Sand, Public domain, via Wikimedia Commons	3D model kindly produced by Sergey Bederov of Cortona3D,	Photo by M. Schwarzkopf, Zürich, Public domain, via Wikimedia Commons

Future evolution evolution of humanity may necessarily embody Harlequin-like characteristics, as with the engagement with hypothesized extraterrestrials (Emergence of Homo undulans -- through a "grokking" dynamic? 2013).

---

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