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15 March 2016 | Draft

Engaging with Hyperreality through Demonique and Angelique?

Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation

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Introduction
Surreal nature of current global governance as experienced
Reality of the demonic and the angelic by comparison with secular understanding of existence
Methodological parallels in identifying insubstantial entities?
Hypercomprehension and radicalisation of identity through paradox -- a new frontier
Mathematical theology as source of mnemonic clues to global comprehension
Hyperbolic space: realm of hypothetical entities?
Global communication patterns in a hyperbolic space of negative curvature
Recognizing failure of viable cognitive self-reflexivity as "angel falling"
Envisaging the global dynamics of a "hyperbolic round table" through tessellation
Traditional modes of cognitive engagement with hyperreality
Hyperbolic reframing of the Dominique and Angelique of tradition
Engaging creatively with hyperreality through music
Towards a Daimonic Comprehendium?
References


Annex of: Variety of System Failures Engendered by Negligent Distinctions:
mnemonic clues to 72 modes of viable system failure from a demonic pattern language
(2016)


Introduction

The apparent incapacity of authorities to engender fruitful new ways of engaging with the crises faced by global governance is emphasized in the main paper. A case was made for configuring these problems as a "demonique" -- suggesting the need for a matching "angelique". The terminology exploits the pattern initiated by the Club of Rome with respect to "problematique" and "resolutique", as separately discussed (Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007).

The argument followed from recognition of the increasing degree of "demonisation" of others -- duly reciprocated -- and the suggestion that this process might be better focused on the problems with which governance sought to engage. Given the continuing widespread belief in angels, a corresponding case was also made for the "evangelisation" of solutions. In both cases this switches attention away from those controversially held to embody the demonic or the current focus of evangelical preoccupations.

Such a radical shift in perspective follows from the seeming inadequacy of think tanks to reframe the processes of comprehension and engagement in a period of crisis. Priority is seemingly given to fruitless military solutions, as separately argued ("Tank-thoughts" from "Think-tanks": metaphors constraining development of global governance, 2003). The issue is evident in the limited learning from analysis of failures of remedial initiatives of the past in framing those proposed for the future (Recognizing the Psychosocial Boundaries of Remedial Action: constraints on ensuring a safe operating space for humanity, 2009; Remedial Capacity Indicators Versus Performance Indicators, 1981). This has notably been the case with respect to progress on the UN Millennium Development Goals, now replaced by the UN Sustainable Development Goals. Might the hypothetical be better understood as entities "dancing" in hyperbolic space -- namely one of negative curvature?

Arguably the policy sciences now bear a strong resemblance to medieval scholasticism. Whilst the question How many angels can dance on the head of a pin? has long been used deprecate and dismiss scholasticism and angelology, the current intellectual bankruptcy suggests that angelology might well be fruitfully revisited -- but otherwise.

The approach taken here is framed by recognition of the cognitive challenges to comprehension of the complexity with which models of reality purport to deal. Unfortunately these challenges are only too evident with respect to the models themselves. These notably fail to address issues of their own relative incomprehensibility and lack of popular uptake (as argued in the main paper). As a potentially appropriate discipline, mathematical theology therefore embodies to a suitably extreme degree the cognitive incommensurabilities by which current global decision-making could be said to be effectively "bedevilled".

In an era of ever increasing populism, there is a case for starting from what many people already believe in -- and not with what some would have them believe in. The focus here is therefore on the challenge of human comprehension of sets of intangibles as they may frame and influence the complexities of global governance, experienced as increasingly surreal. Potentially to be understood as memes, the question is how cultures have chosen to distinguish and order larger systems of intangible qualities that they value and in which they variously believe.

The approach therefore emphasizes the fundamental role of that which cannot be fully comprehended or effectively articulated (Global Strategic Implications of the "Unsaid": from myth-making towards a "wisdom society", 2003). The question is the nature of the interface with (and between) the known knowns, the known unknowns, the unknown knowns, and the unknown unknowns -- given the challenges of their knowability in the light of human cognitive constraints, as discussed in the main paper (Transcending the barrier to global comprehension from attention deficiency and memory erosion, 2016) .

The concern here is not with the existence of the unknown, nor with the nature or truth of the insubstantial, but rather with the challenge to comprehension of patterns whose locus may be usefully understood as in hyperreality. In an era of virtual reality and special media effects, particular attention is given here to the use of visual facilities to enable creative discussion of any such "global" comprehension, however it may be imagined or understood. A focus is therefore given here to representation through tessellation of hyperbolic space as a potentially appropriate contrast to conventional representations in Euclidean space.

Surreal nature of current global governance as experienced

Surreality: There is no lack of recognition of the current reality of national and global governance as being surreal. Originally understood as a movement to "resolve the previously contradictory conditions of dream and reality", it could be said that surrealism has become a defining characteristic of international politics, if only to some degree (as argued on a website Surrealism in Politics).

Especially interesting are the media echoing that perspective:

In that vein, as queried by Christopher Fear, do academic discussions of political theory belong to the world of make-believe? (Down the rabbit hole, Prospect: Harvard University Press, 1 October 2013). From a perspective of international relations theory, for Victoria Lennox, as a graduating student:

Although particular theoretical paradigms within IR emphasise specific characteristics of global governance, no single paradigm has been capable of fully capturing the complexity of global governance. Resultantly, the conceptualisation of global governance requires a combination of particular aspects of realism, institutionalism, constructivism, and pluralism, that embraces and goes beyond Rosenau's conception of global order to enhance the comprehensiveness and sophistication of the conceptualisation of global governance by infusing it with meaning and value. (Conceptualising Global Governance in International Relations, E-International Relations, 3 October 2008)

Consistent with its origins, the engagement with the surreal is evident both in the arts and in the special effects of popular entertainment and marketing. In the following exploration of its relation to hyperreality, it is also evident in the unconstrained use of hyperbole and spin in many domains, most notably politics.

Rationality is indeed a strategic option -- but it is no longer the only one, as embodied by the campaigning success of Donald Trump as variously noted, despite repeated comments on the absurdity of his proposed strategies:

With such examples, and the associated mystification of processes of governance, it is appropriate to ask whether the future will find past and present preoccupation with angels and demons to be equally questionable -- and whether angelology offers some relevant insights into new forms of engagement with the "wicked problems" of the policy sciences.

Absurdity: Associations with the surreal are consistent with recognition of the absurd (Michael Foley, The Age of Absurdity: why modern life makes it hard to be happy, 2010). This is further associated with both nihilism and the challenge to many of the "nothing" they face in their lives (Emerging Significance of Nothing, 2012).

Most curious is the extent to which global governance is currently framed by a preoccupation with terrorism -- through the Global War on Terror -- engendered by the world view of suicidal jihadists.

Contrasting worldviews: The surreal condition of the times can be usefully caricatured by the following schematic indicating the relation between the worldviews of the World Economic Forum (notably held annually in Davos) and the World Social Forum (initially held in Porto Alegre). Use is made in the schematic of reference to Dante's famous contrasting representations of Paradiso (Heaven) and Inferno (Hell).

Contrasting perspectives of conventional archetypes

Such a polarity frames the question of the locus of Purgatorio as Dante's intermediary state. A previous exercise in reframing the archetypal dynamics between the two worldviews was articulated through a musical metaphor (All Blacks of Davos vs All Greens of Porto Alegre: reframing global strategic discord through polyphony? 2007). The schematic indicates the extent to which each is quite unable to encompass or justify the reality of the other -- each desiring the disappearance of the other. The pattern is similarly evident in the relation between science and religion -- neither apparently capable of acquiring the skills or insights to encompass that dynamic (or aspiring to do so).

Palestine-Israel: The pattern is otherwise exemplified in the archetypal relation between Palestine and Israel as a driving dynamic of regional conflict -- also variously recognized as surreal in a supposedly knowledge-based global civilization:

"Reality distortion"? In the world of high-tech, the fundamental significance of the surreal has been noted with respect to the manner in which leading creative entrepreneurs engender a so-called reality distortion field by which managers and leaders try to convince their employees to become passionately committed to projects without regard to the overall product or to competitive forces in the market. It has been most notably applied to the ability of Steve Jobs, founder of Apple, to convince himself and others to believe almost anything with a mix of charm, charisma, bravado, hyperbole, marketing, appeasement and persistence. It was said to distort an audience's sense of proportion and scales of difficulties and made them believe that the task at hand was possible. The same phenomenon is now associated with Elon Musk (Drake Baer, What It's Like Inside Elon Musk's 'Reality Distortion Field', Business Insider, 1 January 2015). For Jobs, it was a bubble he lived in that shielded him from the normal fears that held back so many others.

It is difficult to avoid use of the term "radical" with respect to the modalities and mind-set of extremely innovative leaders, such as Jobs and Musk. Especially relevant to the nature of radicalism is the critique of Michael Shermer (The Reality Distortion Field: Steve Jobs's modus operandi of ignoring reality is a double-edged sword, eSceptic, 10 July 2013). Shermer suggests that these might be more fruitfully understood through frameworks such as that provided by Robert Triversin (The Folly of Fools: the logic of deceit and self-deception in human life, 2011). Such a critique completely avoids the issue of widespread appreciation of the product of such thinking, most evident in the Apple products -- presumably also appreciated by sceptics -- and of whether there is proven capacity to engender products of such quality otherwise.

Clearly there is considerable relevance to the question of Markus Giesle (Is There a Science Behind Reality Distortion Fields? The Huffington Post, 16 September 2015). Curiously any reference to such a reality bubble is reminiscent of the filter bubble cultivated for each through personalization of search engine results (Eli Pariser, The Filter Bubble: what the internet is hiding from you, 2011). Definitive closure by some on the nature of "reality" might be better understood as premature -- even a reality distortion field in its own right (Einstein's Search and the Illusion of Reality, Knowledge of Reality Magazine, 2005, 11).

As might be expected, the process is now explored with respect to the campaigning success of Donald Trump, as the possible superpower leader of global civilization (Matt K. Lewis, The Trump Reality Distortion Filter: why so many US Republicans are suspending their disbelief, The Telegraph, 28 November 2015).

Virtual reality: Much is now being made of virtual reality and its potential impact on society. This follows from speculation about artificial worlds in science fiction, extensive development of virtual worlds in cyberspace (Second Life, etc), and the familiarity with designing and embodying avatars in online gaming -- whatever extraordinary forms they may take. Current possibilities suggest that there will be little difficulty in creating virtual reality environments populated by angels and demons, whether embodied as avatars or otherwise.

More intriguing is the possibility that some institutions with offices in virtual worlds (as is already the case) may choose to use reception (and other) personnel configured as angels -- as religious institutions might have wished to do in the past. The other extreme may well be adopted, using demonic configurations to threaten visitors -- perhaps through failure to fulfill financial or other commitments, or as a modality for responding to deprecated critics. Politicians and others may choose to configure themselves according to occasion, much as dress is currently selected.

Further adaptations are to be envisaged. These might include a facility of individual viewers to reconfigure the images of politicians, celebrities and other TV personalities -- to dress them otherwise, modify their racial type or skin colouring, and change the tone of voice. Viewers would then acquire a greater degree of control of their environment -- with the possibility of allowing such changes to occur dynamically, perhaps in continuing reaction to appreciation of the viewing experience.

Reality of the demonic and the angelic by comparison with secular understanding of existence

Belief: Clearly extreme importance continues to be associated with faith-based perspectives in relation to global governance -- exemplified by the conflicts that they inspire, inform and are held to justify (Future Challenge of Faith-based Governance, 2003).

This emphasis is despite extreme deprecation of any such framing from a scientific perspective by a number of authors (Richard Dawkins, The God Delusion, 2006; Christopher Hitchens, God is Not Great: how religion poisons everything, 2009). Belief in the reality of angels and demons is variously associated with such religions, however their existence may be understood. How these beliefs are promoted (or overlooked) by religions and ideology may well be irrelevant to popular belief.

With respect to the Abrahamic religions, as noted by Erran Bar Shushan (The Angels of God), belief in angels is one of the six pillars of the Islamic faith, without which there is no faith ( Islamic view of angels). Angels have been embraced by Christianity, like no religion before it (see Wikipedia: Christian angelic hierarchy). Forbidden as they are to create any image of any such entities, Judaism makes use of Hebrew letters in order to represent deity and the aspects of deity (see Wikipedia: Angels in Judaism). This is evident from the value variously attached to angels. Equivalent entities, termed otherwise, are recognized by other major religions of the world, notably as the devas of Buddhism and the devas of Hinduism.

However, clearly more controversial for those deprecating religion, is the sense in which science and atheism can themselves be understood as belief systems -- and held to be functioning as such for their adherents (Top 10 Reasons Science Is Another Religion, Listverse, 15 December 2012; Paul Bloom, Scientific Faith Is Different From Religious Faith, The Atlantic, 24 November 2015; Bill Meacham, Is Science a Religion? Philosophy for Real Life, 16 January 2012; Mike Dobbins, Atheism as a Religion: an introduction to the world's least understood faith, 2014). What are the entities which evoke such belief? In what space might they be represented, especially if they are primarily to be characterized by processes?

Image: Although the secular world endeavours to distance itself from such matters, it is curiously embroiled in the cultivation of image -- whether individually or collectively (Kenneth Boulding, The Image: knowledge in life and society, 1956). Image may be everything in the promotion of individual careers and the success of collective initiatives. Considerable resources may be devoted to such cultivation -- and to reframing the image of opponents negatively, most obviously through negative campaigning (Wendy L. Patrick, Why Bad Looks Good -- The Optics of Politics: appearances are not always reality, Psychology Today, Ben Zimmer, Optics, The New York Times Magazine, 4 March 2010).

In what manner does any such image "exist" -- especially in the light of understanding of its existence at the focus of optical systems?

Divine surrogate: As indicated with respect to the increasingly surreal perceptions of processes of governance, and the sense of the absurd, to what extent do the collective processes and entities exist -- most notably when many are defined as "nobodies", or experience themselves to be such? A striking example of this is provided by the so-called international community to which appeals are now frequently made, and whose inaction is frequently deplored. Separately it has been argued that this is increasingly functioned as a divine surrogate (International Community as God or Sorcerer's Apprentice? 2015). To what extent can it be said to "exist"?

Nonsense: Much is made of the international rule of law as the foundation of global civilization (What is the International Rule of Law? Brandeis Institute for International Judges, 2010). Increasingly it is only too evident that its rules are set aside and judicial interpretations in those terms are declared to be "nonsense", "ridiculous" or "absurd". To what extent can its rules be said to "exist"?

Values and goals : Much is also made of human values, especially as enshrined in international declarations. As intangibles, how is their existence to be recognized -- especially when they are beyond any recognition by science? Their nature is a theme of discussion within the Club of Rome (Martin Palmer and Karl Wagner, "Valuesquest": the search for values that will make a world of difference, 2013). Embodied in goals, values were an early focus of the Club of Rome (Ervin Laszlo et al,, Goals for Mankind: a report on the new horizons of the global community, 1977). As noted above they are a feature of UN Millennium Goals and UN Sustainable Development Goals.

In an effort to clarify the nature of the inherent ambiguity of their articulation (at least in English), values were the focus of a Human Values Project, which notably focused on the paradoxical nature of value polarities.

Models and hypotheses: There is clearly a fundamental issue as to the nature of the "existence" of the many conceptual models and hypotheses, whether engendered by science or to frame and facilitate social organization. They are a feature of the psychosocial construction of reality. They could well be understood to be "creatures of hyperreality", readily declared as creations of the imagination -- especially in the course of the speculative processes during which they are engendered

Attribution of real existence to intangibles: An obvious response to such questions is that reality is attributed to such intangibles and action is taken on the basis of their existence. This is especially evident in the case of political campaigns and commercial marketing. A form of existence may also be engendered by fear as in concerns about a culture of fear framed to justify allocation of vast resources in response to terrorism. There is then the strange dilemma of existence presented in the classic tales of The Emperor's New Clothes and of the Boy Who Cried Wolf., as discussed separately (Entangled Tales of Memetic Disaster: mutual implication of the Emperor and the Little Boy, 2009; "Big Brother" Crying "Wolf"? But them "wolves" are a-changin' -- them's becomin' "werewolves"! 2013).

Considerable significance is clearly attached to the intangible in relation to jihadist suicide bombing and the promise that the perpetrator will go immediately to heaven -- whatever further attraction is offered to males by the prospect of 72 virgins. With respect to the latter, it is noteworthy that it has been argued that "virgin" is a mistranslation of "angel" (Ibn Warraq, Virgins? What virgins? The Guardian, 12 January 2002). The understanding of virgin has however been otherwise confused with houri and even with raisin (Rebecca Sato, The Great Koran Controversy: will Muslim martyrs get 72 raisins instead of virgins, and other speculations, The Daily Galaxy, 16 January 2008)

Fiat realities: The extraordinary degree of economic reliance on fiat money -- now framed as quantitative easing -- offers a striking case of questionable existence, especially since its use was previously condemned as evidence of incompetent governance. More generally this can be understood as an instance of fiat reality, as framed by the formulation of edicts by authorities, most notably religious authorities (Papal bull, fatwa, Posek, etc). Especially striking is the sense in which

Myth: Despite their deprecation by a secular society, such arguments are consistent with recognition of the fundamental role of myth (Joseph Campbell and Bill Moyers, The Power of Myth, 1988; Karen Armstrong, A Short History of Myth, 2005). There remains the possibility that cultures of the past, in the absence of science, may have had other modes of sensitivity to the challenges of comprehension and representation of the subtle. There is clearly a sense in which it is perceptions, however cultivated, which frame reality in terms of which action may be taken -- irrespective of how such perceptions are related to evidential reality and requirements for concrete proof, themselves questionable (10 Demands for Concrete Proof by We the Peoples of the World, 2012).

Possession? Frequent reference to "having a problem" or to "being a problem", whether in the case of an individual or a collective, frames the question of how this may be further associated with demonisation, even possession by a demon. References to "having an inspiration" or to "being an inspiration" recall the references by the creative to the influence of a benevolent daimon -- notably amongst the Greeks of antiquity. Related reference is however also made to "being an angel", whether as a benefactor or in romantic situations. Bother extremes raise the question of how such qualities are imbued or embodied.

Collective perceptions: It was in terms of the perceptions of international constituencies that extensive efforts were made to profile the problems of the world in the Encyclopedia of World Problems and Human Potential. Fundamental to the methodology was recognition that the existence of any one problem, as claimed by one constituency, might well be called into question by another as misleading or a delusion. A similar situation obtained in the case of the remedial strategies advocated by one constituency -- readily perceived and deprecated as problematic by another.

The dynamics of global civilization could well be understood as determined by subtle perceptions -- however much these may be claimed to be factual and evidence-based by constituencies promoting belief in their reality.

Of particular interest in relation to any sense of angels "falling" into a demonic condition (or "rising" from one) is the sense of time with respect to the appreciation of particular subtleties, namely their period of credibility to some constituency. This can be usefully framed in terms of the so-called half-life of knowledge , as variously explored (Samuel Arbesman, The Half-life of Facts: why everything we know has an expiration date, 2012; John Ioannides, Why Most Published Research Findings Are False, PLOS, 30 August 2005).

With respect to deities engendering belief over the history of human civilization, many could now be considered "dead" (Michael Jordan (Encyclopedia of Gods: over 2,500 deities of the world, 1993; Patricia Turner and Charles Russell Coulter, Dictionary of Ancient Deities, 2001). Similar consideration consideration could be given to the theories, hypotheses and models of science, now considered obsolete. Wikipedia offers a checklist of superseded scientific theories.

Collective entities -- de facto or de jure: The extent to which any such entities can be held to "exist" remains a matter of debate -- despite the reality attributed to them in practice. Especially significant at this time is the frequency of appeals to the international community, raising the question as to whether it has become a divine surrogate or an act of deception, as argued separately (International Community as God or Sorcerer's Apprentice? 2015)

Methodological parallels in identifying insubstantial entities?

Correspondences? The following table suggests a degree of equivalence between different memes -- as experienced, sensed, comprehended or embodied -- including the more subtle and mysterious. The juxtaposition recalls insights from the sciences, arts and symbolism into the nature of correspondences (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).

Memes "Angelic"
("positive", "constructive")
"Demonic"
("negative", "destructive"
problems as necessary challenges to creativity as seemingly unjustified cause of pain
memories good memories, evoking nostalgia memories of past injustice
images attractive images repulsive images
issues as a focus of creativity as a focus of concern
values, principles values framed as positive values framed as negative
strategies remedial strategies disruptive strategies
goals global collective goals (MDG, SDG) questionable goals ("total system dominance")
concepts, models, hypotheses creative concepts destructive concepts
explanations inspired, imaginative unimaginative, restrictive
(dis)agreement consensus, harmony disagreement, discord
(mis)understanding sharing an understanding misunderstanding
findings (judicial, scientific) balanced, insightful biased
collectivities benevolent collectivities malevolent collectivities
decisions good decisions bad decisions
identity developed, fulfilled constrained, demeaned

Documenting the insubstantial: With respect to the existence of angels and demons, a remarkably balanced commentary is provided by Gustav Davidson, as compiler of a massive compendium: A Dictionary of Angels -- including the fallen angels (1967). The commentary might well have been adapted to that relating to the existence of problems (as "demons") and remedial strategies (as "angels") in the above-mentioned Encyclopedia (on which extensive methodological comments are separately provided).

Relevant to this argument are the following statements by Davidson on his experience as compiler, individually annotated here (as indents) with respect to conventional problem and strategic preoccupations:

The comparisons made above can be reframed and reviewed in terms of cultural memes (D.J.H. Brown, Cultural Memes and their Inhibitory Impact upon Energy Policy Change, 2008; Robert Walker, Cultural Memes, Innate Proclivities and Musical Behaviour: a case study of the western traditions, Psychology of Music, 2004; Alexis Morris, et al. The Evolution of Cultural Resilience and Complexity, 2011). Engendered and sustained by belief, both solutions (angelic or otherwise) and problems (demonic or otherwise) merit consideration as creations of the human spirit which may "rise" and "fall" under circumstances -- potentially defying forms of comprehension the future may find appropriate.

Hierarchical orders? Conventional articulations of the clustering into angelic and demonic orders and hierarchies call for continuing critical review regarding the implied prioritization -- notably given the ineffectual and unsustainable consensus they tend to inspire. Especially intriguing with respect to angelic and demonic perspectives are the distinctive modes of cognition they purportedly imply when ordered in this way.

How does nesting within the lower orders of any hierarchy imply a form of "subunderstanding" -- following the argument of Magoroh Maruyama (Peripheral Vision: polyocular vision or subunderstanding?, Organization Studies, 25, 2004, 3, pp. 467-480)? More challenging is the nature of of the insight implied by the highest levels of any proposed cognitive hierarchy upheld as a focus for belief. To what extent does such positioning -- as the "archangels" of secular society -- go beyond facile manipulation of conventional labels: environment, security, employment, education, resources, and the like?

Named otherwise, the angels and demons in the Dictionary resemble strangely the entities profiled in the Encyclopedia as problems, strategies, values and concepts. Other labels for the insubstantial -- whose nature also remains elusive

Given the limited set of archangels recognized, there is of course the provocative case for recognizing that functionally their secular equivalent could be compared to the limited sets of corporate entities that "rule the world", as implied by the argument of Pier Giuseppe Monateri (Rational Angels: understanding the theological background of economic rationality, Cardozo Electronic Law Bulletin, 2011; La Natura Angelica della Corporation). This argument is distinct from recognition of "corporate angels" in various forms (Corporate Angel Network; Types of Business Angels; Brian E. Hill, Attracting Capital from Angels, 2002).

The contrasting perceptions of their ruling function could be compared with deprecation of the various hierarchical orderings of angels (Swiss study shows 147 corporations rule the world, 12 October 2012; Revealed -- the capitalist network that runs the world, New Scientist, 19 October 2011; The 50 Corporations that Rule the World, WhiteOut Press, 16 September 2012; The Four Companies That Control the 147 Companies That Own Everything, Forbes, 26 October 2011; 10 Corporations Control Almost Everything You Buy, Information Clearing House, 4 November 2013).

Dynamics? Missing from Davidson's very comprehensive Dictionary of Angels is any sense of how particular angels interact systemically with other angels in Heaven or in Hell. This dynamic is only implicit, if at all, in the hierarchical orders and arrays presented. An analogous dynamic has been a preoccupation of the Encyclopedia and the visualization of the networks of entities it profiled. Such systemic insights are under exploration for possible future evolution of the Encyclopedia (Encycling Problematic Wickedness for Potential Humanity, 2014).

To the extent that problems and remedial strategies can be imaginatively framed as demonic and angelic in nature, as memes they offer another way of speculating about the nature of the final battle envisaged by some religions in their end times scenarios, notably the Battle of Armageddon. Should such a battle be understood in the light of what is now being termed memetic warfare (Missiles, Missives, Missions and Memetic Warfare: navigation of strategic interfaces in multidimensional knowledge space, 2001; Brian J. Hancock, Memetic Warfare: the future of war, Military Intelligence, 36, 2, 2010, pp. 41-46)

Evocation and invocation: The correspondences implied above offer the ironic implication that the deprecated traditional magic rituals of "evocation" of demonic and angelic entities could be fruitfully compared to the practices and settings in which modern day politicians ceremoniously invoke values in response to selected issues.

As exercises in public relations, image management and "talking up" strategic options, such processes could well be compared in systemic terms with traditional magical operations -- effectively "summoning" demons (notably through fear mongering) and invoking angelic assistance in response.

Especially intriguing in any such operational correspondence are efforts to control the demonic in a cultivated culture of fear -- drawing on myth when appropriate.

Quest for angelic language: There is an evident absence of any unifying language or discipline -- highlighted by the failure of unified science and philosophy (Nicholas Rescher, The Strife of Systems: an essay on the grounds and implications of philosophical diversity, 1985). It is then appropriate to note the traditional quest for the language of angels as variously imagined (Aaron Leitch, The Angelical Language, 2010).

As argued separately, traces of it may now be speculatively recognized in use of Twitter (Re-Emergence of the Language of the Birds through Twitter? 2010). The associated set of communication concerns is contrasted there with the quest for a language adequate to governance, as notably recognized by Umberto Eco in exploring an appropriate language for "The Making of Europe" (The Search for the Perfect Language, 1993). He summarizes the insights of philosophers, theologians, mystics and others for at least two millennia about the idea that there once existed a language, notably a "language of the birds", which perfectly and unambiguously expressed the essence of all possible things and concepts. He explores the history of that idea and of its profound influence on European thought, culture, and history.

The question here is whether global use of Twitter -- exemplifying many of the current communication concerns -- can be understood as enabling the emergence into practice of a form of that mythical language of the birds, common to the memory of many cultures. These include Islam, Hinduism, Judaism, Christianity and shamanism, as indicated by Philip Coppens (Tweet Tweet: the language of birds). Despite that title however, no reference is made to Twitter, whose potential relationship to any "language of the birds" does not seem to have been considered by others.

Hypercomprehension and radicalisation of identity through paradox -- a new frontier

Cultivation of surrealism and hyperreality in fiction and imagery : Clearly the sense of the surreal is fundamental to the original aesthetic and symbolic motivation of surrealism -- exemplified by the Surrealist Manifesto (1929). The more recent manifestation of the surreal and the hyperreal in fiction continues to be evident in science fiction and fantasy -- and the remarkable uptake of such works of the imagination via blockbuster movies.

A related genre is that of works whose uptake is dependent on a degree of tension-inducing mystification, achieved through a pattern of combining both logical non-sequiturs and their use as a literary device. As psychological thrillers, examples might include: The Leftovers (2015), Blair Witch (2000). Of interest is the appreciation of such representations of reality in contrast to those conventionally favoured and promoted.

Hypercomprehension: The argument made above emphasized the challenge of complex dynamics to human comprehension. In terms of the complexity sciences and chaos theory, this challenge can be partially framed in terms of the comprehension of hyperreality -- however that might be understood and articulated.

Arguably there is a case for some form of "hypercomprehension", as can be variously argued:

Paradox and value polarities: value polarities -- indications of hyperreality

The challenge could also be framed in terms of the transcendence of the binary distinction too readily made between the demonic and the angelic. This would be somewhat consistent with traditional recognition of demons as "fallen angels" -- that "fall" being from a transcendent cognitive comprehension of the complex complementarity of their systemic relationship.

Daimonic: As noted above, the relevant traditional Greek term for this transcendent (unpolarised) condition is associated with daimon -- when understood, and experienced, as epistemologically prior to the distinction made between angel and demon. Angels and demons are then better recognized as ambiguously paradoxical interfaces to the systemic complexities of hyperreality -- however this might come to be understood.

Such transcendent insight necessarily involves some form of cognitive transformation from conventional binary thinking, most typically recognized and experienced in the processes of creativity.

That transformation of locus is usefully understood in terms of higher orders of self-reflexivity, notably as indicated in the work on knowledge cybernetics of Maurice Yolles (as discussed below).

In classical Greece, creativity was understood as involving communication a strange form of discourse with one's daimon, possibly reframed as a muse -- the goddesses of the inspiration of literature, science, and the arts. The latter term has been adopted with respect to modern artistic inspiration, as well as being extended to political inspiration (Adam J. Sorkin, Politics and the Muse: studies in the politics of recent American literature, 1989).

"Identique" and imaginative radicalisation? The cognitive transformation can be framed as a form of radicalisation, even a transformation and relocalisation of the focus of identity and how it is imagined (Radicalisation of Existence and Identity: recognizing the global emergence and influence of daimonic dynamics, 2015).

This suggests a further extension of the Club of Rome pattern to encompass the complex variety of forms of identity -- perhaps as an "identique", as separately argued (Elaborating a richer "global identique", 2008; Living as an Imaginal Bridge between Worlds: global implications of "betwixt and between" and liminality, 2011)

Mathematical theology as source of mnemonic clues to global comprehension

As noted above, mathematical theology embodies to a suitably extreme degree the cognitive incommensurabilities by which current global decision-making could be said to be effectively "bedevilled". Its potential significance has been highlighted separately (Mathematical Theology: Future Science of Confidence in Belief -- self-reflexive global reframing to enable faith-based governance, 2011) with respect to a proposed International Institute of Advanced Studies in Mathematical Theology (2011).

Clearly extreme importance continues to be associated with faith-based perspectives in relation to global governance -- exemplified by the conflicts that they inspire, inform and are held to justify. This is despite extreme deprecation of any such framing from a scientific perspective (Richard Dawkins, The God Delusion, 2006; Christopher Hitchens, God is Not Great: how religion poisons everything, 2009). Set against this is the remarkable ability of the sciences, most notably mathematics, to articulate new ways of understanding the subtleties of extreme complexity. (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

Comprehension of uncommon ground: More remarkable, however, is the complete lack of capacity of the sciences to enable modes of comprehension adequate to the crises of governance -- exacerbated as they are by competing belief systems -- including those of the sciences and philosophy more generally (Nicholas Rescher, The Strife of Systems: an essay on the grounds and implications of philosophical diversity, 1985). It is in this sense that the argument of Dawkins could be reframed in terms of any prospect for global consensus (The Consensus Delusion: mysterious attractor undermining global civilization as currently imagined, 2011).

By combining incommensurables -- as exemplifying requisite variety -- mathematical theology frames the challenging quest for "uncommon ground" (In Quest of Uncommon Ground: beyond impoverished metaphor and the impotence of words of power, 1997). It draws on the fundamental importance of number in the theology of many religions, as well as in the sciences (and most notably mathematics). The fundamental issue is how any form of unity is to be comprehended and articulated, whether as an inspiration for belief or in its primary role in the logic on which the sciences are based.

Transcendental unity: Clearly religions encourage a mystical belief in a transcendental unity of some kind -- purportedly accessible to experience in some measure, as appreciated in the mystics held to embody it to a degree. Their articulation is however almost completely divorced from that of mathematicians otherwise purportedly preoccupied with fundamental comprehension of the infinite in some form. A degree of compatibility is only occasionally recognized, as reviewed in the work of Sarah Voss (What Number Is God? Metaphors, Metaphysics, Metamathematics, and the Nature of Things, 1995). How might understandings of "heaven" be compared with the fundamental framings offered by mathematicians -- notably given the discussion by Voss of the infinite sets of Georg Cantor? To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications - he identified the Absolute Infinite with God.

Curiously, as has been an inspiration to some mathematicians, that discipline enables the articulation of fundamental insights which mystics are challenged to match. However the latter claim an experiential engagement with their belief which is seemingly meaningless to mathematicians, despite any reference to intuitive comprehension (Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981). Arguably, however, it is mystics that inspire a greater number of followers through the manner in which they embody their belief.

Angels and the transfinite? As summarized by Joad Raymond (Protestant Culture: Milton's Angels, History Today, 60, December 2010), though Protestants sought to distance themselves from Roman Catholics on the subject, angels played a key role in Protestant culture as a means by which to understand humans and their place in the universe. In the commentary of Peter Collins, however, angels in a sense represented a mythical way of representing the unconscious layers of the psyche, with the "bad" angels or demons pointing -- by contrast -- to the deep shadow elements (of the unreformed psyche). That "shadow" side is then to be understood as being increasingly exposed in demonic fashion throughout the world.

Collins considers that a substantial transformation in consciousness will eventually emerge, but not before much greater crises eventually convince us of the genuine poverty of our present responses. He argues in a chapter on Angels on a Pinhead (1994) that -- when rightly appreciated -- the burning issue in medieval theology of how many angels can dance on a pinhead? -- is far from trivial. He argues that:

In fact it is exactly the same issue that Cantor addressed with respect to "how many numbers exist within a small interval of the number system?". This of course led to his discovery of different orders of transfinite numbers and culminated in the "Continuum Hypothesis", which was the very first on Hilbert's list of important unsolved mathematical problems (at that famous conference in 1900). ( The Number Paradigms: the remarkable complementarity of mathematics and transpersonal psychology , 1994)

With respect to a section on Qualitative Numbers: transfinite, Collins introduces that chapter as follows:

Though their relevance has now greatly faded, angels in the past played an extremely important part in religious experience. There are in fact over 300 references in the Bible to angels. Also, it is clear that Angelology was an indispensable component of medieval theology finding its most developed expression in the system of St. Thomas Aquinas. From a psychological perspective it is quite clear that angels were essentially religious projections of profound archetypal significance providing a convenient figurative device for portraying the vital role of the unconscious mind. Thus the biblical war as between good and bad angels can be seen as a metaphor for the ongoing struggle between opposing forces in the unconscious mind....his In the Bible angels were frequently used to deliver important messages of great spiritual significance. Again this relates well to how so often, especially at critical periods in life, intuitively inspired inspirations - springing from the unconscious - mysteriously support people in the taking of key decisions.

Collins cites Etienne Gilson in a chapter on The Philosophy of St. Thomas Aquinas, to the effect that

For Collins, with an appropriate change of a few key words, this becomes a statement regarding the number system:

Embodiment of metaphor: The argument above frames the relevance of the explorations of cognitive psychologists such as George Lakoff, especially given the challenge of women to theology ( Women, Fire, and Dangerous Things: What Categories Reveal About the Mind.1987; with Mark Johnson, Philosophy In The Flesh: the Embodied Mind and its Challenge to Western Thought, 2000; with Rafael Núñez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, 2000) From that perspective it is questionable whether mathematicians are able to live fully by the metaphors they exploit so skillfully (George Lakoff and Mark Johnson. Metaphors We Live By, 1980) -- or derive significance from their understanding of limits and finality in the process of dying (Metaphors To Die By: correspondences between a collapsing civilization, culture or group, and a dying person, 2013)

** meta-maths

Questionable hierarchical determinism: This question is discussed below.

Avoiding premature closure: The methodological concern here is to avoid any form of definitional closure by using the challenge of comprehension as a means of ensuring appropriate cognitive distance from any premature conclusion. One question is therefore how belief with regard to what is comprehended is framed, especially when this is fundamental to an understanding of reality and identity within it. As an example, one inspiration for closure avoidance, in relation to cognitive fusion (however envisaged), is offered in metaphorical terms by the design requirements for nuclear fusion reactors. In that case it is vital that the nuclear plasma not come into contact with the walls of the toroidal container. (Enactivating a Cognitive Fusion Reactor, 2006)

Arguably it is premature closure on contrasting understandings which is most likely to engender and sustain conflict -- especially when these fail to reflect higher or more fundamental orders of comprehension, to the extent that these can be articulated and communicated meaningfully.

Requisite variety versus Contextual comprehension capacity: The focus here on comprehension of patterns of demons and angels -- understood as a demonique and an angelique -- therefore offers a means of considering the articulation of memorable distinctions in relation to what transcends conventional modes of comprehension.

As indicated in the main paper, if comprehension requires a grasp of more than 20, 50, or more factors (or dimensions), how is this to be achieved when cognitive capacity is constrained to less than 20, and possibly less than 12?

Hyperbolic space: realm of hypothetical entities?

Exploring triangular curvature: A useful contrast can be made between conventional focus on the Euclidean triangle and two non-Euclidean equivalents distinguished as by Riemannian manifolds of constant curvature. Through the process of triangulation, the first is clearly fundamental to surveying. It has its implications in considering the interlocking of networks in systems and cyberspace. So depicted, the triangle is fundamental to various symbol systems and associated sets of categories, as explored by Paris Arnopoulos (Sociophysics: cosmos and chaos in nature and culture, 2005) and separately discussed (Triangulation of Incommensurable Concepts for Global Configuration, 2011; Triangulation, Connectivity and Stitching: enabling coherent global system dynamics, 2012).

The most obvious challenge to use of Euclidean triangles is evident in the navigation of the globe. This requires insight into spherical triangles and those in elliptic space.

Contrasting spaces of constant curvature
Hyperbolic space
(negative curvature)
Euclidean space
(zero curvature)
Elliptic space
(positive curvature)

Beyond conventions of the familiar, flat Euclidean triangle, a global civilization could be expected to require equivalent elliptic insights for its navigation, as discussed separately with respect to the mathematical discovery of the Pentagramma Mirificum (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

In distinguishing the three forms of triangle depicted above, it is typically noted that the angles of the Euclidean type sum to 180 degrees, and those of the elliptic type to more than 180 degrees. It is the third variant which is of interest here, namely the hyperbolic triangle, in which the angles sum to less than 180 degrees (possibly to be understood as zero, given the asymptotic convergence at the vertices). The first is characterized by zero curvature, the second by positive curvature and the third by negative curvature. With respect to any "agreement", it could be said that in the first case the results of triangulation "add up". More complex insight is required to recognize such coherence in the second. However in the hyperbolic variant, meaningful coherence and consensus dependent on such forms is far less obvious.

Negative curvature of hyperbolic space: Such negative curvature may well have implications for understanding the shape of the universe, as discussed separately (Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships, 2013). Recent research by Stephen Hawking and colleagues (Accelerated Expansion from Negative Lambda, 2012), has shown that the universe may have the same surreal geometry as some of art's most mind-boggling images (Lisa Grossman, Hawking's 'Escher-verse' could be theory of everything, New Scientist, 9 June 2012). This offers a way of reconciling the geometric demands of string theory, a still-hypothetical "theory of everything", with the universe as observed -- through a negatively-curved Escher-like geometry (essentially a hyperbolic space). These clues were used in relation to the aforementioned discussion of cognitive epicycles -- on the assumption that a more elegant "heliocentric" focus may emerge (Cognitive epicycles -- prefiguring more fundamental insight? 2012; Identification with a sustaining "heliocentric" locus? 2012)

The recent results of astrophysics rely on a mathematical twist previously considered impossible, namely the use of a negative cosmological constant rather than a positive one. The new approach provides a description of "all the possible universes that could have been -- including ones in which the solar system never formed, or in which life might have evolved quite differently". Making conventional use of a positive cosmological constant, it had proven impossible to describe universes that were "anything more than clunky approximations to reality." A plethora of universes have now been generated from wave functions with negative cosmological constants.

Reframing the focus of integration: As presented schematically below, a hyperbola may be used to offer an alternative means of expressing the complementary nature of incommensurables -- effectively framing the paradox of the challenge of modernity,

Use of the hyperbola to frame cognitive paradox
   

Re-presentation of symbols framing paradox: The mystery attributed to the Trinity and to the Star of David may derive primarily from the hyperbolic equivalents they imply rather than from their conventional Euclidean form. Given the argument above for mathematical theology, the fundamental importance of this paradox is remarkably illustrated by the use of the set of 3 entangled 3 Borromean rings for both a medieval depiction of the Christian Trinity and through its recent selection as the most appropriate symbol for the International Mathematical Union.

Borromean rings in 2D and 3D

Christian Trinity in 2D
(image reproduced from Wikipedia)

3D Logo of the
International Mathematical Union
2D Trinity as Borromean ring 3D Borromean ring logo of International Mathematical Union

Of relevance to this argument is the cognitive implication in such patterns, as discussed separately (Representation of Creative Processes through Dynamics in Three Dimensions: global insight from spherical reframing of mandalas, the zodiac and the enneagram, 2014). The potential of those patterns -- as articulated and tested with respect to patterns in "external" material form -- thus derives from "internal" patterns of thinking. This perspective has been extensively argued, from a cognitive psychological perspective with respect to mathematics more generally by George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

5-pointed star versus 6-pointed star: It is curious to recognize the extent to which global dynamics at this time is significantly determined by principles associated with such distinctive star symbols. The schematic above using Dante's distinctions can be usefully adapted as below.

Contrasting conventional interpretations of 5-star and 6-star symbolism

.For the purpose of this argument, the locus of the mysterious integrative insights implied by the contrasting star symbols can be explored in terms of configurations of hyperbola. Each configuration frames that central focus -- but from an unconventional perspective. Note that in "fitting" the 2 curves of a hyperbola to the stars:

Fundamental star symbols speculatively reframed using hyperbola (animations)
5-pointed Star of Islam
(featured on many Islamic flags)
6-pointed Star of David
(featured on the flag of Israel)
Animation of 5-pointed Star of Islam Animation of 6-pointed Star of David

Given the problematic perceptions between those associating with the symbolism of each, it is intriguing to note that both are intimately related to life processes. The hexagonal structure is fundamental to organic matter through the so-called benzene molecule. The pentagon is fundamental to Eastern and Western traditional understanding -- through Hygeia and Wu Xing -- of 5-fold organization of life processes (Memorable dynamics of living and dying: Hygeia and Wu Xing, 2014).

The possibility of reconciling such seemingly incompatible cognitive geometry was the focus of a separate exercise (Middle East Peace Potential through Dynamics in Spherical Geometry: sngendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks, 2012). This noted that the contrasting patterns could be uniquely reconciled within the spherical form of the truncated icosahedron -- a seemingly obscure geometrical form which is ironically familiar worldwide as the common football.

Recognizing the truncated icosahedron stitching pattern of a soccer ball
(adapted from Wikimedia entry, to highlight the alternative colour variant)

Using the above hyperbola-based adapations of the 5-star and 6-star forms, these can be applied to the truncated icosoahedral pattern. As shown below, the stitching pattern of the common football then suggests a possible means of exploring a form of reconciliation between the contrasting symbolism and the conflicts to which their interaction otherwise gives.

Application of hyperbola-based 5-star and 6-star forms to the truncated icosoahedron
3D configuration Unfolded representation in 2D
3D mapping of 5-star and 6-star patterns on truncated icosahedron 2D mapping of 5-star and 6-star patterns on unfolded truncated icosahedron

Reconciling Heaven and Hell?: Given the important role that the the art of M. C. Escher played in early promotion of explorations of hyperbolic configurations, a valuable introduction to hyperbolic geometry is provided, with extensive illustrations, on the Math and the Art of M C Escher Wiki (at Hyperbolic Geometry and at Euclidean and Non-Euclidean Geometry).

Those insights are extensively explored by others (B. Van Dusen and R. P. Taylor The Art and Science of Hyperbolic Tessellations Nonlinear Dynamics, Psychology, and Life Sciences, 17, 2013, 2, pp. 317-323) notably by Douglas Dunham (M.C. Escher's Use of the Poincaré Models of Hyperbolic Geometry; Transformation of Hyperbolic Escher Patterns; H.S.M. Coxeter and Tony Bomford's Colored Hyperbolic Rugs).

Experimental animations indicating the "dance" between angelic and demonic forms
Based on the classic Circle Limit IV (Heaven and Hell) (1960) of M.C. Escher
Coloured to highlight demons (dark) Coloured to highlight angels (dark)
Colour animation of  Escher Circle Limit etching Colour animation of  Escher Circle Limit etching

Angels and Demons "dancing on a pin head" in negative space? The introduction mentioned the question How many angels can dance on the head of a pin? -- as long used to deprecate and dismiss scholasticism and angelology. The following animations suggest a way of thinking about intuitions regarding the manner of existence of the kinds of insubstantial entities indicated above (whether understood "positively" or "negatively").

Animations suggestive of the dynamics of hyperbolic space
Dynamics of the Angelique ? Dynamics of the Demonique ?
Animation suggestive of the dynamics of the angelique in hyperbolic space Animation suggestive of the dynamics of the demonique in hyperbolic space

At this stage it is appropriate to interpret the shapes as sets of categories or concepts, whether 3-fold, 4-fold, 5-fold, or more -- in each case functioning as a form of integrative cognitive container. The schematics are then indicative of the manner in which such sets are "invoked" -- for a period -- for purposes of explanation, or as offering guiding principles, as previously discussed (Representation, Comprehension and Communication of Sets: the role of number, 1978; Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980; Spherical Configuration of Categories to reflect systemic patterns of environmental checks and balances, 1994; Comprehension of Numbers Challenging Global Civilization, 2014).

These animations anticipate further comment below, especially with regard to their asymmetry as polygons in hyperbolic space. Their animation is suggestive of the dynamics of global communication in which such categories are called into play for a time, only to be replaced by other sets -- also for a time only.

Global communication patterns in a hyperbolic space of negative curvature

Section uncompleted: work in process !!!!

Comprehension and communication: Following the reference above to the inspiration offered by Escher drawings, the argument may be further developed through mathematical consideration of the Poincaré disk and the tessellation of the hyperbolic plane with which it is so closely associated.

The focus in this argument is on communication and comprehension as they may be more appropriately and richly envisaged using hyperbolic tessellation and the Poincaré disk model. Contrasting images can therefore be usefully represented to suggest the degree of connectivity between participants (or perspectives) at any table around which they are gathered (as discussed below).

The question to be explored is the degree of complexity which can be individually or collectively comprehended and communicated. This relates to the concern in the main paper with the number of factors to be understood to avoid the failure of a complex viable system. How to engage with a greater number of factors when human cognitive and communication capacity is constrained to seven (plus or minus two)?

Expressed otherwise, the issue is the number of distinctions required to ensure sustainability -- and how they might be organized for optimal memorability. The limited number conventionally presented in relation to global governance at this time can be fruitfully called into question by the larger sets typical of traditional articulations in various cultures. Given the present dynamics between perceived problems and advocated solutions, traditional articulations of sets of 72 demons and sets of 72 angels suggest that a number of that order may enable intuitive insight into comprehensible patterns of greater complexity. Hence the framing here of a Demonique and an Angelique as potentially offering more widespread comprehensibility and engagement.

Mathematical intuition: For Sz?vós Eszter (On the boundaries of mathematics and psychology. Poincaré's notion of intuition, Magyar Pszichológiai Szemle, 19 November 2013):

"Intuition" is a central term in Henri Poincaré's writings on the philosophy of mathematics. Poincaré -- as opposed to the logicists (above all Russell) -- states that mathematics cannot be reduced to logic: without its intuitive component, mathematics would become tautological and sterile. But what exactly does the Poincaréan term of intuition mean and what sort of role does it play in mathematics and in mathematical history, study, and invention?In the present analysis -- due to the complexity of the concept "intuition". I shall review the different meanings of the expression and the linkages between these meanings. Furthermore, I shall point out that Poincaré's view on mathematical invention and intuition very much resembles that of several psychologists of his age and that Myers' theory of the unconscious had a strong influence on Poincaré's reasoning. Lastly, I shall examine the question of how closely Poincaré's theory is linked to psychology. Can his theses function within the philosophy of mathematics, or are they committed -- from a philosophical point of view -- to psychologism? I shall argue that in Poincaré's writings mathematical and psychological issues are not to be confused: some important and relevant arguments in the philosophy (rather than the psychology) of mathematics are grounded in the Poincaréan concept of intuition.

Tessellation: There are many commentaries on hyperbolic tessellation. That of Dmitry Brant (Hyperbolic Tessellations 2007), with helpful images, includes the comment:

A tessellation refers to a uniform tiling of a plane with polygons, such that an equal number of identical polygons meet at each vertex. For example, the tiles in a bathroom, the squares of linoleum on an office floor, or the honeycomb pattern in a bees' nest are all tessellations of the Euclidean plane.

However, tessellations are also possible on non-Euclidean spaces, such as the elliptic plane (like the stitching pattern on a soccer ball), and the hyperbolic plane (like... nothing you'd find around the house). In fact, the Euclidean plane has only three regular tessellations (with squares, hexagons, and triangles), while the hyperbolic plane can be tessellated in infinitely many ways.

Since we do not exist in hyperbolic space, we cannot truly "see" hyperbolic tessellations. We can only "represent" them in Euclidean form. A common way of doing this is on the Poincaré disk, which is a finite circle that represents the boundary of the (infinite) hyperbolic plane that is contained inside. The image on the right is a hyperbolic tessellation drawn on the Poincaré disk.

Since tessellations of the hyperbolic plane are especially interesting and mesmerizing to look at, I wrote a small program that generates them, with a great deal of configurable options.

For A. A. Goodenough and A. Schlamm (Interactive visualization of hyperspectral images on a hyperbolic disk. Connecting Minds for Global Solutions, 2011):

Visualization of the high-dimensional data set that makes up hyperspectral images necessitates a dimensionality reduction approach to make that data useful to a human analyst. The expression of spectral data as color images, individual pixel spectra plots, principal component images, and 2D/3D scatter plots of a subset of the data are a few examples of common techniques. However, these approaches leave the user with little ability to intuit knowledge of the full N-dimensional spectral data space or to directly or easily interact with that data. In this work, we look at developing an interactive, intuitive visualization and analysis tool based on using a Poincaré disk as a window into that high dimensional space. The Poincaré disk represents an infinite, two-dimensional hyperbolic space such that distances and areas increase exponentially as you move farther from the center of the disk. By projecting N-dimensional data into this space using a non-linear, yet relative distance metric preserving projection (such as the Sammon projection), we can simultaneously view the entire data set while maintaining natural clustering and spacing. The disk also provides a means to interact with the data; the user is presented with a "fish-eye" view of the space which can be navigated and manipulated with a mouse to "zoom" into clusters of data and to select spectral data points. By coupling this interaction with a synchronous view of the data as a spatial RGB image and the ability to examine individual pixel spectra, the user has full control over the data set for classification, analysis, and instructive use.

For Vladimir Bulatov (Tilings of the hyperbolic space and their visualization Corvallis, Oregon, USA Joint MAA/AMS meeting, New Orleans, January 7, 2011):

Visual representation of tiling of 3D hyperbolic space attracted very little attention compare to tilings of hyperbolic plane, which were popularized by M.C.Escher circle limit woodcuts. Although there is a lot of activity on theoretical side of the problem starting from work of H.Poincaré on Kleinian groups and continuing with breakthrough of W.Thurston in the development of low dimensional topology and G.Perelman's proof of Poincaré conjecture. The book "Indra's Pearl" have popularized visualization of 2D limit set of Kleinian groups, which is located at the infinity of hyperbolic space. In this talk we present our attempts to build and visualize actual 3D tilings. We study tilings with symmetry group generated by reflections in the faces of Coxeter polyhedron, which also is the fundamental polyhedron of the group.

Poincaré disk: The hyperbolic plane cannot be metrically represented in the flat Euclidean plane. Poincaré described ways that it can however be conformally represented in the Euclidean plane -- known as the Poincaré disk (see also Poincaré Hyperbolic Disk, Wolfram MathWorld). It is one of the most common models used to visualize hyperbolic geometry (and to print it in 2D).

The disk is a 2-dimensional model of n-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk and the straight lines consist of all segments of circles contained within that disk that are perpendicular to the boundary of the disk, plus all diameters of the disk. A straight line in the hyperbolic plane is thus represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.

The Poincaré disc maps the point at infinity of a hyperbolic space to a circle where hyperbolic lines are represented as arcs of circles intersecting the éé disc at 90 degrees. As we move away from the origin of a hyperbolic space, the space itself expands due to negative curvature, so as we reach the perimeter of the Poincaré disc, the scale of the space changes dramatically, subdividing into an infinite number of pieces.

A Poincaré disk is a circle that represents an infinite region of space. As the circular edge is approached, the images diminish at such a rate that they appear to be infinitely small and be infinitely close to the circle's edge without ever touching it. By using this Poincaré disk model, it is possible to give the impression of an infinite array of tile images within a limited space and, unlike other disk models, the shape of the tiles stays recognizable as they approach the circular boundary.

As succinctly summarized by Wikipedia with regard to the Poincaré disk model: In plain English, this means you can squash an infinite 2D plane into a circular disc! (also see images on Wikipedia at Media in category "Poincaré disk models").

Other resources relating to Poincaré disk:

A valuable overview is offered by David E. Joyce (Hyperbolic Tessellations: some printable tilings, 2002) through a page providing access to a range of images -- and enabling users to display their own -- vertices etc

 
Regular tilings
3 4 5 6 7 8 10 12
3         {3,7} {3,8} {3,10} {3,12}
4     {4,5} {4,6} {4,7} {4,8} {4,10}  
5   {5,4} {5,5} {5,6}        
6   {6,4} {6,5} {6,6}        
7 {7,3} {7,4}     {7,7}      
8 {8,3} {8,4}            
10 {10,3} {10,4}            
12 {12,3}              

Quasiegular tilings
34567 81012
3         {3|7} {3|8} {3|10} {3|12}
4     {4|5} {4|6}        
5   {5|4} {5|5} {5|6}        
6   {6|4} {6|5} {6|6}        
7 {7|3}       {7|7}      
8 {8|3}         {8|8}    
10 {10|3}           {10|10}  
12 {12|3}              

Animation of set of regular tilings (above) Animation of set of quasiregular tilings (above)
Animation of set of regular tilings Animation of set of quasiregular tilings

W. Goldman, Ultraideal Triangles, 2004

This graphics package draws objects in the hyperbolic plane using the Poincaré model. The pictures are drawn in the unit disc, but the calculations occur in the upper half plane, where the isometry group is PSL(2,R).

Ultraideal Triangles, 2004 Escher

 

Poincaré Disk
(adapted from Wolfram Demonstration Project)
     
<    ?    >

Recognizing failure of viable cognitive self-reflexivity as "angel falling"

Much is made by religious tradition of the falling of angels and the nature of "fallen angels". However they may be understood to exist, those entities "falling" in this way can indeed be fruitfully recognized as thereby cognitively transformed into "demonic" forces -- namely the wicked or rebellious angels that have been cast out of "heaven" -- however that is to be understood as a context for existence. By implication, "heaven" is then the worldview within which all is harmoniously reconciled -- in contrast to the "hellish" fragmentation of a global civilization in crisis.The latter might then be understood as a condition tortured by unanswered questions, with the former one in which all questions have been answered.

Temptation and sin: Fundamental to any process of falling (or failing) is clearly the recognition of temptation, especially as articulated by religion, and variously framed as sin. As the pressure to give in to a desire for immediate gratification, it is recognized through synonyms such as allurement, enticement, and seduction.

Curiously the antonyms are less convincing as descriptors of the conation of the unfallen, except in relation to vigilance, prudence and precaution.

Hyperbolic discounting: With regard to any form of temptation, the extensive literature on hyperbolic discounting, is of particular relevance to the problematic implications of the quest for collective short-term gratification -- perhaps to be provocatively understood as "collective sinning". This is especially curious in relation to the argument here for insight in hyperbolic terms. The process refers to the tendency for people to choose a smaller-sooner reward over a larger-later reward. As described by Wikipedia, in economics, this is understood as a "time-inconsistent" model of discounting. Given two similar rewards, humans are recognized to show a preference for one that arrives sooner rather than later. They are then said to discount the value of the later reward, by a factor that increases with the length of the delay.

This process is conventionally modeled in form of exponential discounting, as a "time-consistent" model of discounting. By contrast, hyperbolic discounting is a mathematical model devised as an improvement over exponential discounting, in the sense that it better fits the experimental data about actual behaviour. However the time inconsistency of this behavior has some quite perverse consequences. In hyperbolic discounting, valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods. Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time -- they make choices today that their future self would prefer not to have made, despite using the same reasoning.

Four prominent models of delay discounting have been usefully compared: a one-parameter exponential decay, a one-parameter hyperbola, a two-parameter hyperboloid in which the denominator is raised to a power, and a two-parameter hyperbola in which delay is raised to a power (Todd L. McKerchar, et al, A Comparison of Four Models of Delay Discounting in Humans, Behavioral Processes. 81, 2009, 2, pp. 256-259).

As clarified by J. Doyne Farmer and John Geanakoplos (Hyperbolic Discounting is Rational: valuing the far future with uncertain discount rates):

Conventional economics supposes that agents value the present vs. the future using an exponential discounting function. In contrast, experiments with animals and humans suggest that agents are better described as hyperbolic discounters, whose discount function decays much more slowly at large times, as a power law. This is generally regarded as being time inconsistent or irrational. We show that when agents cannot be sure of their own future one-period discount rates, then hyperbolic discounting can become rational and exponential discounting irrational. This has important implications for environmental economics, as it implies a much larger weight for the far future.

The significance is evident from the summary by Murray Bourne (Hyperbolic and exponential discounting SquareCirclez, 8 April 2015):

Hyperbolic discounting has many implications in the areas of savings rates (small pain now for future gain), climate change (impact of energy policy now on future environmental conditions), attitudes to health screening (some discomfort now for future health), lifestyle choices (amount of exercise now for reducing obesity) and behavior due to weather predictions (how many crops to plant for future benefit).

Present bias and collective learning: Time-inconsistency in decision-making has not featured significantly in public debate, despite efforts to highlight its implications (W. Kip Viscusi and Joel Huber, Hyperbolic Discounting of Public Goods, 2006; Larry Karpand Yacov Tsur, Climate Policy When the Distant Future Matters: Catastrophic Events with Hyperbolic Discounting, eScholarship, February 2007; Julie Rehmeyer, 'Discounting' the Future Cost of Climate Change, ScienceNews, 21 May2010; Joseph H. Guth, Resolving the Paradoxes of Discounting in Environmental Decisions, Climate Legacy Initiative 2009).

Irrespective of comprehension of the issue by authorities, the matter has been questionably framed in terms of voter bias (Robert Axtell and Gregory J. McRae, Changing How We Discount to Make Public Policy More Responsive To Citizens' Time Preferences, Brookings, March 2006; Alberto Bisin, et al, Government Policy with Time Inconsistent Voter).

References to "present bias" and "intertemporal" perspectives recall concerns with the implications of short-termism as the bane of sustainability (David Foulke Hyperbolic discounting and Present bias, AlphaArchitect, 22 May 2014; Petra M. Geraats, Intertemporal Substitution and Hyperbolic Discounting, 2006; Roger L. Martin, Yes, Short-Termism Really Is a Problem, Harvard Business Review, 9 October 2015).

There is clearly a paradoxical context to be recognized, especially in the light of the arguments of John Ralston Saul (The Unconscious Civilization, 1995). If, as noted in the main paper, learning is especially necessary in order to avoid systemic failure, time consistency merits recognition as jeopardizing the requisite double-loop learning. The paradox is further complicated by the case for an existential focus on "now" (Peter Russell, The Spirit of Now; Eckhart Tolle, The Power of Now, 1997). The dilemma can be explored as Presenting the Future (2001).

Logico-mathematical formalization of "sin"? Given the discussion in the main paper of the possibility of fruitful ordering of the patterns of failure of a viable system, it is useful to see such failure as encompassing the failure associated with succumbing to short-term temptation. The temptation process may itself be susceptible to a formalization, as suggested by the research on hyperbolic discounting, and as speculatively argued separately (Towards a Logico-mathematical Formalization of "Sin": fundamental memetic organization of faith-based governance strategies, 2004). This included the following sections:

Confusing variety of sins Sins and logical fallacies
Sins in relation to axes of cognitive bias
Sins as catastrophes
Sins and the mathematics of harmony
Sins as disruptions of the seamlessness of the cosmic plenum
"Sins" understood through the drama of psycho-social dynamics
"Sinful questions"
"Sins" in the light of number theory
Trigrams: a coding system for "sins"?
Enneagram of sin
Sins inherent in value polarities
"Sins" in the light of a Theory of Everything
"Wrongness": as "sins" of structural design and aesthetic composition
Statistical indicators of "sin"
Social process triangles as a potential framework for "sins"
Vector equilibrium as a dynamic configuration of tendencies to disorder or "sin"
Day of Judgement: multi-dimensional accounting for sin?
"Redemption of sins" and "healing"
Implications for faith-based governance
Role of mathematics in support of faith-based governance

Failing cognitively: Rather than projecting the process of falling by insubstantial entities into some distant (and hypothetical) time and realm, there is a case for exploring how this may offer a metaphor for erosion of a pattern of insight, namely a loss of some special form of present moment connectivity which it is a challenge to name and describe. The question of who or what does the falling may be far closer to home than is conventionally assumed -- whether it applies to an individual, a collective, or even to a global civilization.

As queried in the main paper, if there are as many as 72 "angelic" forces vital to the viability of sustainable global governance (as explored here), what is the danger that many will "fall through" any "cognitive safety net"? Possibly even all of them? What intuitive understanding is associated with the significance so extensively accorded to "fallen angel" in film, literature, music and song, as noted in a checklist in Wikipedia?

Such a safety net might indeed be understood in more conventional terms as the connectivity implied by concept maps, cognitive maps, mind maps, network diagram, or even maps of web hyperlinks. The loss of connectivity might then be understood as loss of conceptual links, social relationships, and erosion of memory, with their implications for creative inspiration. This could be readily recognized in the evident loss of capacity as a consequence of aging, dumbing down, psychic numbing, or the like -- through which solitude, loneliness and loss of self-esteem are variously engendered.

As cited with respect to collective memory (Societal Learning and the Erosion of Collective Memory, 1980), there is a striking parallel to be drawn between the many attempts by the UN Secretary-General (and others), to communicate to the world the urgency of humanity's present situation and the following fictional account by the Nobel Laureate Doris Lessing. This describes the encounter between a "galactic development agent" and the inhabitant of a "developing planet":

To say that he understood what went on was true. To say that he did not understand -- was true. I would sit and explain, over and over again. He listened, his eyes fixed on my face, his lips moving as he repeated to himself what I was saying. He would nod: yes, he had grasped it. But a few minutes later, when I might be saying something of the same kind, he was uncomfortable, threatened. Why was I saying that? and that? his troubled eyes asked of my face: What did I mean? His questions at such moments were as if I had never taught him anything at all. He was like one drugged or in shock.

Yet it seemed that he did absorb information for sometimes he would talk as if from a basis of shared knowledge: it was as if a part of him knew and remembered all I told him, but other parts had not heard a word. I have never before or since had so strongly that experience of being with a person and knowing that all the time there was certainly a part of that person in contact with you, something real and alive and listening -- and yet most of the time what one said did not reach that silent and invisible being, and what he said was not often said by the real part of him. It was as if someone stood there bound and gagged while an inferior impersonator spoke for him". (Re: Colonised Planet 5 - Shikasta, 1979, pp. 56-57).

Feedback processes and angelic orders: In the modern jargon of management cybernetics, one overly simplistic approach to the experience of "angels" and "demons" is in terms of positive feedback and negative feedback respectively. Here the sense of "positive" and "negative" is borrowed from their use in describing "good" feedback (especially of an inspiring, empowering nature) in contrast with "bad" feedback (especially of a critical, disempowering nature).

Missing from this interpretation is of course the sense in which positive feedback may further exacerbate a problem and negative feedback may be vital to any corrective learning. The ambiguity is evident in use of metaphors that challenge the appearance of good (as with a wolf in sheep's clothing) or bad (as with a benefactor in dubious disguise). The point has been argued by Barbara Ehrenreich (Smile Or Die: How Positive Thinking Fooled America and the World, 2009).

As is evident in cybernetics, the control of any complex system -- and the achievement of sustainability -- requires a complex balace of positive and negative feedback, as discussed separately (Being Positive Avoiding Negativity: management challenge of positive vs negative, 2005). Exposure to the "demonic" may be vital for new forms of learning as a complement to the inspiration and support offered by the "angelic".

Presented in this way, the traditional pattern of thinking regarding the angelic and demonic orders could be recognized as an early effort to organize engagement with collective issues -- now framed by cybernetics in systemic terms. Both the earlier framing and the modern one offer challenges to comprehension, but of a different nature. Where are checklists now to be found of the forms of positive feedback and of negative feedback required for the achievement of sustainability? This is now acquiring an aura resembling goals of the past (In Quest of Sustainability as Holy Grail of Global Governance, 2011; In-forming the Chalice as an Integrative Cognitive Dynamic: sustaining the Holy Grail of global governance, 2011).

How many factors are considered appropriate for such a checklist -- given the relatively limited number highlighted in relation to the UN's Sustainable Development Goals? Where is the systemic map of their interrelationship -- in the light of the degree of organization sought and presented in the case of angelic and demonic orders? What subtler factors are to be distinguished, despite their elusive nature -- in contrast with those which lend themselves readily to representation in terms of sets of socio-economic and environmental indicators?

Erosion of self-reflexivity and learning capacity: The question here is whether the cognitive failure is of a quite different nature, better framed by the significance of the mirror self-recognition test. Rather than limiting this to its simplest conventional form, are higher orders of "mirror test" to be envisaged? This was previously explored with respect to the hypothetical encounter with extraterrestrials who, in cognitive terms, might well be comparable to angels (Self-reflective Embodiment of Transdisciplinary Integration (SETI): the universal criterion of species maturity? 2008; Sensing Epiterrestrial Intelligence (SETI) Embedding of "extraterrestrials" in episystemic dynamics? 2013).

Following the arguments in the main paper, such "higher orders" can be explored in the light of knowledge cybernetics, effectively as higher orders of self-reflexivity, as articulated by Maurice Yolles and Gerhard Fink (A General Theory of Generic Modelling and Paradigm Shift: cybernetic orders, Kybernetes, 44, 2015). Varying degrees of self-reflexivity are distinguished meriting exploration and recognition. As phrased by the authors:

The obvious difficulty is that such language is far from enabling the understanding of self-reflexivity suggested by the possibility of more complex forms of mirror test. As noted, further difficulties are suggested by the process of "falling" from a higher to a lower order, as might be recognized (and experienced) through the aging process or the use of narcotics. Especially perverse is the deliberate instigation of any such fall through dumbing down.

An issue central to this argument is the relative incomprehensibility and meaninglessness of a higher order of self-reflexivity from the perspective of a lower order. Expressed otherwise, insights deriving from higher order understanding are extremely difficult to communicate into frameworks relying on lower orders of reflexivity and connectivity. The further challenge is to represent such challenges to the communication/comprehension process. Hence the merit of adapting insights associated with the Poincaré disk.

Curiously appropriate, two noted specialists in cybernetics have resorted to a quite distinctive mode in order to communicate their insights. W. Ross Ashby made a systematic collection of 159 aphorisms (published by the Cybernetics Society). Russell L. Ackoff made use of fables (Ackoff's Fables: irreverent reflections on business and bureaucracy, 1991; Jonathan H. Klein, Ackoff's Fables revisited: stories to inform operational research practice, Omega, 2009). This follows in the long tradition of many cultures, as discussed separately (Reframing connectivity through metaphor, 2011). As yet unaddressed is how these sets, whether aphorisms or fables, are effectively pattern languages -- precautionary or otherwise -- meriting particular attention from the perspective of requisite self-reflexivity.

"Demon raising"? Ironically, to be recognized as the converse of "angel falling", is the process of "demon raising". This could be understood as the transformation from the most limited forms of self-reflexity to those of higher orders -- however these might be understood, as tentatively explored separately (Engaging with Insight of a Higher Order: reconciling complexity and simplexity through memorable metaphor, 2014).

It is of course the case that understandings of these processes have been articulated both within mystical traditions and through a variety of spiritual disciplines, as profiled in the Human Development Project. Ken Wilber has been most assiduous in endeavouring to order and reconcile the contrasting languages used by them (Integral Psychology: consciousness, spirit, psychology, therapy, 2000; The Spectrum of Consciousness, 1977). Especially problematic is the degree to which those identifying with particular disciplines and languages endeavour to lay special and exclusive claim to the originality of their insight. Aspects of this tendency are -- perhaps necessarily -- evident in those promoting the integral theory pioneered by Wilber. Perhaps necessarily again in systemic terms, this is indicative of the constraints on any universal uptake.

As stressed above, it would appear that there is a particular requirement for avoidance of cognitive closure, whether premature or otherwise, in imagining the nature of the self-reflexivity of angels. The following are therefore merely suggestive indications, potentially framing further exploration.

Knots -- ideal and otherwise: The cognitive failure of self-reflexivity associated with "falling" can be fruitfully explored in terms of knots, as has been done from a psychoanalytic perspective (R. D. Laing, Knots, 1972) as discussed separately (Knot Being: to be or knot to be, 2001). The knot metaphor featured in Laing's development of the so-called "double bind". As explained by Christine Wertheim (To Be or Knot to Be, Cabinet, 22, Summer 2006), Jacques Lacan appropriated geometry, topology, and eventually knot theory for his notion of the psyche, making particular use of the Borromean knot (depicted above). This offered a model in which psychic reality is (re)presented as a complex of three separate, but interrelated spheres: the Real, the Symbolic, and the Imaginary. ?

With respect to any hypothetical "angelic" condition, and the relation between "angels", the knot metaphor can be explored further in terms of the understanding of the ropelength with which a link or knot is associated in physical knot theory. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are termed ideal knots and ideal links respectively. This could be presumed to have implications for patterns of hyperlinks and their comprehensibility.

Such perspectives raise the possibility that the vicious cycles associated with wicked problems might be fruitfully explored in terms of knots.

Envisaging the global dynamics of a "hyperbolic round table" through tessellation

A symposium of the wise, to celebrate the sesquicentennial of Boston University, selected a tessellation as the metaphor that best captured the spirit of the times -- as noted by Lance Morrow (Metaphors of The World, Unite!, Time, 16 Oct. 1989). This is consistent with the explosion of interest in networking over recent decades, and the articulation of discourse that this has offered over the internet.

Archetypal round table: With respect to encounters of the wise, the archetypal metaphor continually favoured is probably the round table. This implies connectivity beyond the conventions of discourse in relation to a square table (as discussed in the main paper) The constraints and possibilities of discourse configured in this way are discussed separately (Implication of the 12 Knights in any Strategic Round Table, 2014).

In the light of the case for exploration of hyperreality, the question arises as to how a "round table" might be fruitfully reframed beyond the constraints and particularities of Euclidean geometry. How might the "roundness" be understood in terms of a sphere, for example? This would presumably facilitate discourse with respect to issues of a "global" nature (Future Generation through Global Conversation, 1997).

Hyperbolic round table: It could be imagined that some such spherical configuration is already implied in the global connectivity offered by the internet (Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998). . However the question remains as to the pattern of communication between participants when any such table is finite but unbounded -- as might be the case when configured around a sphere, impractical as this may be in reality. One approach is through exploring how mathematicians have sought to represent non-Euclidean connectivity on a flat plane, within a circle. This could then be recognized as effectively a hyperbolic round table. Might this enable higher orders of connectivity to be envisaged?

The pattern of discourse, as a whole, in the case of a conventional round table could indeed be represented by some form of tessellation or tiling, as can be variously envisaged (Interweaving Thematic Threads and Learning Pathways, 2010). In an ideal case the pattern might then be composed of regular polygons (triangles, squares, pentagons, etc) reflecting lines of agreement enclosing areas of consensus. In the regular tessellations of a plane, six equilateral triangles meet at each vertex, four squares meet at each vertex, and three hexagons meet at each vertex. Wikipedia offers an array of images contrasting Euclidean, spherical and hyperbolic tilings (Uniform tilings in hyperbolic plane).

In the images below, the nature of communications suggested by the Poincaré pattern is associated using a superimposed radial gradient indicative of the contrast in clarity and comprehension between the centre and the periphery in that depiction of hyperbolic space. The centre is then reflective of the higher connectivity of the insights of the "angelic" condition.

Exploratory association of radial gradient with use of Poincaré disk pattern
   

In the following images, the colouring of the radial gradient is reversed to suggest the implication of "demonic" ignorance with respect to the subtler forms of "angelic" connectivity. The centre is then reflective of the distorted connectivity of the "demonic" condition associated with "fallen angels".

Exploratory association of radial gradient with use of Poincaré disk pattern
   

Communicability around a radially zoned "Dyson sphere" of comprehension: The radial gradients recalls the layers of the atmosphere: troposphere, stratosphere, mesosphere, thermosphere, exosphere. Of fundamental significance to radio communication is the ionosphere, made up of the last three layers. It is ionized by solar radiation and forms the inner edge of the magnetosphere. Its importance to global communications lies in the ability of radio waves to bounce off the ionosophere to more distant locations around the globe. Known as skywave or skip, this refers to the propagation of radio waves reflected or refracted back toward Earth from the ionosphere. The phenomenon is illustrated by many images.

The argument can then be developed by imagining the communication challenge to be "around a globe" rather than "at a table". In this sense the radial gradient is to be understood in terms of nested spheres rather than nested circles. This recalls the hypothetical Dyson sphere imagined as a shell completely encompassing a star (and capturing most or all of its power output).

With respect to the comprehension challenge, the above schematics are then indicative of a reversed condition. The stakeholders seated around the inner surface of the outermost sphere then effectively propagate waves of communication which are bounced off the outer layers of inner spheres. These layers of the "cognitive ionosphere" are spheres "within" rather than "without".

Clearly the smaller the diameter of that layer -- namely the closer to the centre -- the greater the ability to enable communication with more distant stakeholders positioned around the inner surface of the outer shell. Thus the smaller the diameter, the lesser the degree of global incomprehensibility, and therefore the greater the communicability. Just as in radio communication, there is then recognition of a skip zone, or zone of silence, namely a region where communications cannot be received. The larger the diameter, however, the more reduced the communicability enabled.

Indication of communicability "around" a radially zoned
"Dyson sphere" of comprehension
(communicating stakeholders indicated on outer circle)
Communicability in a radially zoned Dyson sphere

To the extent that any inner sphere serves only as a reflecting mirror to communication, it also implies the incomprehensibility of more fundamental aspects of globality. The nesting of the spheres is also usefully indicative of increasing or decreasing orders of self-reflexivity (as discussed above with respect to cybernetic orders of feedback).

As a consequence of its remarkable recent development, the radio communication metaphor offers a powerful language for the exploration of the communicability of global comprehension amongst stakeholders. Noteworthy is the contrast between shortwave communications (as reflected off the ionosphere) and longwave communication (understood as "hugging" the surface as a groundwave). This could enable discussion of the contrasting modes of communication of comprehensibility using the extensive language of radio propagration and the associated radio frequencies and wavelengths.

One understanding of the distinctive "inner spheres" is offered by the colour-coded contrasts of spiral dynamics, notably as summarized from an integral theory perspective by Ken Wilber (Ken Wilber Summary of Spiral Dynamics model).

  1. Beige: Archaic-Instinctual
  2. Purple: Magical-Animistic
  3. Red: Power Gods
  4. Blue: Mythic Order
  1. Orange: Scientific Achievement
  2. Green: The Sensitive Self
  3. Yellow: Integrative
  4. Turquoise: Holistic

A visual presentation of these as nested levels or stages might then take the form of the image on the right (below) -- although the software used did not permit the nested spheres to be coloured distinctively according to such a convention.

Images suggestive of the challenge of deriving and communicating insight
(generated using Stella Polyhedron Navigator)
Communicability constrained
by nested spheres of comprehensibility
Array of "panelists"
in quest of focus and inspiration?
Communicability constrained by nested spheres of comprehensibility

Consistent with earlier discussion of the truncated icoshedron, this has been used as the approximation to the outermost sphere on which the stakeholders are held to be positioned. The advantage of the use of a spherically symmetrical polyhedron is that it is also suggestive of the territory with which each stakeholder is identified (from a flat Earth perspective) -- namely the set of polygons constituting the faces of the polyhedron.

Partial unfolding of that polyhedral net (in the image on the right) suggestively recalls the array of elements in a parabolic antenna ("dish antenna") characteristic of a satellite dish and of some radio telescopes. Clearly the processes of dialogue may well be configured to focus insight associated with thre stakeholders represented there -- as "panelists" (Meeting Focus: a Description -- Towards Transformative Conferencing and Dialogue, 1984). The unfolding configuration also recalls the arguments for the Dymaxion global map of Buckminster Fuller, projected onto an icosahedron.

The partial unfolding around the nested spheres also recalls the form of a petalled flower as a traditional metaphor for inspiration, especially in the light of the study by Keith Critchlow (The Hidden Geometry of Flowers: living rhythms form and number, 2011)

Integrating stakeholder perspectives: Both sets of the earlier images are helpful in contrasting quasi-hierarchical order (images on right) with the potentially more appropriate hyperbolic order (images on left). The latter is a particular challenge to conventional comprehension but is representative of that challenge to an interesting degree. The regularity of the hierarchical form (on the left) is that to which many initiatives aspire, whether as the unity of science or as the heavenly order of religion. It reflects the synthesis which is the aspiration of dialogue at a round table of the wise -- or deprecated as the demonic order ruled by Satan (as implied by the lower right-hand image).

A hyperbolic understanding of the global integration of a "round table" (images on the left) is especially suggestive of the nature of connectivity of communication around the table. In the radial zones closest to the periphery, the primary mode of communication could then be understood as being in binary 2-fold terms. Zones towards the centre are then progressibely more indicative of the increasing subtlety of 3-fold, 4-fold, and n-fold communication. This understanding offers the possibility of further articulation (in a subsequent discussion).

Especially useful is the sense in which the proximity of points around the circumference of the table is indicative of the greater facility of dialogue -- in binary terms, between those points adjoining each other. The facility of communication between points more distant is thus to be understood as more constrained -- that passing in larger arcs, across the table, being more especially so. The manner in which all arcs meet the circumberence at right angles is helpfully indicative of how absolutely "right" is held to be the associated perspective.

There is then the question of the global connectivity of those "seated" around the circumference of the table (as stakeholders of contrasting qualities) -- potentially framed in terms of the 72 angels or demons, or perhaps 144 (given both angelic and demonic possibilities). Is the perspective from each seat to be identified with one such quakitative insight or pattern -- or perceived to be so identified? More intriguing is the degree of integration implied by the non-binary connectivity of points within the table and by the areas thereby circumscribed. Understood as dynamc containers of complex insights, these would be indicative of subtler and more challenging insight towards the centre of the table -- implying ever greater degrees of self-reflexivity.

Paradoxical comprehension: Visualizing the challenge to comprehension can be taken further by interrelating the contrasting understandings -- specualitively -- within a pattern of interlocking Möbius strips as indicated below. This recalls the challenge suggested above by Borromean rings.

Experimental integration of 4 contrasting perspectives
within 4 Möbius strips
Framework of interlocking Mobius strips

Use was previously made of interlocking Möbius strips (in a schematic Denkmodel), to frame issues of some relevance to the above argument, as discussed separately (Psychosocial Energy from Polarization within a Cyclic Pattern of Enantiodromia, 2007)

Experimental visualization of three-stage transformation between "global" and "local"
. Stage 0/1 Stage 2 Stage 3/0
Explication Three-stage transformation between "global" and "local"
Ambiguity
Contradiction

 

 

Implication

 

 

Such schematics are suggestive of the requisite cognitive complexity for engagement with "global" integration. They recall the many efforts to represent schematically the adaptive cycle proposed by the Resilience Alliance, and promoted as necessary for strategic resilience by Thomas Homer-Dixon (The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization, 2006), as discussed separately (Adaptive Hypercycle of Sustainable Psychosocial Self-organization, 2010) .

Representation of adaptive cycle in complex systems
(with many variants available via Google images)
(adapted from a variant of Thomas Homer-Dixon,
The Upside of Down, 2006)
(as reproduced from Panarchy,
as described by The Sustainable Scale Project)

Traditional modes of cognitive engagement with hyperreality

The argument above has focused on use of verbal argument and visualization to clarify possibilities for engagement with hyperreality. It is however appropriate to recognize that other modes are widely favoured, in the light of traditions of the past. These modes could be distinguished as variously emphasizing decision-making, game-playing, use of circlets of beads for mnemonic guidance in prayer, and focusing devices for meditation. (Correspondences between Traditional Constellations and Pattern Languages: requisite simplexity for sustainable comprehension of complexity, 2014)

Classical Chinese encodings of engagement with hyperreality: Of particular interest in terms of their systemic organization are several classical Chinese encodings of a set of conditions -- I Ching (Yijing), T'ai Hsüan Ching (Tai Xuan Jing) -- and the specific manner in which transformation is possible between them. These are variously discussed separately (Sustainability through Magically Dancing Patterns, 2008; 9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes. 2003; 9-fold Magic Square Pattern of Tao Te Ching Insights experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006).

Of particular relevance to further reflection on the dynamics of a "hyperbolic round table" (as discussed above), is the pattern of hexagrams of the I Ching and the transformations between the conditions they represent, most notably with respect to decision-making (whether deliberate or inadvertent). Especially striking, in the light of the visualization of hyperbolic tessellation on the Poincaré disk are the arcs indicative of links between proximate hexagrams -- reminiscent as they are of the arcs on that disk.

Screen shot of circular configuration of 64 hexagrams of I Ching
(formed by animation of inner circle of trigrams
relative to outer circle of trigrams)
Circular configuration of 64 hexagrams of I Ching

The schematic above was used as the framework for a previous exercise (Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation, 2008).

Engaging with hyperreality through traditional circlets of beads: Meriting particular attention is the significance widely associated with circlets of beads. Their use in ritual practice can be recognized as a form of cyclic engagement with an elusively implied integrative understanding -- one which is not exclusively encompassed by the mnemonic associations of any one bead. The manner of the use of such circlets suggests an intuitively appreciated engagement with a connecting meta-pattern, as discussed separately.(Designing Cultural Rosaries and Meaning Malas to Sustain Associations within the Pattern that Connects, 2000)

Clearly such a circlet may have only a few 'beads' as mnemonic aids -- as is the case with some charm bracelets. Alternatively, it may be of much larger size, as the following examples indicate::

Mandalas as a focus for meditation: As with the resemblance of the above schematic to the Poincaré disk, mandalas and rose windows could be interpreted as structured to provide a focus for engagement with hyperreality.

Symbolic games and game symbolism: Many games have been elaborated to enable engagement with subtler insight than is immediately obvious -- or have lent themselves to such interpretation. These notably include chess and go, and their many equivalents, as well as games using sets of cards. Especially relevant to this argument are:

Given the acceptance in popular culture, it is appropriate to ask how further use might be made of games (a process now recognized as gamification), possibly with extensive sets of cards, to enable greater comprehension and organization of factors relevant to sustainability. Especially intriguing with respect to sustainability is the transformstion from the win-lose focus of conventional games to that skillfully articulated by religious scholar James P. Carse (Finite and Infinite Games: a vision of life as play and possibility, 1987).

Of particular interest is enabling understanding in terms of patterns of feedback dynamics -- such as might be relevant to recognizing the modes of failure of viable systems (discussed in the main paper). Such possibilities are illustrated by the approaches variously taken with respect to representation of the map of metabolic pathways (WikiPathways for the People; The Pathway Localization database (PathLocdb); General Overview of the Major Metabolic Pathways). With respect to sets of cards, one opportunity is to enable the cards to be variously juxtapositioned (even fitted together as in a jigsaw puzzle) such as to visualize systemic pathways between conditions enabling or undermining sustainability. Why this has not been done is a mystery in its own right.

The cognitive-aesthetic combination in games has been extensively described by allusion, most elegantly in the Glass Bead Game (1943) by Hermann Hesse, where he comments:

I suddenly realized that in the language, or at any rate in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbol led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with truly a meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang holiness is forever being created.

These and other associations offer curious echoes to the elaboration of a "game of spheres" by Nicholas de Cusa (De Ludo Globi, 1463), written as a contribution to both a literature and a practice of moral game-playing. This formed part of the tradition of the forgotten chess-like game Rithmomachia (The Battle of Numbers or Rythmomachy), which combined the pleasures of gaming with mathematical study and moral education.

Intellectuals of the medieval and Renaissance periods who played this game were not only seeking to master the principles of Boethian mathematics but were striving to improve their own understanding of the secrets of the cosmos (Ann E. Moyer, In The Philosophers' Game, 2001). It has been an inspiration to the magnum opus of Hesse, as noted by Todd R. Harris (The Interplay of Opposites, the Language of Experience, and the Geometry of Ascent: a comparison of Hermann Hesse's "Das Glasperlenspiel" and Nicholas of Cusa's "De Ludo Globi", 2001).

Hyperbolic reframing of the Dominique and Angelique of tradition

Sustainability as Heaven? The argument can be further developed by recognizing the extent to which "sustainability" has now been framed as a secular understanding of the condition traditionally associated with "heaven". This is matched by the "hell" with which global civilization is confronted if it fails to get its act together appropriately -- and the condition with which many are already confronted in their daily lives.

Through the pattern of angels and demons, the order governing these conditions has been traditionally intimated -- elusively and mysteriously. However the requisite order would seem to be equally mysterious to global governance of the present, notably given the variety of contradictory advice by experts regarding its nature. Achieving sustainability appears to be as essentially elusive as access to heaven.

The merit of the pattern of angels and demons lies in part in its complexity which would seem to exceed, possibly by an order of magnitude, the somewhat simplistic pattern through which the condition of sustainability is articulated in academic and political discourse. Of particular interest however (as noted above) is the extent to which the angels and demons continue to evoke popular attention worldwide, whereas the articulation of sustainability is far from achieving such credibility. Especially relevant is the manner in which their roles and functions are personalized as a feature of individual experiential reality. Promotion of sustainability struggles to achieve this. That belief in angels and demons is deprecated as ridiculous by secular authorities, needs to be set against the surreal absurdity of global governance which those authorities have engendered and continue vigorously to sustain (as noted above).

Use of "pattern" implies the possibility of depiction in 2D, whereas the sets of angels and demons might well call for articulation as a complex in 3D, 4D, or more -- justifying introduction of the terms Angelique and Demonique (as an elaboration of associations with the Club of Rome's resolutique and problematique). These can be understood as respectively characterizing the dynamics of "heaven" and "hell" -- whether recognized in secular terms or otherwise.

Requisite complexity: Especially as being common to the Abrahamic religions in long-standing conflictual relationship, there is therefore a case for exploring the pattern of 72 angels (and the matching pattern of 72 demons) as an alternative lens, engendered by past struggles for coherence. In terms of credibility, it could be said to have stood the test of time to a degree which the quest of sustainability has yet to match. However there is also a case for questioning the nature and adequacy of that pattern in the light of recent systemic insights and the richer articulations suggested by mathematics. Essentially there would appear to be a cognitive challenge to the meaningful comprehension of complexity in experiential terms -- and not merely in the cognitively dissociated terms promoted as scientific objectivity. Integrating design and ethical dimensions, comprehension of sustainability might then be understood through the challenges of appropriateness (Comprehension of Appropriateness, 1986).

The focus here on 72 constitutes an interesting challenge to the conventional articulations of sustainability through sets of factors of a much more limited size and a far lesser degree of systemic organization. Mathematically 72 has particular properties, notably as 23 x 32, which suggest that the set as a whole might then be especially conducive to comprehension. Clearly smaller sets may be even more conducive, but without necessarily offering the requisite variety for understanding the dynamic nature of "heaven" -- or sustainability. This suggests exploration of optimum set size as a balance between comprehensibility and distinguishable variety. The size of some circlets of prayer beads as mnemonic aids extends to 108, for example (as mentioned above).

The traditional set of 72 is described in the Wikipedia entry on Shemhamphorasch with reference to a "hidden name of God" in Kabbalah (including Christian and Hermetic variants), and in some more mainstream Jewish discourses. It is composed of either 4, 12, 22, 42, or 72 letters (or triads of letters), the last version being the most common as the 72-fold angels of the Shemhamphorash. It is derived from Exodus 14:19-21 read boustrophedonically to produce 72 names of three letters, as explained in some detail by Lon Milo DuQuette (Angels, Demons and Gods of the New Millennium: musings on modern magick, 1997). Every other line of writing is flipped or reversed, with reversed letters -- a common way of writing in stone in Ancient Greece, with a number of modern examples (including a web application). It is related to the mirror writing used by Leonardo da Vinci. The key to sustainability could itself be fruitfully explored as cognitively "hidden", justifying use of such devices.

The triadic naming of the angels is curiously reminiscent of the trigrams of the 64 I Ching hexagrams and the subdivision of each into 3 pairs.

Forms of presentation: lists, tables geometry: The concern here is with how a pattern of 72 might be distinctively represented as indicative of a unique form of coherence -- essential to secular comprehension of sustainability. The many traditional lists of 72 angels and demons are not helpful in this respect. More interesting is their representation in 8x9 table form -- notably with respect to the sense of 9 choirs of angels (discussed below). However tables of those proportions are themselves not especially conducive to credible integrative comprehension -- whatever that may be held to be. The experiential cognitive challenge could be more fruitfully framed metaphorically as one of juggling 72 balls.

A further possibility is to switch to visual presentations in geometrical or topological form, preferably in 3D. This would be consistent with insights into globality -- historically subsequent to the tabular presentations of tradition. It may be readily assumed that there is a quality to the spherical form of the planet which is at least indicative of the integrative connectivity of the dynamics of sustainability -- and of heaven. The challenge to comprehension of globality is however highlighted by the extent to which many are obliged to think in terms of the flatness of the Earth which they most commonly experience -- despite representations of it and images from the Moon (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008; Future Generation through Global Conversation, 1997).

In this respect it is intriguing to note the variety of approaches to depicting the global nature of the Earth (in 3D) as a map on a flat surface (in 2D) -- namely the process of map projection. Wikipedia offers a List of map projections, including the following:

The notable limitation of these projections is that they are necessarily designed to represent a globe in static terms -- with the necessary distortion of each constituting a further challenge to comprehension. Sustainability (and heaven) are however necessarily, and essentially, dynamic in nature, as separately argued (Dynamic Transformation of Static Reporting of Global Processes: suggestions for process-oriented titles of global issue reports, 2013).

It might be asked what form temporal mapping projections of the globe could take -- and whether a range equivalent to the spatial mapping projections is explored, for analogous reasons (see Wikipedia discussion of animated mapping). The design of some prayer wheels may reflect a quest for such representation. [It is appropriate to note that a number of the spatial projections are of a form which recalls the depictions of winged angels -- a resemblance potentially even more probable in the case of the dynamics of any temporal projection.]

Comprehension through alternation?: The argument above endeavours to emphasize that a pattern of 72 factors -- potentially vital to governance and sustainability -- poses a major challenge in terms of human cognitive constraints. Representations of the traditional angels and demons can be used to highlight the difficulty, as in the following.

Animation of sequence of 72 Angel names from the Shemhamphorasch
(in two contrasting representations)
Animation of sequence of 72 Angel names from the Shemhamphorasch Animation of sequence of 72 Angel names from the Shemhamphorasch

Variously understood as corresponding to the 72 angel names are those of the set of 72 demons (List of demons in the Ars Goetia; Goetia Demons). Especially interesting is the sense in which the diagrams below might in each case be understood as systemic pathways or patterns within the larger set, as discussed separately (Evil Loops and Sigils as a Pattern Language, 2016). As noted there, the sigils could well be understood as corresponding to the individual patterns of a pattern language -- through which the multiple faces (facets) of evil manifestation are to be recognized. The pattern language could then be considered as corresponding to any proposed pattern with respect to wicked problems featuring within the global problematique, as indicated with respect to Patterning the Problematique.

Animation of sequence of 72 demonic sigils from the Ars Goetia
(with matching reversed images)
Animation of sequence of 72 demonic sigils from the Ars Goetia Animation of sequence of 72 demonic sigils from the Ars Goetia

It is noteworthy, as implied by such animations, that there is a difficulty in representing a pattern of 72 distinctions in 2D. This "difficulty" may be significant in its own right, justifying any bias to limit representations to sets and patterns of a more limited size only -- a bias constraining any capacity to encompass adequately the problematique or the resolutique as total systems.

Comprehension through mapping in 2D? One experimental layout in 2D is illustrated in the following for the case of the demons. An equivalent could be presented for the 72 angels of the Shemhamphorasch (possibly as animated alternative). These have some resemblance to the Tree of Life which is the focus of the traditional Kabbalah. The angels are held to be the inhabitants of that 10-fold pattern, with its 22 interconnecting pathways.

Alternative experimental configurations alternating between the 72 angels and demons
Animation of 8 sets of 9
(enlargements for detail: angels / demons)
Animation of 9 sets of 8
(enlargements for detail: angels / demons)
Experimental configuration alternating between the 72 angels and demons Experimental configuration alternating between the 72 angels and demons
The allocation of sets to the star "tables" in the above schematics 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 questionable, demanding further consideration.

The patterns above can also be understood as a systematic representation of modes of failure -- the demons of governance of a viable system (as discussed in the main paper). Governance could then be understood as the art of juggling the 72 forms of negative feedback (demons) and the 72 forms of positive feedback (angels) -- recalling various traditional references to 144. The traditionally recognized functions and qualities of both could be fruitfully explored for clues to a modern articulation (possibly in management jargon) as has been done with the 64 hexagrams of the I Ching (Transformation Metaphors derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997).

An equivalent mapping challenge is evident in the representation of the periodic table of chemical elements, now numbering over 100. Whilst its conventional representation as a table is a convenience offering a variety of insights, it is of limited value in enabling comprehension of the pattern as a whole -- as might be desirable in the case of the pattern of governance-significant factors. Using that table as a metaphor, aspects of the challenge are discussed separately (Tuning a Periodic Table of Religions, Epistemologies and Spirituality -- including the sciences and other belief systems, 2007).

Comprehension through mapping in 3D? As noted above, any 2D representation only faclitates "global" comprehension to a limited degree. A fruitful 3D possibility is to associate the demons or angels with polyhedra. A number of these have 72 features (notably vertices or faces). This was the approach previously taken with the 64 hexagrams of the I Ching as traditionally indicative of conditions of change (Proof of Concept: use of drilled truncated cube as a mapping framework for 64 elements. 2015; Eliciting Memorable Spheres and Polyhedra from Hyperspace: integrative connectivity of problems, strategies, themes, groups or people, 2015).

The polyhedral library of Stella Polyhedron Navigator enables display of the following polyhedra onto whose features angels or demons might be mapped:

The suitable polyhedra are themselves relatively complex as mnemonic aids. Another approach is to make use of simpler polyhedra with different constraints, enabling the display of subsets of demons on several variants, as shown below using the polyhedron best known for its pattern in football stitching (as noted above).
Experimental use of truncated icosahedron to display 2 sets of 32 demons
linked by an octahedron displaying a set of 8 demons (thus totalling 72)
(generated using Stella Polyhedron Navigator)
Version A mapping of 32 Mapping of 8 Version B mapping of 32
Experimental use of truncated icosahedron to display a sets of 32 demons Experimental use of truncated icosahedron to display a sets of 32 demons

Clearly a similar approach can be taken to the display of any set of images of 72 angels. Of interest from a traditional perspective is the degree of resemblance to the vajra (Sanskrit) or dorje (Tibetan). This is used as a ritual object to symbolize both the properties of a diamond (indestructibility) and a thunderbolt (irresistible force). This is equivalent to the sceptre traditionally emblematic of a ruler, as with the ceremonial mace (for which an official, appointed as Black Rod, is responsible in some modern parliamentary procedures). Curiously the form also recalls both the dumbell used in weight training, as well as the use of that term to refer to the form of a 2-lobed atomic orbital.

Use of the intermediary octahedron with its 8 faces potentially reconciles the set of 64 (mapped onto the two truncated icosahedra) with the set of 72. The set of 64 has of course been long associated, through use of the I Ching (as presented above), with an understanding of governance and decision-making in China. However as a distinctive set, in addition to the 64, the pattern of 8 can be understood as a supernal or controlling pattern, perhaps to be associated with the functions and insights of "archangels" -- the ninth "choir" in the pattern of 9x8 angels (as discussed below).

With respect to holding together cognitively both the 72 patterns of the angelique and the 72 patterns of the demonique, a further possibility is indicated by the 4-fold dorje. Two vertices of the octahedron might then be linked to two more truncated icosahedra to enable mapping of both sets. However, with respect to the cognitive challenge of governance (for which any such symbol has mnemonic functions), the relation between angelic and demonic is essentially dynamic, potentially better illustrated by having the images alternate on the equivalent of a 2-fold dorje.

A suggestion of this alternation can be mnemonically enabled by rotating the 2-fold complex on an axis through the octahedron. With any such dynamic, the 2-fold mapping surface then holds the complete set of patterns for the sustainable governance of a viable system. The cognitive significance and requisite skills may well be implied by variants of the prayer wheel, baton twirling, pen spinning, and astrowheeling.

The range of possibilities for holding complexities memorably -- in terms of the set of 72 (or 144 or 288) is usefully illustrated by the following "mapping surfaces" which enable various "stories" to be explained in terms of positive and negative dynamics. Following the argument above, given its 32 surfaces, use is made of the truncated icosahedron (a "football") in various configurations. Two such polyhedra may be linked by an octahedron to give 72 surfaces. Four such polyhedra may also be linked, with the addition of two tetrahedra, to give 144 surfaces.

Exploration of possibilities of holding complexity memorably in terms of 72 angels and/or demons
(visualizations constructed using X3D to enable interactive virtual reality display and/or use of WRL browser capacity)
72 surfaces (64+8)
(interactive X3D / WRL)
136 surfaces
(interactive X3D / WRL)
144 surfaces
(interactive X3D / WRL)
Rotational pathway?
(interactive X3D / WRL)

Each of the above can of course be rotated around the central axis through the octahedron (as shown in animations). Of particular interest, benefitting from design/display possibilities, are the lighting effects causing the colours to switch and flash during rotation, as partially illustrated in the screen shots. This helpfully suggests the dynamic between angelic and demonic dynamics (without the need to alternate the colours programmatically). The relevant files can be readily modified to change the proportions, colours and transparency. Due to such transparency, the image on the far right partially illustrates the manner in which the tetrahedra -- as "archangels" -- transcend and interpenetrate the other elements. Through the addition of the circle linking the truncated polyhedra in that image, a suggestion is offered of the pathway of those polyhedra during rotation.

In addition to rotation around the central axis through the octahedron, the truncated polyhedra can also be individually rotated (possibly with counter-rotation). Such effects could also be achieved with colour alternation -- even to all white, or all black -- according to variants of the "story" regarding the relation between angelic and demonic dynamics. A more extensive range of colours could be used. It is also appropriate to note that, with the exception of the cube, the mapping complex uses all the Platonic solids, explicitly or implicitly (the truncated icosahedron being a conflation of the icosahedron and the dodecahedron).

"Technological theology" or "Theological technology" ? Rather than the emphasis here on mathematical theology, the degree to which such a structure holds the extremes of the symbolism associated with twirling and the form of a frequently imagined space station (image on far right above) suggests that such a dynamic meta-pattern has an intuitively attractive appeal -- possibly far more fundamental than is readily appreciated. It is even possible that enthusiasm for investment in either extreme, especially a permanent space station in orbit, constitutes a form of cognitive displacement from the possibility of engagement with hyperreality -- understood otherwise.

Currently of potentially greater significance is the invasive role that it is expected that drones -- now widely available -- are expected to perform. There is considerable irony to the possibility that the future will shortly see use of personal drones as guardian angels, hovering protectively above the wealthy (at least), with their analogues in cyberspace. More curious is that their form may readily recall depictions of the halos associated with the holy.

The necessary rotation of any outer space vehicle offers a frame for discussion of "cognitive gravity" and the possibility of its transcendence in orbit -- notably in terms of the viability of such an environment as a habitat. Of relevance in that context, and at a time of detection of "gravity waves", is potential recognition of Arrogance as an analogue to gravity -- equally fundamental and mysterious (2015). The challenge of "arrogance waves" could be considered fundamental to the future organization of a time-bound civilization in that everyone perceives themselves to be normally "right" -- if not unquestionably "upright" -- irrespective of their respective orientation to the sun and the shadow thereby cast.

The space station form readily recalls a rotating mandala, variously valued as enabling "ascent", if not "escape" from planetary constraints -- as discussed in Clues to 'Ascent' and 'Escape' in (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002). With respect to "ascent" and transcendence, the last point recalls the inspiration for the designer of the Bell helicopter, Arthur Young, for the possible design of a "psychopter" -- as separately discussed (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011).

Given the requisite complexity (as implied above by interlocking Möbius strips), of particular interest from a mathematical perspective is the figure-eight twirl characteristic of variants of the plate trick (or belt trick). As shown below, this is used as a demonstration that a double rotation is a loop in rotation space that can be shrunk to a point.

Demonstration of how the end of a set of belts can be continuously rotated without becoming twisted or tangled.
(Screen shots of an animation by Jason Hise via Wikimedia Commons)
Plate trick demonstration Plate trick demonstration Plate trick demonstration Plate trick demonstration

Curiously this 8-fold pattern (as with that of the intermediary octahedron above) recalls the framing by Stephen Prothero (God Is Not One: The Eight Rival Religions That Run the World -- and Why Their Differences Matter, 2010). Failure to comprehend the integrative pattern implied by the angelic dynamic effectively transforms it into the more explicitly tangled demonic equivalent.

Comprehension through mapping in 4D? Given the constraints on integrative displays and global comprehension, it is is appropriate to note the 4D polyhedra indicated by Stella Polyhedron Navigator as being potentially useful for display of 72 demons or angels -- or the set of 144. Their four-dimensional nature is consistent with the sense that the sets of angels and demons are effectively four-dimensional entities and can only "exist" in time rather than in any static depiction. However the following example of such a "pseudo-static" structure also makes the point that these remain a challenge to comprehension.

4D Polyhedron: 68-Odho with 72 faces
72 faces of 3 types (3-gon, 4-gon, 6-gon); 96 edges; 24 vertices
Cat4-60-Ico-regiment
(derived from 4D Library of Stella Polyhedron Navigator)
4D Polyhedron: 68-Odho with 72 faces

Analogous to polygons and polyhedra, forms such as the above are 4-polytopes, or polychora. These can be tentiatively explored in terms of their potential temporal significance (Comprehending the Shapes of Time through Four-dimensional Uniform Polychora, 2015). The latter formed part of a discussion with respect to governance and sustainability (Enhancing Strategic Discourse Systematically using Climate Metaphors, 2015).

Although visually ncomprehensible to humans, it is appropriate to recall their potential future relevance in the light of foressen emergence of higher orders of artificial intelligence (Is A.I. an existential threat to humanity? Quora; Kevin Drum, Why Artifical Intelligence Is Closer Than We Realize, Mother Jones, 17 November 2011).

"Hyperbolic comprehension" in the light of time inconsistency? The argument above noted the extensive literature on hyperbolic discounting in responding to the challenge of time inconsustency in anticipating the future, most notably in decision-making. As a cognitive constraint, this adds to the challenge of any dependency on alternation or 4D representation. Further insight may however be derived from the acknowledged relevance of the hyperbolic form.

Especially interesting is that both angels and demons, as indicative of problematique and resolutique, may be best understood through a hyperbolic framing of human experiential engagement with time. Understood as a complex of positive feedback loops, the set of angels then indeed merit recognition as the Angelique (as argued above) -- but one in which a hyperbolic framing is fundamental, most probably of higher dimensionality. The same may then be said of negative feedback loops with respect to a Demonique. Explicit integration of temporal inconsistency, framed in hyperbolic terms, would seem to be especially appropriate in the light of the role it plays in the global governance of the civilization described by John Ralston Saul (The Unconscious Civilization, 1995).

There is a delightful irony to the possibility that the form of both angels and demons could well be associated, if only for mnemonic purposes, with the hyperbols. As temporal entities, transcending human cognitive incosnsistency in some way, they could well be understood as "existing" at the nexus with which the origin of the hyperbola is associated. Their "wings" are then to be recognized in the two curves of that form -- potentially indicative of learning pathways in time, distinguished directionally to reflect the angelic and demonic feedback loops.

Imagining angels and demons in mathematical terms? Some consideration has been given to the "physics" of angels, as noted above (Rupert Sheldrake and Matthew Fox, The Physics of Angels: exploring the realm where science and spirit meet, 2014). Given the arguments above, how might the mathematical form of angels and demons be imagined -- when informed by cognitive psychology (George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000)? What recent mathematical insights could fruitfully "transform" angelic and demonic depictions beyond their anthropomorphic limitations?

One interesting example is provided in an interactive applet by Jon Rogness (The Hyperbolic Paraboloid, Math Insight) as part of an Interactive Gallery of Quadric Surfaces

Hyperbolic paraboloid as suggestive of angelic form
Screen shots of an interactive applet by Jon Rogness
Hyperbolic paraboloid as suggestive of angelic form Hyperbolic paraboloid as suggestive of angelic form Hyperbolic paraboloid as suggestive of angelic form Hyperbolic paraboloid as suggestive of angelic form

The applet permits the form of the hyperbolic paraboloid to be variously modified. The central image is indicative of the underlying structure. That image usefully recalls both the detailed organization of antenna arrays (radiotelescopes, etc) and the wings of birds and insects.

Implications of the "shape of the universe" for the shape of the "Angelique" and "Demonique"? Astrophysics has long been preoccupied with the seemingly extraordinary question of the shape of the universe -- in terms of its curvature and topology. Hypotheses have notably explored the possibility of a universe with zero curvature, a universe with positive curvature, and a universe with negative curvature.

Use is made of "universe" as a metaphor (Clare Beghto, From the Universe of Knowledge to the Universe of Concepts: the structural revolution in classification for information retrieval, Axiomathes June 2008; Charles van den Heuvel, Elementary Structures: universe of knowledge or universe of concepts? 2011). This then raises the question of whether the "universe of meaning" merits similar exploration, as may be variously argued (Towards an Astrophysics of the Knowledge Universe: from astronautics to noonautics? 2006; Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships, 2013; Samuel Reiss, The Universe of Meaning, 1953).

The question here is whether the problematique, understood experientially as the Demonique, could be usefully explored in terms of its curvature and topology. The same applies to the Angelique as the complex of inspirations associated with the resolutique. What indeed might be the "shape" of the global problematique -- especially as it is mysteriously embodied cognitively in the Demonique?

Engaging creatively with hyperreality through music

Angelic and demonic music? Whereas demonic entities figure extensively in video games, it is somewhat extraordinary to note the multitude of references to angels in popular song -- and to a far lesser degree in traditional hymns. The question is what intuitions are meaningfully cultivated and sustained through music in a world informed by the rational explanations of science? Any response must necessarily go beyond those offered with respect to the traditions of religious music and chant. Reference is appreciatively made to angelic music and to angelic song. The image is offered of heaven being characterized by the song of angels, notably framed as a choir of angels (William Saunders, Choirs of Angels, Catholic Exchange, 18 September 2014). According to Catholic tradition and theological commentary, nine choirs of angels are distinguished as being associated with the different orders of angels (Angels, Archangels, Principalities, Powers, Virtues, Dominions, Thrones, Cherubim, Seraphim).

With respect to the demonic, also of relevance is the extensive condemnation from a religious perspective of music and song deemed problematic, as with earlier Christian concerns with diabolus in musica (The Christian's Guide to Detecting Demonic Music, The Landover Baptist Church Forum, 2012; David J. Stewart, How Can Music Be Demonically Inspired? Jesus-is-Savior; Donald Phau, The Satanic Roots of Rock, Dial-the-Truth Ministries).

Of relevance here is the nature of the insight into hyperreality enabled by music and song. Clearly the cognitive appeal of a "ninth choir" merits careful consideration, beyond the use of a linking octahedron in the mapping explored above. Extensive consideration has been variously given to the enneagram as a mapping of a 9-fold typology, including that relating to angels by Peter Collins (Angels on a Pinhead, 1994), and that of Anthony Blake (The Intelligent Enneagram, 1996). In a separate exercise, a visual rendering is provided of an enneagram embedded in 3D within an icosahedron in terms of circles (Correspondences between Traditional Constellations and Pattern Languages: requisite simplexity for sustainable comprehension of complexity, 2014). How such circles might be cognitively related to the sense of "choir", and musical cycles, is a matter for further consideration.

Sonification of the problematique (the "Demonique") and the resolutique (the "Angelique")? Whilst religions have for centuries made use of music to enable insight and discourse with regard to the heavenly, however explained, relatively little use has been made of music and song to convey insight with regard to remedial action in response to the problematique. It might have been expected that the resolutique would have evoked as much insight into the role of song as has been evident with respect to theological framing of angelic choirs. Although the question could be asked to what degree that framing inspired musical insight rather than verbal commentary and imagery.

Problems do indeed evoke songs, as do calls to action. This medium is however strangely dissociated from the modalities of international institutions in practice -- despite appropriation of Beethoven's Ode to Joy as the Anthem of Europe. However there is little evidence of music or song at international summits where they might have been used to articulate the concerns and opportunities in a complementary mode -- one potentially more widely comprehensible and communicable. This argument has been developed separately (A Singable Earth Charter, EU Constitution or Global Ethic? 2006). The latter notes as an the 12 songs of The Globalization Saga: Balance or Destruction, 2004, as a CD accompaniment to a book by Franz Josef Radermacher (of the FAW - Institute for Applied Knowledge Processing, in association with the Global Marshall Plan Initiative).

Given the importance attached to Beethoven's musical insight, the argument above invites attention to his capacity to generate the 32 Variations on an Original Theme in C minor (1806) following the earlier initiative of Johann Sebastian Bach (30 Goldberg Variations, 1741). These were a notably feature of the study on self-reflexivity by Douglas Hofstadter (Gödel, Escher, Bach: an eternal golden braid, 1979). As a pattern language, how is any "angelic" pattern of 30 to be distinguished from one of 32? From what is the coherence of the pattern composed and produced by composer Benjamin Mapochi (La Folia 72 Variations)?

The point may be made otherwise by reference to the increasingly recognized role of sonification as a means of enabling meaningful comprehension of patterns of a high degree of complexity, as promoted by the International Community for Auditory Display. Given the capacity of the ear to detect such patterns, it is regrettable that the focus on angelic choirs has been limited to overly simplistic hierarchical configurations, at a time when engagement with higher orders of complexity is seemingly required. It might even be argued that music perceived as demonic is currently valued because it reflects such complexity to some degree.

In a global civilization in which many constituencies favour expression with one voice -- their own, however -- the question of how many "voices" merit expression in a complex society suggests the possibility of articulating the challenge in musical terms. If the 9 angelic choirs represent distinct "voices", what insights are associated with their respective singing -- perhaps as archetypes implied by the Specialized Agencies of the United Nations? The argument with respect to the number of voices and their relationship in multi-part singing is explored separately (Enabling a 12-fold Pattern of Systemic Dialogue for Governance, 2011). Of further interest is understanding the conditions and modalities enabling improvisation (Multivocal Poetic Discourse Emphasizing Improvisation: clarification of possibilities for the future, 2012).

Just as experiments are undertaken with sets of musical variations, so could they be undertaken with classic patterns, as separately explored (Musical Articulation of Pattern of Tao Te Ching Insights: experimental sonification based on magic square organization, 2016).

Reversal of music: Known as reverse tape effects, as might be expected, experiments have been made with reversing the direction in which music is played, including the controversial use of so-called backmasking, as popularised by the Beatles. More recently this has become possible with the aid of reverse music players, notably as apps for smartphones. Given the argument above for reading boustrophedonically, the possibility has inspired experimental music (Boustrophedon by Evan Parker - Transatlantic Art Ensemble), with a website offering music under that name (Boustrophedon), and another offering classical music played in reverse (Reverse Music).

Classic composers, including Mozart, Bach and Haydn created pieces in which time, pitch and/or melody was reversed at some point -- a technique known as crab canon -- leading to studies on retrograde performance (Yingshou Xing, et al, Mozart, Mozart Rhythm and Retrograde Mozart Effects: evidences from behaviours and neurobiology bases, Scientific Reports 6, 2016). As noted by the Riemenschneider Bach Institute, in three canons, including the Crucigeros, Bach employs an especially rigorous procedure in which intervals not only reverse direction but also strictly maintain quality as well as number. The follower voice of these "mirror" canons may be discerned, quite literally, in the reflected images of their leaders (The Crown of Thorns, 1997). Given the significance of the sets of variations, it is unclear what musical significance their reversal would have -- bringing the total to 60 or 64, in the light of the above argument.

Despite the experiments, the issue is only partially addressed by the response of Bill Morrissey as one of many relevant comments on the question (Why does music sound bad when played backwards?, Music Practice and Theory Stack Exchange, January 2012 ):

"Why does music sound "negative" when played backwards. This allows a much simpler explanation being "because it is "negative" The same as a ligt image reversed is a "negative" image. Looking "Bad" "Strange" and "Demonic". Nothing is created in its negative state. Music is creation. It is created. When played or listened to backwards what we are experiencing is negativity which is degenerative,we are hearing creation decay or more descriptively speakin death. We as living creatures seek life and accept life. Death, decay, pestilence, disease, all negativity is rejected and knoen to be "bad" "Strange" and even "demonic". It sounds that way because it is.

In a period in which there is concern with cultivation of a culture of fear, the use of music to that end merits exploration, as in the compilation by Neil Lerner (Music in the Horror Film: listening to fear, 2009). The Oxford Companion to Consciousness (2009) makes the point:

The musical function of an event also depends on its rhythmical position. Modifying the temporal structure of a melody suffices to alter the musical function of its notes, and this results in a radically new percept. For example, the pitches B C D E F G imply a major key, with the B as a unstabile leading note and the C as the most referential tonic. Played in reverse order, G F E D C B, these pitches imply the key of B major, with the C being perceived as an unstable tone anchored in the referential tonic B. (p. 459)

Complementary insights are offered in the detailed analysis by Stephen McAdams (Psychological Constraints on Form-bearing Dimensions in Music, Contemporary Music Review, 1989). An online experiment has been offered by Daniel Ross (Can you tell what pieces of classical music we've reversed? Classic fm, 18 January 2016).

Given the role of Beethoven's 9th Symphony -- an "Ode to Joy" -- as the anthem of Europe, it could be fruitfully asked what might be learned collectively by playing it in reverse, especially in a time of crisis of evident disconnect of governance from experiential reality. Perhaps as a "Requiem to Joy " or a "Lament for Misery"? The experiment could be relevant to the use of many national anthems in times of crisis -- as with the significance of flying a flag upside down. A version reversed in this way is now readily accessible on YouTube. (where other classical pieces played in this way may also be heard).

Despite the continuing controversy about such reversal, might critical appreciation of the alternative perspective then suggest means whereby Europe could come fruitfully to terms with its own demons? As previously suggested by Jacques Attali, social organization is significantly determined by that of classical music (Noise: The Political Economy of Music, 1977). A form of "pattern exhaustion" may becoming evident (William Caplin, Classical Form: a theory of formal functions for the instrumental music of Haydn, Mozart, and Beethoven, 1998). Is it possible that there are strategic dependencies which merit being called into question -- as explored in the parallels between music and quantum theory (Scott Shoger, Quantum Theory and Classical Music, Nuovo, 23 April 2014)?

Possible learning is indicated by Kate Bevan-Baker (Critical Backwards-Listening, Ampersand, 15 November 2015) with respect to performance of a folk song, The Blacksmith:

Though I have only selected one verse above, listening to it in its entirety provided me with a new view of the song, and I will never perform it the same way again now that I have heard and seen what it sounds and looks like in reverse... In a recording, the lyrics are totally morphed and cannot be made out in reverse. It gives the piece a completely different character when the lyrics cannot be heard or understood. Instead of telling a story through words, the piece becomes purely a musical statement on a different level.

Amongst other sound samples, that document provides access to an unusual piece by Mozart that can be played in either direction:

There have been pieces written that sound the exact same played in reverse -- musical palindromes. Mozart even composed a duet that can be played with one player reading the music right-side-up, and another reading from upside-down.

Reverse speech and doublespeak? If the current inadequacy of political and other authorities is in some way due to having their thinking "back-to-front". this may be strangely related to what has long been labelled doublespeak (Enabling Suffering through Doublespeak and Doublethink, 2013; Occupying the Moral and Ethical High Ground through Doublespeak, 2013). Intriguing in that regard is the research on reverse speech inspired by controversial backmasking in music and song. This research, itself controversial, was originated by David John Oates (About Reverse Speech, ReverseSpeech; Beyond Backward Masking, 2013). The criticism of that research and the response are themselves interesting (John Shirley, Reverse Speech, Sceptical Inquirer; Matjaz Kranjec, Response To The Skeptical Believer Article About Reverse Speech, Reverse Speech).

Musical tradition: Of great potential value to any clarification of the distinctive voices are the studies by the musicologist Ernest G. McClain (The Myth of Invariance: the origin of the Gods, Mathematics and Music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978). These were initially inspired by work on the tonal "languages" of the Rg Veda by Antonio de Nicolas, as reviewed by the latter (Meditations through the Rg Veda: a retrospective, Philosophy East and West, 1999), and discussed separately (Tonal patterns of Rg Veda poetry, 1984).

McClain's discussion of the "tone number field" and the "tone mandala" is an inspiration to further research. He notably discusses how a tonal system is engendered by prime numbers 2, 3, 5, and 7, especially in their relation to polyhedral patterns framed by 2p3q5r < 60. He frames the tone number field and mandala by what he terms the Holy Mountain, presented in various charts (Meditations through the Quran: tonal images in an oral culture, 1981). Central to this are the five tones C, G, D, A, E and their smallest integer form as being the only invariances in the mandala (1976, p. 75-76):

ratios 72 81 96 108 128 144
rising D E G A C D
falling D C A G E D

Elsewhere he notes, with respect to a more complex version of that table, that:

To define that same scale as string-length ratios on the monochord requires in terms of smallest integers a set of Platonic "friends" in the 72 : 144 octave. Thus New Jerusalem's "wall" of 144 cubits is musically inseparable from the Rig Veda's opposite limit of 30; anyone concerned with numerical sequences which increase and decrease symmetrically had to know both the basic Hindu set and its "Christian" transformation (1976, p. 109)

He notes that 72 begins this symmetric pentatonic scale form, and 10 72 = 720 defines the "Just" limits within which it arises. These digits play a prominent role in Hindu, Hebrew, Christian, and Chinese arithmology. With respect to the tonal form of the Holy Mountain, he later draws attention to the manner in which 2x36 = 72 elements are paired around its centre, and hence to its importance (1981, p. 143).

McClain comments further that 72 figures more frequently in Jewish legends than in the Bible (1981, pp. 143-144):

The implication of "translation" merits consideration in terms of "transformation" -- as in the role of "voices" with distinctive functions in a musical work. As with the work of McClain, such arguments are given further significance by the recent discovery of Jay Kennedy (Plato's Forms, Pythagorean Mathematics, and Stichometry, Apeiron: a journal for ancient philosophy and science, 2010) of the use by Plato of a regular pattern of symbols to give his books a musical structure (Manchester historian deciphers hidden 'Plato Code', BBC News, 29 June 2010; Julian Baggini, Plato's Stave: academic cracks philosopher's musical code, The Guardian, 29 June 2010).

Geometry of musical chords: A distinct, but implicitly related, approach to music has been elaborated by Dmitri Tymoczko as follows:

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses. (The Geometry of Musical Chords. Science, 313, 2006):

In terms of any engagement with hyperreality, the application to music theory associates orbifolds with hyperbolic space, as indicated in Wikipedia.

The approach was later developed in book form (A Geometry of Music: harmony and counterpoint in the extended common practice, 2011), reviewed by Steve Olson under the title A Grand Unified Theory of Music (Princeton Alumni Weekly, 9 February 2011), making the point that chords have not only sound but shape. Tymoczko has subsequently developed his work in terms of the hypercube (The Generalized Tonnetz, Journal of Music Theory 56, 2012, 1)


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