Challenges to Comprehension Implied by the Logo
of Laetus in Praesens
Laetus in Praesens Alternative view of segmented documents via Kairos

8 June 2020 | Draft

Global Insight from Crown Chakra Dynamics in 3D?

Strategic viability through interrelating 1,000 perspectives in virtual reality

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Introduction
Simpler crown chakra models
Design options for a crown chakra with more complex dynamics
Framing crown chakra dynamics in relation to symmetrical polyhedra
Rendering crown chakra dynamics through interlocking tori
Perspectives, projections and learning processes in 3D modelling
Cognitive embodiment in the crown chakra?
From helicopter to "psychopter": the role of anti-torque?
Crown chakra understood as an axial turbofan -- an "attention breathing" jet engine?
Balancing duality: objectivity vs subjectivity? 
Complementarity of pattern languages -- an all-encompassing meta-pattern?
Dependence of viable global governance on pattern management?
Meta-pattern, Meta-poesis and Music
References

Introduction

The argument for imaginative exploration in 3D of the so-called "1,000-petalled lotus" -- the Crown Chakra or Sahasrara -- was developed in an earlier exercise (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? Experimental representation of crown chakra in virtual reality, 2020). Various models were presented as animations in 3D, illustrated by video and accessible interactively according to the X3D protocol.

Subsequent experiment has resulted in the direct presentation of a series of such 3D models in web pages, as indicated separately (Eliciting Insight from Mandala-style Logos in 3D: interactive engagement with mandalas and yantras in virtual reality, 2020). Distinct from the earlier approach, the models were directly interactive rather than requiring plugins or separate operations.

The focus there on logos was inspired by the tendency of many organizations, and notably international institutions, to frame their identity through symbolic, mandala-like, imagery in 2D. This is consistent with traditional use of heraldic imagery -- notably emblazoned on banners through which followers were led into battle. Many make use of a centro-symmetric image, most obviously a representation of the globe (World Guide to Logotypes, Emblems and Trademarks of International Organizations, 1997). Despite their static nature, enabling them to be reproduced in printed form, this is readily held to be indicative by inference of a sense of strategic coherence and integration. The most controversial example is obviously the swastika.

The purpose here is to note further possibilities in 3D, building on the models previously presented -- and offering interactive access to them over the web. The various design metaphors explored here frame the question as to how global integration might be comprehended in dynamic terms in virtual reality -- especially given other possibilities that such models might suggest.

Understood in terms of pattern language, and depicted as composed of 20 rings of 50 "petals", the "1,000-petalled lotus" offers a challenge. It is potentially indicative of the most complex pattern whose coherence is susceptible to comprehension and communication. The question is whether it can be represented in ways which reconcile these constraints, thereby facilitating the cognitive embodiment with which that pattern has been traditionally associated.

As a comprehensible pattern, the "1,000-petalled lotus" could be contrasted with that celebrated in the mathematics of group theory as the so-called "monster group". The group is regarded by many mathematicians as a mysteriously beautiful object -- intriguingly framed by the monstrous moonshine conjecture.

As seemingly beyond normal human comprehension, it might then be asked how that "mathematical object" relates to the significance otherwise attributed to the "1,000-petalled lotus" (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008). However comprehensible, is the "1,000-petalled lotus" to be understood as another form of "monstrous moonshine"? Are both suggestive of the illusory nature of global governance as optimistically imagined?

With respect to both such complex patterns, Is the challenge one of recognizing the complementarity of pattern languages, namely some kind of meta-pattern -- even one of which the self-reflexive aesthetics of meta-poesis is indicative? Is it indeed a meta-pattern which defines the vast generalization that it is patterns which connect, as argued by Gregory Bateson (Mind and Nature: a necessary unity, 1979)? Noteworthy, from that perspective he then warned that: Break the pattern which connects the items of learning and you necessarily destroy all quality.

Simpler crown chakra models

The following models are reproduced from the earlier exercise in order to frame the subsequent argument. They can now be viewed interactively in an X3DOM framework on web pages, as indicated (Anestis Koutsoudis, Bring Cultural Heritage 3D Content on the Web Using the X3DOM Framework, Connecting ARchaeology and ARchitecture in Europeana / Athena Research and Innovation Centre).

NB: The animations visible on this page, especially the more complex, are typically visually inadequate (jerky, etc) and therefore purely indicative. Those embedded with X3DOM on the linked pages are of much higher visual quality (use the links below marked "interactive web"). However these are subject to other constraints in comparison with the original X3D models to which links are provided (for those who have access to software to render them).

Crown chakra with rotation of 20 rings of 50 petals ("1,000-petalled lotus")
(fastest rotation in the smallest rings at the centre)
Counter-rotation Wireframe counter-rotation Rotation into plane
Counter rotation of 20 rings of 50 heart-patterns Horizontal counter-rotation of 20 rings of 50 hearts moving independently Vertical rotation of total set of 20 rings of 50 hearts
interactive web -- mp4 -- x3d interactive web -- mp4 -- x3d interactive web -- mp4 --  x3d

These imply particulat importance to the central rings, potentially emphasized by relative rate of rotation, Rather than being presented as an array of of rings, another design option explored is "stacking" the rings, thereby emphasizing an additional hierarchical understanding. This is potentially reminiscent of the symbolism of traditional temple designs.

Experimental "temple" stacking of 20 rotating rings crown chakra each of 50 heart-patterns
(side view of animations pesented above; (fastest rotation in the smallest rings at the centre)
Counter-rotation Wireframe rendering Rotating into plane
Crown chakra as stack of 20 rotating rings

Crown chakra as stack of 20 rotating rings

Rotation of stack of 20 crown chakra into the plane
interactive web -- mp4 -- x3d interactive web -- x3d interactive web -- mp4 -- x3d

Design options for a crown chakra with more complex dynamics

Consideration can be given to configuring the rings with the intrioduction of more complex dynamics. The point to be emphasized is that a degree of coherence and comprehensibility is retained, despite the effective increase in complexity. In a sense the coherence is "transferred" from a relatively static configuration in which subtler comprehension is implied (inferred or intuited) to one in which this is held by the dynamics. Comprehensible patterns are recognizable in the dynamics rather than in the relatively static configuration.

NB: The coherence and aesthetics of the following models are far more readily appreciated through access to the interactive displays (marked "web" in each case, meaning embedded within the X3DOM protocol). As displayed here, and via video, the representations suffer from numerous constraints, especially when slower or faster movement is preferred. As a consequence of technical constraints regarding web security, a few of the web displays may not work at this time.

Crown chakra with rotation of 20 rings of 50 petals ("1,000-petalled lotus")
(fastest rotation in the smallest rings at the centre)
Ring stack inversion "Gyroscopic" movement Movement of 20 rings framing a torus
Inversion of stack of 20 crownm chakra rings in 3D Representation of crown chakra with 20 rings rotating on various axes Movement of 20 crown chakra rings rings framing a torus
interactive web -- mp4 -- x3d interactive web -- mp4 -- x3d interactive web -- mp4 -- x3d

Of particular interest, and especially appropriate given their multi-petalled form, are the animations produced by John Edmark, now widely available on the web (Golden Angle; Blooms: Phi-Based Strobe Animated Sculptures). Their remarkable evolving dynamics result from combining images of static sculptures rotated precisely 137.5 degrees successively.

Framing crown chakra dynamics in relation to symmetrical polyhedra

Chiliagon and chiliahedron? Reference here to the "1,000-petalled lotus" from an Eastern tradition invites dismissal of this argument as ridiculous from a Western perspective. It is therefore very curious to note the extent to which a range of Western philosophers have referred to the little-known chiliagon or 1000-gon. This is a polygon with 1,000 sides. Such philosophers refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Because 1,000 = 23 × 53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon is not a "constructible polygon".  Geometrically it can be constructed as a truncated 500-gon, or a twice-truncated 250-gon, or a thrice-truncated 125-gon.

In its extensive discussion of the chiliagon, Wikipedia notes:

René Descartes uses the chiliagon as an example in his Sixth Meditation to demonstrate the difference between pure intellection and imagination. He says that, when one thinks of a chiliagon, he "does not imagine the thousand sides or see them as if they were present" before him – as he does when one imagines a triangle, for example. The imagination constructs a "confused representation," which is no different from that which it constructs of a myriagon (a polygon with ten thousand sides). However, he does clearly understand what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. 

The entry notes reference to the chiliagon by other philosophers, including  Immanuel Kant,  David Hume,  Gottfried Leibniz,  John Locke, Henri Poincaré -- thereby inspiring 20th century philosophers to make such reference. By contrast there is little reference to the "chiliahedron", namely a polyhedron bounded by 1,000 plane surfaces -- posing even greater problems of construction, although there is mathematical speculation in that regard, notably with respect to the subdivision of the surface of a sphere. There is also the possibility that some such form might be constructed in four dimensions (or more).

With respect to the "myriagon" of 10,000 sides, it is appropriate to recall the "ten thousand things" of objective reality featured in the much-cited first stanza of the Tao Te Ching of Chinese Taoist philosophy. Surprisingly, reference to "myriahedron" is now the focus of a new class of mapping methods for the globe (Jarke J. van Wijk, Unfolding the Earth: Myriahedral Projections, The Cartographic Journal, 45, 2008, 1). The methods described suggest other approaches to configuring a chiliahedron (James Lane Conkling, Tools to generate and project a myriahedral grid onto the globe, GitHub, 2020).

An approximation to a 1,000-sided polyhedron is possible by generating a geodesic sphere, as shown below. Another approach is to consider the 20-rings of the "1,000-petalled lotus" to be great circles of a polyhedron. Ironically "20 great circles" features in a patent for the design of the common golf ball (Process for designing dimple pattern of golf ball, US patent 20130102417A1, 24 October 2012).

Great circles are of particular relevance in that they have been fundamental to the development of global navigation, as discussed separately with respect to the spherical geometry of the so-called Pentagramma Myrificum (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

Platonic polyhedra? Exploration of the manner of configuring such a set of great circles features in the argument of Gerald de Jong (The Geometry of the Time Star). The latter consists of five regular tetrahedra, configured so that their vertexes are located at the centers of the twenty triangular faces of the icosahedron -- in either a left-handed or right-handed configuration. As described, this gives rise to 20 great circles -- which the author notes as reducing to 10 because of pairing.

In quest of a pattern of 20 great circles and a 1,000-sided chiliahedron
Time Star configurations From Icosahedron to 980-faced Geodesic sphere (7 freq.)
Time Star (Gerald de Jong) Time Star (Gerald de Jong) Icosahedron rotating with great circles 980-faced Geodesic sphere
From Gerald de Jong (The Geometry of the Time Star) Made with Stella Polyhedron Navigator

Particular design advantage can be taken of the 20-ring configuration of the "1,000-petalled lotus" in the light of the 20-fold ordering of two of the most fundamental symmetrical polyhedra, namely the dodecahedron (of 20 vertexes) and its dual the icosahedron (of 20 faces).

Treating the 20 rings of the crown chakra as great circles, they could possibly be given a dynamic independent of either of those polyhedra. This design metaphor, and the dynamic, is of relevance given the current launching of a configuration of thousands of satellites as previously described (Satellite Constellation and Crown Chakra as Complementary Global Metaphors?, 2020).

Another approach is to give the rings cn a dynamic from the centre of the polyhedron to each vertex or to each face -- possibly reversed back through the centre to the opposing vertex or face. The rings then emerge from the centre or withdraw into it -- suggestive of other cognitive implications. There is the additional possibility that their diameter might be reduced to zero as they reach the centre.

Dynamics of 20 rings of crown chakra configured in relation to dodecahedron and icosahedron
("1,000-petalled lotus")
Ring movement in relation to 20 vertex axes of dodecahedron Rotation of dodecahedral array with ring movement in relation to vertex axes Ring movement in relation to 20 faces of icosahedron
Movement of 20 crown chakra rings in relation to 20 vertex axes of dodecahedron Rotation of movement iof 20 crown chakra rings in relation to 20 vertex axes of dodecahedron Movement of 20 crown chakra rings in relation to 20 faces of  icosahedron
interactive web -- x3d interactive web -- x3d interactive web -- x3d

An earlier argument treated the 10 dodecahedral axes through the vertexes as the basis for an emphasis on the emergence and withdrawal of "spikes", as these might be relevant to an understanding psychosocial processes (Spike-endowed Global Civilization as COVID-19: humanity "bristles" as the world "burns", 2020). The two visual metaphors might then be combined as indicated below -- with the possibility of adjusting those dynamics to either coordinate their movement, to render them contrapuntal, or random.

Dynamics of 20 rings of crown chakra configured in relation to dodecahedron and icosahedron
-- with spike dynamics indicated on left
Dodecahedron alone
(20 spikes through 20 vertexes)
Dodecahedron with spikes
(20 spikes through 20 vertexes)
Icosahedron with spikes
(20 spikes through 20 faces)
Dynamics of 20 spikes in relation to dodecahedron vertexes Dynamics of 20 crown chakra rings (with spikes) in relation to dodecahedron vertexes Dynamics of 20 crown chakra rings (with spikes) in relation to icosahedron faces
interactive web -- x3d interactive web -- x3d interactive web -- x3d

The "spike-based" argument for such a 20-fold configuration derived in part from the as yet unexplained (unconscious?) enthusiasm for that pattern in the articulation of strategic practice (Requisite 20-fold Articulation of Operative Insights? Checklist of web resources on 20 strategies, rules, methods and insights, 2018).

The fundamental nature of that pattern is evident in that of the 20 protein amino acids vital to human life, suggesting a speculative comparison in memetic terms (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). The basis for that pattern merits particular attention (A. L. Weber and S. L. Miller, Reasons for the occurrence of the twenty coded protein amino acids, Journal of Molecular Evolution, 17, 1981, 5, pp. 273-284).

Rendering crown chakra dynamics through interlocking tori

A design option presented above uses the movement of the 20 rings, with an associated change of diameter, in order to frame a torus. The rings pass successively through the central hole of the torus then around the torus -- with rings and torus all sharing a single common axis.

Interlocking: A case made separately argued for the significance of coupling two tori as indicated below. It is their orthogonal relationship which is then particularly suggestive as a means of holding fundamental duality, notably that of objective and subjective modalities -- each frquently deemed meaningless to the other.

Torus coupling or interlocking with distinctive movements
Dynamic virtual reality model of intertwined tori Dynamics of interlocking tori

Discussed with respect to the Fearful attraction of a hole (In: Visualization in 3D of Dynamics of Toroidal Helical Coils: in quest of optimum designs for a Concordian Mandala, 2016)

Red torus has a vortex (smoke ring) dynamic in the model; Blue torus has a wheel-like dynamic in the model

VRML animation produced by Bob BurkhardtX3D and VRML variants produced by Sergey Bederov of Cortona3D

Ring containment: Each torus could then be used as a container for 10 rings, as previously illustrated by use of a torus as a container for the classic Zen images of 10 Bulls (Zen of Facticity: Bull, Ox or Otherwise? Herding facts and their alternatives in a post-truth-era, 2017). The image on the right is an experimental combination of the animation of a schematic Ouroboros with the 10 Zen ox-herding images. The cognitive implication then derives from how the coherence for the complex interlocking of two tori might be comprehended, as discussed separately (Cognitive Osmosis in a Knowledge-based Civilization: interface challenge of inside-outside, insight-outsight, information-outformation, 2017).

Animation of simplest torus as container for dynamic cognitive relations
10-fold pattern of Zen bulls
8-fold BaGua pattern of trigrams
Ourobouros through images
Zen ox-herding images in torus animation Bagua trigrams in torus animation Ouroboros and  Zen ox-herding
mp4  -- wrl -- x3d  mp4 -- wrl -- x3d x3d

Examples of models using two tori: The significance of such interlocking was the focus of an earlier discussion (Enabling Wisdom Dynamically within Intertwined Tori: tequisite resonance in global knowledge architecture, 2012).

Indicative interlocking torus dynamics with 20 crown chakra rings -- 10 rings per torus
10 rotating rings "flat" in each torus 10 rings turning in each torus 10 rings circulating thru each torus
Horizontal rotation of 20 crown chakra rings within interlocking tori Overlapping movement of 20 crown chakra rings within interlocking tori Circulation of 20 crown chakra rings within interlocking tori
interactive web -- x3d interactive web -- x3d interactive web -- x3d

The animation on the right above is potentially more interesting when the rate of movement of the circles through one torus is much greater than that through another -- best seen through another interactive variant. (due to the technical challenges of representation). This animation is especially relevant to the subsequent argument.

Range of design options with two tori:

Torus movement options
"Horizontal" torus static moving (faster/slower) static moving (faster/slower)
"Vertical" torus static static moving (faster/slower) moving (faster/slower)
Torus visibility options
"Horizontal" torus visible invisible invisible invisible
"Vertical" torus visible visible visible invisible
Ring colour options
"Horizontal" torus

same colour
(for both tori)

distinctive single colour same 10 colours
(for both tori)
20 distinctive
colours
"Vertical" torus distinctive single colour
Ring dynamics options
Ring orientation Ring dynamic Ring movement within torus
Horizontal
(plane of torus)
static turning on
ring axis
rotating
parallel
to torus
axis
stationary moving along torus in sync moving along torus different rates (passing through each other) some moving in reverse direction (colliding)
"Vertical"
(to plane of torus)
static turning on
ring axis
rotating
parallel
to torus
axis
stationary moving along torus in sync moving along torus different rates (passing through each other) some moving in reverse direction (colliding)

Perspectives, projections and learning processes in 3D modelling

Learning processes: As in any creative process, the result viewed, imagined as an objective, distracts from the process by which it was achieved. In the case of 3D models, it is the result which is viewed in a few seconds and found to be interesting or otherwise. The process of achieving that result may take many tedious hours and can be considered to be of little interest. There is little trace of that process, except in the learning involved.

In the case of the models presented here, it might be assumed that they derive from a degree of expertise in 3D geometry and the relevant software applications. Typically this may be far less than is assumed. The creativity may well lie largely in navigating a variety of forms of ignorance. These may relate to use of the application (in this case X3D-Edit, or X3DOM) for which many levels of expertise may exist elsewhere, and relatively inaccessibly. The applications may not function as assumed in engaging with them. Their correct use, and their constraints, have to be discovered progressively. As noted, the applications may be in process of development. Guidance as to their use may be inadequate or out of date.

With respect to the geometry, many of the issues may well be trivial from a mathematical perspective -- if one had access to that and could comprehend it. In thse exercises, any such knowledge has been long forgotten by the author and effortd to apply any remembered principles in the light of guidance in that respect may be experienced as frustrating. Under such circumstances there is a need to adopt a crudely experimental approach of lengthy trial and error -- readily understood to be ridiculous from a professional perspective.

Despite such constraints and inadequacies, trial and error can be recognized as eliciting insight which would not necessarily be available otherwise.It can then be usefully asked how the crown chakra implies an accumulation of learning processes. Of particular interest is the manner in which a learning process may give rise to options -- and models -- whose existence had not been envisaged.

Perspectives: It could be argued that building relatively complex 3D models requires consideration of multiple perspectives. This is the case with the "1,000 petalled lotus". It raises the question as to how many perspectives have to be considered. The simpler concentric models, whatever their dynamics, necessarily call upon a more limited range of perspectives.

As the complexity of the models increases, and especially their dynamics, more perspectives have to be recognized -- progressively. Typically these are not adequately understood when embarking on their construction with software tools -- even though the final result can be imagined or intuited to some degree.

It can then be asked how many perspectives might bre usefully embodied in any model of a "1,000-petalled lotus": why 1,000? Arguably such a model strikes a creative balance between the increasing number of perspectives (with the complexity implied) and the comprehensibility of the model through a sense of its coherence. Clearly this has much to do with symmetry, but not to the point of oversimplification. What forms of complexity are held coherently by speculations of physics regarding super-symmetry? To what degree are they readily comprehensible?

Simplification is a weakness inherent in singular conceptual frameworks -- and any sense of "singularity". References to "love", "unity", "consensus", "wisdom", "insight" or "inspiration" might be considered examples. Given the limited ability (or desire) to articulate these comprehensibly and credibly in practice. as a consequence they tend unfortunately to take the form of slogans in which people are enjoined to believe -- irrespective of any evident constraints on implementation.

The challenge could then be understood as building multiple perspectives -- diversity -- into coherence. A particular opportunity is offered by dynamics rather than relying on static forms of structural symmetry. The question is then what do models of this kind imply for an appropriate balance in comprehensiuon and social organization -- with "organization" then potentially to be understood in its form as a verb, as argued separately (Engaging Playfully with Coronavirus through "Organizing" Global Governance? 2020; Envisaging a Comprehensible Global Brain -- as a Playful Organ, 2019)

Projections: It is intriguing to note the long-standing struggle to represent the planetary globe -- as a form in 3D -- on maps. The latter are characteristically 2D in nature, with a third dimension creatively implied by various means through "projections" -- of which there are many (List of map projections). There are echoes of this challenge in conceptualizing the "shape of the universe" and depicting it on maps of the cosmos -- a matter of concern to astronomers and cosmologists.

Missing is any analogous reflection on the "shape of society" or the "shape of knowledge space". In the case of the former, the focus is on references to "global society", with the implkications that it is somehow speherical, but with little consideration of what this might then imply -- as suggested by the contrasting perceptions of night and day, especially as a consequence of annual variations. In the case of knowledge space, its organization typically takes the form of a nested hierarchy of topics with little effort to consider more complex patterns of order. This is ironically true of the Mathematics Subject Classification whose topics imply a wide variety of subtle approaches to order.

Witbh respect to the structure of knowledge space, for example, a proposal by Thomas E. Portegys (Morphognosis: the shape of knowledge in space and time, 2017) presented to the 28th Modern Artificial Intelligence and Cognitive Science Conference (MAICS 2017) argues that:

Its basic structure is a pyramid of event recordings called a morphognostic. At the apex of the pyramid are the most recent and nearby events. Receding from the apex are less recent and possibly more distant events. A morphognostic can thus be viewed as a structure of progressively larger chunks of space-time knowledge. 

As a step beyond any nested hierarchy, consideration can be given to the form of a periodic table, as argued separately (Towards a Periodic Table of Ways of Knowing -- in the light of metaphors of mathematics, 2009). Curiously, in the case of the archetypal Periodic Table of Chemical Elements, this too has given rise to an extensive database of alternative modes of organization (The INTERNET Database of Periodic Tables). The relevant mathematics is also a matter of debate (D. H. Rouvray, et al, The Mathematics of the Periodic Table, 2006).

Such considerations can therefore be seen as informing any approach to the organization of the "1,000 petalled lotus" in the light of its implications for higher orders of coherence and comprehensibility -- potentially its requisite variety in cybernetic terms.

One approach explored here is a toroidal form -- which happens to be one of the candidate forms for the "shape of the universe". As a step beyond the "global" form, its possible relevance has been argued separately (Imagining Toroidal Life as a Sustainable Alternative: from Globalization to Toroidization or back to Flatland? 2019). It is somewhat obvious that people do not inhabit a static globe, since it is subject to daily and annual movements -- if nothing else. Less obvious is that the latter traces out a torus -- if not a helix -- with which people have a degree of familiarity, in the absence of much reference to how people live on it.

The particular relevance of the torus is seen as a central to the current design and construction of a nuclear fusion reactor -- promoted as the hope for sustainable energy in the future. An analogue can be speculatively discussed (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006). Also of relevance are the designs promoted for outer space habitats and vessels to other planets -- both of which have long featured in science fiction.

The models presented here take the argument further through imagining the interlocking, or coupling, of two toruses. (***).

Cognitive embodiment in the crown chakra?

Cognitive embodiment: The approach taken here is one of a series of modelling experiments. This implies an assumption that the "1,000-petalled lotus" can be usefully recognized as symbolizing an ultimate form of cognitive organization. It derives from an Eastern tradition which has long reflected on the subtleties of some such implication.

Obviously missing from any approach to "modelling" is the process of cognitive implication. This is a matter of contrioversial debate in Western traditions and most precisely in physics, especially given its enthusiastic construction of models of reality. Where is the observer in relation to the model that can be "viewed" only too conveniently by ignoring assumptions by which that process can be challenged?

With respect to the argument of the previous section:

The crown chakra is traditionally associated with the many possibilities of "ascent", most obviously as described in mystical terms. Arguably there is a case for complementary insights into whatever may come to be understood as cognitive development. A variety of possibilities can be recognized (Clues to 'Ascent' and 'Escape', 2002). The latter featured in Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement (2002).

Facilitating the process of ascent may also be framed through understandings of "uplift" -- favoured by some Christian ministries as central to biblical narrative. Uplift notably frames explorations by science fiction writers, most explicitly with respect to the Uplift Universe of David Brin with its focus on biological uplift. This is paralleled by the Stellaris grand strategy video game of which one focus is the uplift of "presentients".

Given the importance which continues to be attached to the possibility of a civilizational "Renaissance", consideration necessarily extends to the metaphor of "rebirth" (Varieties of Rebirth: distinguishing ways of being "born again", 2004; Challenges of Renaissance: suggestive pattern of concerns in the light of the birth metaphor, 2003).

From helicopter to "psychopter": the role of anti-torque?

With respect to "ascent", an obvious metaphor has been extensively explored from a philosophical perspective by Arthur Young, following his role as designer of the first Bell Helicopter. He framed this as the possibility of developing a "psychopter" (Geometry of Meaning, 1976; The Bell Notes: A Journey from Physics to Metaphysics. 1979).

The case for exploring any such metaphors follows from that made for biomimetics, as discussed separately (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011).

There is the ironic possibility that the crown chakra could be recognized -- speculatively at least -- as embodying the functions of a "psychopter", as the future might choose to imagine it. Relevant to that argument would be the role of the dynamics of interlocking tori, as suggested above.

As traditionally imagined, the crown chakra would appear to reinforce a form of illusion. This corresponds usefully to the assumption that a helicopter needs only the more powerful operation of a single rotor in order to rise vertically. Given the nature of the focus on "cognitive ascent", could it be said to be ignoring a factor usefully illustrated by the helicopter?

Cognitive anti-torque? Most helicopters are characterized by having a single main rotor. Its operation engenders torque as a consequence of aerodynamic drag. For the helicopter to operate successfully, this  must be countered by an opposing torque -- typically achieved by a tail rotor, pushing or pulling against the tail boom to counter the torque effect. Other designs include the use of counter-rotating ducted fans (dependent on the Coandă effect), or the use of two or more horizontal rotors turning in opposite directions.

One difficulty with respect to potential "cognitive development" in achieving "ascent", as speculatively imagined to be the primary achievement of a "psychopter", is the manner in which the operation of the primary rotor is framed as "positive". Any alternative is then readily framed as "negative". Positive is good -- negative is necessarily bad. Little if any consideration is given to the possibility that "positive" may exert a form of "aerodynamic drag" in psychosocial terms -- engendering a phenomenon that might be recognized as "cognitive torque".

This subtle effect might be recognized in anecdotal references to "too good to be true", in song, and in aspects of the opposition paradoxically engendered by those promoting "positive" agendas (John Maynard Smith, Too good to be true, Nature,  400, 1999, 223; Toni Vogel Carey, The Better-Best Fallacy, Philosophy Now, 2008; John Randolph Price, Nothing Is Too Good to Be True, 2003). Traditions have recognized the need to render creativity slightly imperfect to avoid evoking the "jealousy of the gods" (Mara Michelle Kutter, Emotion in Politics: envy, jealousy, and rulership in Archaic and Classical Greece, University of Michigan, 2018).

As an "unpopular" perspective, this effect is argued otherwise by Barbara Ehrenreich (Bright-sided: How the Relentless Promotion of Positive Thinking Has Undermined America, 2009;  Smile Or Die: How Positive Thinking Fooled America and the World, 2009). To the extent that the Crown Chakra is indicative of an embodiment of "enlightenment" as so widely associated with human perfection, it is appropriate to explore the little mentioned role of "endarkenment" in relation to that process in practice (Enlightening Endarkenment: selected web resources on the challenge to comprehension, 2005).

The manner in which the two tori are dynamically interlocked in the models above suggests an approach to thinking about "cognitive torque" and the manner by which it might be compensated or counter-acted (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006).

Cognitive torque? Whilst "cognitive anti-torque" is particularly elusive, the notion of "cognitive torque" or "psychological torque" has been variously recognized (Cognitive torque and fruitful associations, 2009). The following paragraphs derive from a discussion with regard to Reality distortion, psychosocial torsion and psychological torque (2019).

There is a degree of experience of being "twisted" by circumstances, of "being bent" in a manner which may well be unwelcome -- or of subjecting another to such a force. "Bending the arm" of another is a common phrase with respect to being manipulated -- as with having it "bent". The experience is beyond the focus of the natural sciences and is inadequately described, although it is clearly an effect sought by public relations, propaganda campaigns and brainwashing. As a typically problematic experience it does however offer a means of recognizing the nature and possibility of fruitful toroidal experience.

Of the greatest potential relevance to this argument are the shared associations of "torc" and "torque" -- with the latter best understood in dynamics with respect to a form of twisting. In such terms torque is what causes an object to acquire angular acceleration. As a static object, a torc could be understood as implying some such force -- if only in symbolic terms. The exploration of "toroidal life" is then suggestive of "life with a dynamic twist" -- potentially vital to enabling and sustaining change. A torc can then be considered a traditional reminder of that possibility -- and hence its symbolic importance in some contexts where change is otherwise elusive. As experienced personally, references to "psychological torque" and "psychic torque" are discussed below -- but not in the light of their potential relevance to the twisting experience of structural violence and its variants (cultural violence, emotional violence, spiritual violence).

Reality distortion: Curiously the clearest description of the problematic nature of the phenomenon may be with respect to the so-called "reality distortion field" exerted by charismatic personalities to convince themselves and others to believe almost anything with a mix of charm, charisma, bravado, hyperbole, marketing, appeasement and persistence:

The capacity to engender such a field may now be the essence of leadership -- and the primary requisite for those seeking that role. Comprehending the nature of such reality distortion is necessarily rendered more complex by the phenomenon of fake news (Varieties of Fake News and Misrepresentation: when are deception, pretence and cover-up acceptable? 2019).

Torsion and twistedness: To the extent that such psychosocial experience is described, the language for its description borrows from the dynamics of torque and torsion as well-recognized in mechanical terms. In that context, torsion is the twisting of an object due to an applied torque. The latter, as the moment, moment of force, or "turning effect", is the rotational equivalent of linear force. The power output of an engine is expressed as its torque multiplied by its rotational speed of the axis. In considering the generation of any form of psychosocial power in the implementation of a strategy -- and getting it to "fly" -- there is a case for recognizing the role of torque. Using a typical helicopter as a metaphor, its single main rotor creates torque such that its aerodynamic drag must be countered by an opposing anti-torque rotor -- the smaller rotor in the tail. The question is how such compensation is achieved in psychosocial systems -- and whether many strategies spin uselessly for lack of an "anti-torque" rotor.

The following description of a combination of socio-economic forces as "torsions" by Matthew C. Ally clarifies the matter somewhat, although lacking any reference to corresponding psychosocial forces:

We might call these torsions: economic torsion, social torsion, and ecological torsion respectively, twisting together from the ground up, bending the arm of the status quo. And of course the three heuristic torsions overlap, the economic, the social, the ecological, nudging and tugging at each other in complementary and critical ways through the push and pull of theory and practice... (Ecology and Existence: Bringing Sartre to the Water's Edge, Lexington Books, 2017 p. 485)

Quotations indicative of understandings of "psychological torque", "psychic torque" and "psychological torsion"
(emphasis added)

Psychological torque:

  • Following their lead, one can construct a decision procedure in which one computes a "psychological torque" T (this need not correspond to true physical torque)... (Tony J. Simon and Graeme S. Halford, Developing Cognitive Competence: new approaches to process modeling, Psychology Press, 2015, p. 165)
  • Yet the dynamic psychological torque in this work does not justify assimilating H. D. to the expressive subjectivity we have long associated with lyric poetry. We are not simply in the presence here of a discourse of resplendent or imperiled identity. (Cary Nelson. Repression and Recovery: modern American poetry and the politics of cultural memory, University of Wisconsin Press, 1989, p. 82)
  • As is well known, the cultural travails of middle-class America in the twentieth century produced the separate, sovereign self, the final logical extension of the Protestant Reformation's psychological torque that spun toward more and smaller sects until the irreducible sect of one was eventually reached (Loren Baritz, The Good Life, Knopf Doubleday Publishing, 2013)
  • I want to suggest that the unique psychological "torque" of modern rhetorical power can be explained as a mechanism "funded" by the divided character of modem self-structure. Modern forms of ethos can "divide" us from our habitual values because, as moderns, we are always, in advance, at a deeper level, divided, self-conflicted selves. (Marshall W. Alcorn, Narcissism and the Literary Libido: rhetoric, text, and subjectivity, NYU Press, 1997, pp. 51-52)
  • Ballard has long maintained that... science fiction is the only literature capable of making sense of the moment we live in. It is a moment whose psychological torque is centripetal, not centrifugal – a moment where "social relationships are no longer as important as the individual's relationship with the technological landscape," which is another way of saying that interpersonal psychology has been displaced by a new, cyborgian psychology: the feedback loop between human and machine (Mark Dery, 'Always Crashing in the Same Car': a head-on collision with the technosphere, The Sociological Review, 54, 2006, 1; An Extremely Complicated Phenomenon of a Very Brief Duration Ending in Destruction: the 20th Century as slow-motion car crash, TechnoMorphica, 1997).
  • ... the concept of war becomes an attempt to describe a low-intensity warfare that reconstitutes the most mundane aspects of everyday existence through psychosocial torque and sensory overload. (Steve Goodman, Sonic Warfare: sound, affect, and the ecology of fear. MIT Press, 2012, p. 33).
  • The author succeeded in making his point, a realization of the evil at the center of human experience, without breaking the organic unity of his life like documentary styled narrative and without losing the psychological torque of the tale. (Debraj Moulick, Book Review: Conrad's Heart of Darkness, Intellinotions, January 2014)
  • Torque, hemispheric dominance, and psychosocial adjustment. (Donald Woods, et al, Journal of Abnormal Psychology, 89, 1980, 4).

Psychic torque:

  • So, in pursuit of being lively and clear, the sheer psychic torque of Edwards's emotional range is noteworthy. Hyperbole-like metaphor, a bit of linguistic fantasy in disguise-forces Edwards to an awareness "of the barely thinkable" (Edward Ingebretsen, Maps of Heaven, Maps of Hell: religious terror as memory from the Puritans to Stephen King, Routledge, 2016)
  • The empirical demonstration of Eddie Oshins' work on self-referential motion and the hunt for psychic torque. (Physics of Tao, Eco Echo Invasives Extraction, May 2017)
  • Paul (an engineer by training) and I discuss torque. He explains the mechanics, the opposing forces, the stress factor, the cause and effect … we talk about the psychic torque we are all experiencing … certainly a time shift … without a doubt a culture shift that intensifies the torque … and yet this is what allows you (if you permit it) to become part of the experience … (The Faces of Gujarat: Experiencing India Subliminally, The Philosophical Traveller, May 2009)
  • ... the psychic torque produced by the sheer weight of the emotional pain he confronts through the letters ultimately lays waste to every intellectual defense he can muster to distance himself from that pain. (Michael Anthony Reardon, Becoming Visionary: reading and living in the existential mode, 1994, p. 98)
  • Spillers defines this as 'a locus at which self-interrogation takes place […] Its operations are torque-like to the extent that they throw certainty and dogma […] into doubt.'... All these are markers of what Fanon termed 'existential deviance', which could be related to both as a psychic torque – a forceful twisting (into pre-made definitions of the square hole Real) – that can be very painful (psychologically disturbing) and, yet crucially, an opening towards hybridity. (Howard Slater, Homicidal Melancholics of the World Unite! Mute, 15 August 2017)

Psychological torsion:

  • The psychological torsion that now belongs to this 'activity' captures the precise sense of 'media' (medios, or recursos), sometimes appearing beneath an instrumental determination of aesthetic culture and sometimes as a new form of agency itself. (Gregg Lambert, The Return of the Baroque in Modern Culture, A and C Black, 2004, p. 28)
  • ... she enacts a conflicted embodiment of the monstrosity Eliot repeatedly explores as a renunciation of the human, civilised self. Maggie's taking 'her stand out of herself' not only implies a psychological torsion (twisting towards the eccentric or demented), but also shows her arguably 'most monstrous' (Hollis Berry, Victorian Psychology Monstrous Maidens and George Eliot, Brill, 2019, p. 120)
  • This brings me to my third topic: the torsion exerted on the liberal self by finance. Specifically, I argue that financialization abstracts the formerly immediate and organic relationship between possession and ownership; as the two terms drift apart, the very basis of liberal subjectivity, what C.B. McPherson famously termed possessive individualism, begins to fray. (Matt Kavanagh, Second Nature: American fiction in the Age of Capitalist Realism, 2007)

Psychic torsion:

  • This is one of Wilde's most famous epigrams..., but, in context, it is something more than a subversion of Victorian piety. It captures the peculiar psychic torsion that structures Wildean self-fashioning, within which he recuperates a sense of will, and with it a sense of freedom. (Michael F. Davis, Petra Dierkes-Thrun, Wilde's Other Worlds, Routledge, 2018).

It is then appropriate to ask at this time: Does "global" reality -- and the requirement to believe in it -- involve a very particular and peculiar form of cognitive "torsion" or "twist", in contrast with the daily direct experience of "flat earth" reality?

Crown chakra understood as an axial turbofan -- an "attention breathing" jet engine?

The structure of the crown chakra with 20 rings of 50 "petals" each is reminiscent of the design of an axial turbofan -- as widely used in aircraft propulsion (Turbofan engines, ScienceDirect, 2016). A turbofan can be thought of as a turbojet being used to drive a ducted fan, with both of these contributing to the thrust. Whereas propeller-based engines (like the helicopter) are most efficient for low speeds, turbojet engines for high speeds -- with turbofan engines between the two. Does the crown chakra share principles with gas turbine design?

The question to be explored, suggested by the following illustrations (and their descriptions in Wikipedia), is what principles and constraints are associated with the design and operation of such engines -- with respect to the rotation of the petal-endowed rings of the crown chakra. Further insights with respect to the chakra rings are associated with descriptions of axial fan design. Is the processing of attention to be usefully compared with the operation of an axial compressor (below right), namely a gas compressor that can continuously pressurize gases?

Insights into crown chakra operation from jet engine design?
Turbofan animation
(2-spool, high-bypass turbofan)
Schematic diagram illustrating the operation of a 2-spool, high-bypass turbofan engine
(LP spool in green and HP spool in purple)
Axial compressor
animated simulation
(note alternation of static and rotating blades)
Animation of 2-spool, high-bypass turbofan Schematic of operation of a 2-spool, high-bypass turbofan engine Animated simulation of axial compressor
Zephyris at English Wikipedia / CC BY-SA Rotation of Wikimedia image by
K. Aainsqatsi / CC BY-SA
  NASA - source - public domain

There are many accessible images of turbo fan engines, some taking the form of 3D models in the GrabCAD model liberary. The number of blades may vary as indicated by the images left and right below.

Suggestive correspondence between turbofan and chakra models
Turbofan engine
Aviadvigatel PS-90
 powering the Ilyushin Il-96Tupolev Tu-204Ilyushin Il-76
Crown chakra model Turbofan animation
Turbofan engine Counter rotation of 20 rings of 50 heart-patterns Turbofan animation
Reproduced from Wikipedia (as above) Adapted from GrabCAD model liberary

Framed in this way, are the "petals" of the "1,000-petalled lotus" to be compared with the skillfully curved turbine blades in jet engines? It would indeed appear that the number of blades in each stage may be of the same order as the number of "petals" in a chakra ring (Amin Almasi, Axial compressor considerations: how flows, design and operation differ from other compressor types, Flow Control, 3 October 2016)

It is those blades which are responsible for extracting energy from the high temperature, high pressure gas produced by the combustion process. As extensively discussed by Wikipedia, the turbine blades are often the limiting component of gas turbines. To survive in this difficult environment, the blades often use exotic materials like superalloys and many different methods of cooling that can be categorized as internal and external cooling, and thermal barrier coatings. Blade fatigue is a major source of failure in steam turbines and gas turbines. It is caused by the stress induced by vibration and resonance within the operating range of machinery.

Whereas the central image above has been rotated to suggest the processing of "attention" from understandings of reality of a lower order, it could be reversed to suggest the transfer of energy -- insight -- from understandings of reality of a higher order. The turbojet engine metaphor could be taken further in relation to the six other other chakras, given that these are understood in terms of stages in the processing of energy. Is there a sense in which effective operation of global governance in processing collective attention can be compared to the kundalini process?

Balancing duality: objectivity vs subjectivity? 

As argued above, the challenge of viable helicopter operation depends on a skillful balance between torque and anti-torque through control of the respective rotors.

This frames a way of exploring various dualities, typically experienced as problematic. That between positive and negative offers one example, especially in the light of the extensive insights of cybernetics with regard to the necessity of positive feedback and negative feedback (also termed balancing feedback) required for the sustainable operation of any system.

Potentially of greater relevance is the duality of objectivity and subjectivity, celebrated in the cultural challenge articulated by C. P. Snow (The Two Cultures, 1959). A commentary on this duality  explored it in dynamic terms, rather than as a matter of static perspective (A Subjective Objection: Objecting to Subjection: interplay of questions enabling transcendence of fundamental dilemmas? 2016; Controversy regarding 'subjectivity' vs. 'objectivity' regarding 'beauty' and 'life', 2010). This included their comprehension from a cybernetic perspective -- Interrelating subject and object in terms of knowledge cybernetics -- and reference to the study by Max Deutscher (Subjecting and Objecting: an essay in objectivity, 1983).

The constraints of the operation of a widely familiar helicopter offer the delightful possibility that:

A similar argument might be made for the "positive" as the primary rotor, and its needs for corrective "negative" feedback. The visual image of the distance between the vertically-oriented primary rotor and the horizontally-oriented tail rotor (at the end of the helicopter tail boom) is metaphorically suggestive in its own right.

The animation presented above of two coupled tori would however be indicative of a cognitive refinement in which the two "cognitive rotors" were interlocked in a complex manner recalling the designs for rotorcraft in which the counter-acting rotor is not physically distant from the primary rotor but is integrated with it. The humanities and sciences have yet to engender any such integration -- or even to imagine its possibility. Ironically their relationship could be understood as exemplifying a form of "social distancing" to avoid any risk of mutual infection.

Complementarity of pattern languages -- an all-encompassing meta-pattern?

Contrasting patterns: As noted above, as a potentially comprehensible pattern, the "1,000-petalled lotus" could be contrasted with that celebrated in the mathematics of group theory as the so-called "monster group".Its complexity is indicated by its order, namely the number of elements in its set:

246 x 320 x 59 x 76 x 112 x 133 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71
namely
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
or approximately
8×1053

The group contains 20 sporadic groups (including itself). It is the biggest of the sporadic groups and is equipped with the highest known number of dimensions and symmetries. The set of groups has been termed the "happy family" -- and hence the alternative name for the "monster" as the "friendly giant".  A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

The monster group is regarded by many mathematicians as a mysteriously beautiful object -- intriguingly framed by the so-called  monstrous moonshine conjecture. As seemingly beyond conventional human comprehension, it might then be asked how that "mathematical object" relates to the significance otherwise attributed to the "1,000-petalled lotus" (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008).

Is the "1,000-petalled lotus" to be understood as another form of "monstrous moonshine".

Correspondences: The existence of the monster groups resulted from the recognition of strange and unexpected correspondences, an odd coincidence, as described by Peter Diamond (Mathematicians Chase Moonshine’s Shadow, Quanta, 12 March 2015), noting that its 1053 elements were more than the number of atoms in a thousand Earths. This was originally reported by John Conway and Simon Norton who conjectured that these relationships must result from some deep connection between the monster group and the  j-function (Monstrous Moonshine, Bulletin of the London Mathematical Society, 11, 1979, 3). The deliberate reference to "moonshine" was made because the connection appeared so far-fetched -- beyond any capacity to ever prove it.

Since that time, a numerical proof of the so-called Umbral Moonshine Conjecture proposes that, as summarized by Diamond:

... in addition to monstrous moonshine, there are 23 other moonshines: mysterious correspondences between the dimensions of a symmetry group on the one hand, and the coefficients of a special function on the other.... The 23 new moonshines appear to be intertwined with some of the most central structures in string theory, four-dimensional objects known as K3 surfaces. The connection with umbral moonshine hints at hidden symmetries in these surfaces. 

Aside from this particular focus of mathematicians, a more general question is the nature of "mysterious correspondences" and "deep connections" -- readily to be framed as "moonshine" -- and the manner whereby they may be recognized. As discussed separately, the question can be explored in terms of contrasting understandings of correspondences (Theories of Correspondences and potential equivalences between them in correlative thinking, 2007). The latter was produced in the light of the above-mentioned discussion of the potential psychosocial significance of "monstrous moonshine". It focused on the contrast between the "algebraic" theory of correspondences and the "symbolist" theory of correspondences -- neither of which has the remotest appreciation of the other, although both can be considered fundamental to correlative thinking. 

Curiously however both function as strange attractors -- "mysteriously beautiful" in some way. This is indicative of an aesthetic dimension, readily associated with an appreciation of symmetry, perhaps to be understood as appropriate connectivity..

Fundamental patterns and their correspondences: The point to be emphasized is the future possibility of recognizing other patterns of correspondences and the complementarity between them. Such an exploration would be consistent with arguments made by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999) and George Lakoff and Rafael E. Nunez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

Provocatively it might be asked, for example, whether the relation between the 20-fold set of amino acids constitutes another form of moonshine, given its importance to the mystery of life -- and the complexity of the genetic code. Does that 20-fold pattern bear any relation to the 20 sporadic groups recognized by mathematics -- or the 20 rings of the crown chakra -- if only as a pattern of memetic organization? Why do the dodecahedron and the icosahedron offer complementary means of ordering such a pattern -- with their particular symbolic appeal to the human mind?

Pattern language: A major contribution to further exploration is the seminal work of  Christopher Alexander (A Pattern Language, 1977), following his earlier work (Notes on the Synthesis of Form, 1964), and followed by a major study (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-4). From this he drew further insights (Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger wholeInternational Journal for Unconventional Computing (IJUC), 5, 2009).

The patterns in Alexander's focus on nature and the built environment lend themselves to generalization extending to to the psychosocial environment (5-fold Pattern Language, 1984). That with respect to "harmony-seeking", also invites further relfection (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).

In confronting seemingly disparate patterns, there is a case for drawing on the insights highlighted by Alexander with respect to 15 transformational principles as summarized and discussed in the latter -- and the extension of their interpretation, as discussed in the following sections:

There is a case for comparing such understanding of "transformation" with the set of supersingular primes dividing the order of the monster group  There are precisely fifteen such prime numbers. It could then be asked whether any such 15-fold pattern lends itself in polyhedral form such as to be memorable as a whole, as discussed with respect to their geometrical configuration. This noted that:

There are therefore 15 intersecting golden rectangles, each edge of the icosahedron being defined by an edge of a golden rectangle. The 15 golden rectangles span the interior of the icosahedron. These rectangles have 30 edges, and each edge pairs up with its opposite edge to form a golden rectangle.

Indication of patterns of 15-foldness
Icosahedron
showing single golden rectangle
(made with Stella Polyhedron Navigator)
Icosahedron
showing all 15 golden rectangles
(made with Stella Polyhedron Navigator)
Simplest magic square
(of order 3)
Icosahedron showing single golden rectangle Icosahedron showing all 15 golden rectangles 3x3  Magic square

With respect to use of a magic square, as previously noted (Magic Carpets as Psychoactive System Diagrams, 2010), if any underlying systemic pattern is to be found in Alexander's 15 transformations, a mathematical curiosity of possible relevance is that all the dimensions of the smallest non-trivial magic square total to 15. More subtle understandings of the relevance of magix squares to coherence in governance is evident in their cultivation by Benjamin Franklin (Magic square integrity and implications for the US Constitution, 2015).

Rather than associating any such transformations with 15 golden rectangles as internal features of an icosahedron, another potentially valuable approach is to associate them with the edges of a polyhedron, of which one interesting candidage is the tridiminished icosahedron, as presented below. This has 8 faces (of 4 types), 15 edges (of 5 types), and 9 vertices (of 3 types). Reflection planes are shown in several animations; one shows the stages in (un)folding.

Indicative mapping of supersingular primes onto 15 edges of tridiminished icosahedron
Faces shown Faces transparent Folding Dual
Mapping of supersingular primes onto 15 edges of tridiminished icosahedron Mapping of supersingular primes onto 15 edges of tridiminished icosahedron Mapping of supersingular primes onto 15 edges of tridiminished icosahedron Mapping of supersingular primes onto 15 edges of tridiminished icosahedron
Made with Stella Polyhedron Navigator

In the quest of memorable mappings, the same exercise can be performed with a star polyhedron of 15 vertices (of 3 types), 22 faces (of 6 types), and 35 edges (of 7 types)

Indicative mapping of supersingular primes onto the 15 vertices of a star polyhedron
Faces shown Faces transparent Folding Dual
Mapping of supersingular primes onto the 15 vertices of a star polyhedron Mapping of supersingular primes onto the 15 vertices of a star polyhedron Mapping of supersingular primes onto the 15 vertices of a star polyhedron Mapping of supersingular primes onto the 15 vertices of a star polyhedron
Made with Stella Polyhedron Navigator

Power laws: Considerable importance is accorded in physics to power laws, namely a functional relationship between two quantitites. with one quantity varying as the power of another -- as understood in terms of exponentiation. The exponents defining the monster group and its components are indicative of this. The square-cube law is applied in a number of scientific fields dealing with the natural world. In psychosocial systems, it is of notable importance in proxemics (Scale: power laws exemplified by the square-cube law, 2019).

The latter notes the eexistence of 80 kinds of power laws ranging from atoms to galaxies, DNA to species, and networks to war highlighted by Pierpaolo Andriani and Bill McKelvey (Beyond Gaussian Averages: redirecting international business and management research toward extreme events and power lawsJournal of International Business Studies, 38, 2007, pp. 1212-1230). The authors apply their insights to organization, most obviously to corporations (Perspective: From Gaussian to Paretian Thinking: causes and implications of power laws in organizationsOrganization Science, 20, 2009, 6; From Skew Distributions to Power-law Science, In: Peter Allen (Ed.), The Sage Handbook of Complexity and Management, 2011).

It is therefore appropriate to explore the organization of the crown chakra from a power law perspective, especially given the role of exponentiation in the case of the monster group. The relation to the "10,000 things" of the Tao Te Ching invites similar consideration -- given reference above to the chiliahedron and the myriahedron. With the former as a framing of high order of subjectivity, and the latter of the objectivity of the real world, their entanglement offers a provocative visual mnemonic. This is potentially in the spirit of the preoccupations of the Global Sensemaking Network.

Suggestive pattern of power law relations between subjectivity and objectivity?
Chiliahedron
"crown chakra"
Correspondence? Myriahedron
"10000 things"
Pattern of power law relations in crown chakra / chiliahedron Pattern of power law relations in entanglement of chiliahedron and myriahedron Pattern of power law relations among 10000 things / myriahedron

Dependence of viable global governance on pattern management?

Challenge of numbers: Ironically there is a sense in which global governance would apprear to depend on appropriate "engagement with the monstrous". If the monster group and the crown chakra are indicative of what might be termed cognitive monstrosities, there may be similar conceptual challenges to be addressed. Their problematic nature is recognized by the policy sciences in reference to so-called "wicked problems", as separately discussed (Embodying Strategic Self-reference in a World Futures Conference: transcending the wicked problem engendered by projecting negativity elsewhere, 2015). Speculatively, given the above-mentioned role of aesthetics, the challenge could be framed in archetypal terms (Poetry-making and Policy-making: arranging a Marriage between Beauty and the Beast, 1993).

A primary characteristic of the "monstrous" nature of such numbers is the challenge they potentially represent to global governance, conceptually and otherwise (Comprehension of Numbers Challenging Global Civilization, 2014). How can anyone "re-member" what is experienced as complex?

Just how much can representatives of the people encompass in fulfilling their governance function? How many problems? How many strategic possibilities? Given the 1,000 elements of the crown chakra, how does governance engage with such numbers -- let alone those implied by the monster group -- if either or both have fundamental signifiance to comprehension of appropriate order?

It is ironic, even amusing, that the challenges of governance -- and the problematic understandings of imbalance in the patterns of social control -- are framed as "power problems". At the same time there is scientific recognition of the role of "power laws". It is therefore worth exploring the possibility that pattern recognition and pattern management involve an understanding of power associated with exponentiation. Could it be that memory and comprehension -- "re-membering" -- imply power laws to an unsuspected degree?

Number of parliamentary representatives: The argument can be reframed in terms of reflection on the number of representatives appropriate to any world government and their assembly for decision-making purposes. This has been the focus of an extensive study by a Committee for a Democratic U.N. (Andreas Bummel, The Composition of a Parliamentary Assembly at the United Nations, 2010). Taking the European Parliament as a point of departure, it noted that the apportionment of seats (based on Article 190 of the EC Treaty, and on the Treaty of Lisbon) is not strictly proportional to the population size of the 27 EU member states.

First of all, the EU members have politically agreed to limit the total number of MEPs to 736, to set a minimum number of 5 seats for every state, even the smallest ones, and a maximum number of 99 seats.19 The seats are then distributed according to “degressive proportionality.” In other words, the larger the population of a state, the more people per MEP are represented. The EC treaty does not include a formula... [Under Lisbon, there would be a total maximum of 751 members, the maximum per state would be 96 and the minimum would be raised to 6].

The study offers conclusions for a UN parliamentary assembly according to different models, as indicated below -- with the total number of seats approximating 800. Occupied by "members" of that parliament, and in the light of the challenge of memorability, there is a curiuous sense in which the function of the assembly is to "re-member" the world -- through a framework provocativelyu approximating that of the "1,000-petalled lotus" or the elusive chiliahedron (discussed above).

Hypothetical distribution of UN parliamentary seats in four models
(according to Freedom House categories)
Hypothetical distribution of UN parliamentary seats in four models
From The Composition of a Parliamentary Assembly at the United Nations, Committee for a Democratic U.N., 2010, p. 34

Potentially more intriguing in relation to any such determination is the study by the ACE: Electoral Knowledge Network on Parliamentary Size; (reproduced in Global Advocacy, Electoral Systems Design Componenets). It is framed by the question as to how large should be a country's representative assembly? The arguments made are clearly of relevance to any world assembly.

The question is not trivial. Assembly size has measurable effects on the representation of political parties. Especially, in smaller magnitude systems (such as single-member districts, but also small multimember districts) having more seats means more districts in which smaller parties with localized support have greater chances for representation. An assembly that is too small for the country may thus shut out important interests. Regardless of district magnitude, a small assembly may create a feeling of 'distance' between representatives and voters, even voters who favor large parties. On the other hand, an assembly that is overly large may create an unwieldy legislative process and generate a need for more complex intra-assembly committee structures or encourage the delegation of more legislative authority to the executive branch. Thus the question arises of what is the 'optimal' assembly size for a given country of a given population.

Especially valuable is the focus on the needs for appropriate communication by any legislator, whether with constiuents or other legislators. This would seem to be a question that is notably avoided in any analysis of the operation of assemblies of international significance. How effective is the communication by legislators -- bizarrely highlighted by the manner in which representsatives in the Westminster Parliament are required to stand up to indicate their desire to intervene, whether or not they are then recognized.

Democratic formulae: The ACE study ventures onto highly controversial ground in deriving a model whereby the number of representatives should ideally be the cube root of the size of the population -- recalling the power law focus above. This is then refined by an even more controversial emphasis on the  active population -- "that portion that can be assumed to be actually involved in market exchange and therefore in seeking political representation". The refinement defines the number as the cube root of twice the size of the active population.

A fundamental difficulty with this formula is that determination of the economically active population is itself highly controversial, as indicated by the various methodologies examined, or specifically excluded (ILO Labour Force Estimates and Projections 1990-2030: Methodological description, 2017). Especially problematic is the inadequacy of national reporting procedures and standards -- a factor only too evident in relation to the COVID pandemic. Most controversial is any implication that those not actively working (or seeking work), however that is defined, would have no right to elect representatived to a parliamentary assembly -- notably those rendered jobless by crises such as COVID-19. The only clarity on the matter would seem to be with respect to the lack of clarity.

Hypothetically therefore, even assuming an active population of 50%, the ACE formula when applied to a global population of 8 billion, would give the requisite number of representatives in a democratic world asssembly as (2x4x109)-3, namely 2000, shortly to be increased to 3000 with the predicted rise in world population.

Such numbers are of course common in popular assemblies where there is little requirement for participants to communicate with all present or with one another individually, other than through collective processes (chanting, song, and the like). They raises the question of the challenge they might pose for the communication processes assumed to be the function of democratic representatives.

Comprehensible patterns of clustering: Any dynamics of a "1,000-petalled lotus" then usefully focus attention on the role of patterns of communication as implied by its 20 rings -- potentially to be recognized as factions, thematic preoccupations, or modalities. Some consideration of such patterning can be explored with respect to the European Parliament, as presented separately (Experimental Visualization of Dynamics of the European Parliament in 3D: a 9-fold enneagram of political groups embedded in a 12-fold symbolic icosahedron in virtual reality, 2019).

The question which is not otherwised addressed, and mentioned only in passing in the ACE study, is who needs to communicate with whom in the fulfillment of the expected functions of a legislator -- as potentially to be understood from a cybernetic perspective on systemic viability. More to the point, how is this facilitated or inhibited by conventional procedures and the use of technology?

There appear to have been few studies of this matter in the case of legislative assemblies, the assumption being that assembly of nigh on 1,000 representatives in a hemicycle ensures appropriate processes -- if only in symbolic terms. Arguably this assumption is especially dangerous, given the nature of the challenges with which legislators are expeected to engage. That legislators are specifically remunerated for their pressence in such an assembly is considered irrelevant to evaluation of the efficacy of its operation -- in contrast to the traditional model of acting as an appreciative audience for a leader.

Missing would appear to be the systemic studies of requisite clustering for viable governance, of which the 9-fold pattern above is but one example. A primary concern being the memorability of the patterns deployed, given the need for global coherence -- both within the governance process and its appreciation by the electorate. The question is discussed separately with respect to the currently unexplained enthusiasm for various strategic patterns of N-foldness:

A striking instances are offered by a pragmatic determination of requiste numbers, negotiated by treaty or otherwise (as exemplified by the case of the European Parliament) and the articulation of governing sets of principles, strategic articulations, and the like. These are affirmed in numbered lists, typically without the slightest explanation as to why a set is of any given size. In both cases there is not the slightest concern as to whether the pattern as a whole, or any articulation thereof, is memorable and comprehensible -- despite the importance attached to communicability.

Patterns of N-foldness: There is an as yet unexpllained tendency for authorities to opt for particular numerical constraints on the articulation of strategies and sets of principles. This is also evident in choice of themes, methodologies, values, and in the organization of departments and agencies, as discussed separately (Global Coherence by Interrelating Disparate Strategic Patterns Dynamically: topological interweaving of 4-fold, 8-fold, 12-fold, 16-fold and 20-fold in 3D, 2019; Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980). Of particular interest is how some form of power law may constrain a sense of coherence and appropriateness.

Indicative patterns of coherence and memorability
(see more complete listing: Table of strategic structural attributions by number of elements, 2019)
N-foldness Factors Examples
8-foldness 23 UN Millennium Development Goals; Noble Eightfold Path; Eightfold Way of particle-physics theory; Eightfold Path of policy analysis
9-foldness 33 Planetary boundaries; See checklist of Indicative symbols
10-foldness 2x5 See checklist: Habitual use of a 10-fold strategic framework?
12-foldness 22x3 See: Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources
14-foldness 2x7 Grand Challenges for Engineering in the 21st Century (National Academy of Engineering)
15-foldness 3x5 Global Challenges (Millennium Project); Principles of transformation (Christopher Alexander)
16-foldness 24 UN Sustainable Development Goals (without coordinating 17th); Earth Charter; The Next Generation of Emerging Global Challenges (Policy Horizons Canada)
17-foldness 17 UN Sustainable Development Goals (with coordinating 17th); 17 Things We Don’t Know,,,about Covid-19 (Lisa Rankin); Top 17 Environmental Problems (Renewable Resources Coalition)
18-foldness 2x33 European Convention on Human Rights
20-foldness 22x5 See: Checklist of web resources on 20 strategies, rules, methods and insights
30-foldness 2x3x5

Universal Declaration of Human Rights; note the number of 30-point plans

25-foldness 52 Cairo Declaration on Human Rights in Islam
53-foldness 53 Arab Charter on Human Rights 
72-foldness 23x32 "Demonique": a mnemonic aid to comprehension of potential system failure?; "Angelique": evangelisation of the resolutique in the light of angelology?
82-foldness 2x41 American Convention on Human Rights

Given the very extensive thinking associated with the design of bladed-fans to achieve the requisite thrust for aircraft engines, it is appropriate to explore the nature of thinking in the design of what might be recognized as strategic "fans" to achieve human uplift. Of particular interest is any correspondence to the design factors which mitigate against failure, as especially recognized in the case of multiple stages of rotating bladed-fans under stress from temperature and pressure. Reference to "blades" offers the irony that the term "cutting edge" is used in reference to innovations in both advanced technology and in social change initiatives.

The emphasis on the constructibility of geometric patterns, and their potential relation to comprehensibility and memorability in governance, is further clarified by the following table -- strangely highlighting the unexplained role of 17, now featuring as the primary ordering factor in the 17 Sustainable Development Goals of the UN.

Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)
Extracted from table in Wikipedia by Cmglee / CC BY-SA

Meta-pattern, Meta-poesis and Music

Meta-pattern: Another approach to these questions is through the insight of Gregory Bateson, as a biologist/anthropologist, regarding some form of generative meta-pattern, as discussed by Richard J. Borden (Gregory Bateson’s Search for "Patterns Which Connect" Ecology and Mind, Human Ecology Review, 23, 2017, 2. For Bateson:

The pattern which connects is a meta-pattern. It is a pattern of patterns. It is that meta-pattern which defines the vast generalization that, indeed, it is patterns which connect. (Mind and Nature: a necessary unity, 1979)

And it is from this perspective that he warns: Break the pattern which connects the items of learning and you necessarily destroy all quality (1979, pp. 8-11).

The cognitive engagement with such a meta-pattern invites speculative reflection (Walking Elven Pathways: enactivating the pattern that connects, 2006; Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007; Hyperspace Clues to the Psychology of the Pattern that Connects, 2003).

Since the challenge of recognition of any such meta-pattern could readily be understood as engaging with "monstrous" sets of numbers, is this framing misleading if the issue is more appropriately understood as engaging with patterns through power laws and exponentiation -- as providing a sense of order?

Metapoetics as complement to metalogic? Given the appeal of aesthetic attractors, another approach is suggested by "poesis" -- especially given its significance for autopoesis. In explaining why "we are our own metaphor", Gregory Bateson pointed out to a conference on the effects of conscious purpose on human adptation that:

One reason why poetry is important for finding out about the world is because in poetry a set of relationships get mapped onto a level of diversity in us that we don't ordinarily have access to. We bring it out in poetry. We can give to each other in poetry the access to a set of relationships in the other person and in the world that we are not usually conscious of in ourselves. So we need poetry as knowledge about the world and about ourselves, because of this mapping from complexity to complexity. (Cited by Mary Catherine Bateson, pp. 288-9)

The challenge itself and any potential response can be fruitfully framed through poetry, most obviously that of W. B. Yeats exactly a century ago.

Challenging loss of strategic coherence
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.
W. B. Yeats, Second Coming (1920)
Requisite banners -- with a mnemonic device -- for navigating the measureless?
Kubla Khan Excelsior
In Xanadu did Kubla Khan
A stately pleasure-dome decree:
Where Alph, the sacred river, ran
Through caverns measureless to man
Down to a sunless sea.
The shades of night were falling fast,
As through an Alpine village passed
A youth, who bore, 'mid snow and ice,
A banner with the strange device,
Excelsior!
Samuel Taylor Coleridge (1816) Henry Wadsworth Longfellow (1841)

Curiously it can be argued that insights of a higher order can be embodied and appreciated through aphorisms -- as a necessarily succinct aesthetic form inviting a special form of cognitive engagement -- in contrast with systemic explanations of complex patterns (Andrew Hui, A Theory of the Aphorism, 2019; Victor S M De Guinzbourg, Wit and Wisdom of the United Nations: proverbs and apothegms on diplomacy, 1961). Haiku provide one example of this, notably appreciated by Dag Hammarskjold as an early UN Secretary-Genernal (Ensuring Strategic Resilience through Haiku Patterns, 2006). Recourse to "moonshine" as a descriptor by mathematicians, is indicative of this.

The issue is discussed separately in the light of meta-poetics as a complement to conventions of logic informing preoccupation with metalogic, (as the study of logical systems (Enabling memorability through poetic self-reflexivity: metapoetics, 2016). There is indeed an extensive literature on metapoetics, with a variety of emphases:

Especially helpful in distinguishing the emphasis here are the introductory distinctions made by Huda J. Fakhreddine (Defining Metapoesis in the Abbasid Age, Journal of Arabic Literature, 2011):

Meta-poetry -- poetry about poetry -- is often motivated by poets' preoccupation with their medium and their constant need to examine and justify using it. This is why many meta-poetic compositions often voice the anxieties of a poet towards his/her role and place in a tradition....

However, meta-poetry is not restricted to twentieth-century Modernism and can sometimes be much more than merely addressing poetry and poetics as themes in the poem. This is why it is important here to make the distinction between thematic meta-poetry, i.e. poems whose subject or theme is poetry (poetry about poetry), and a different, harder to define meta-poetry that can be deeper and much more critical in nature although it might not be signaled or marked as clearly as the former. I choose to call this second type referential or contextual metapoesis to reflect a consciousness apparent in the manner in which a poet responds to or engages poetic references, and the context of critical ideas and frameworks in which the poet writes. [emphasis added]

This understanding can be used to clarify the engagement of an audience exposed to poetic improvisation, as with the Basque folk art of bertsolaritza. Fakhreddin comments further:

Contextual metapoesis also requires a specifically alert and expectant audience. Consequently, a major shift in focus occurs in such meta-poetic compositions. The medium of the poem becomes an end and a statement in itself, regardless of the subject matter. This is why for the purposes of this paper, I will define metapoesis as a poet's creative reproduction of, or response to, his poetic heritage. It is a creative state in which a poet's self-awareness as participant in a project for poetic change is evident. However, he is constantly looking over his shoulder to see how his predecessors have done things, not to imitate them or necessarily break away from them but mainly to signal references; references against which his contributions become more obvious and meaningful. (pp. 205-206)

Another desciptions is offered by Hubert Dreyfus and Sean Dorrance Kelly in  concluding that embracing a "meta-poietic" mindset is the best, if not the only, method to authenticate meaning in our secular times:

Meta-poiesis, as one might call it, steers between the twin dangers of the secular age: it resists nihilism by reappropriating the sacred phenomenon of physis, but cultivates the skill to resist physis in its abhorrent, fanatical form. Living well in our secular, nihilistic age, therefore, requires the higher-order skill of recognizing when to rise up as one with the ecstatic crowd and when to turn heel and walk rapidly away. (All Things Shining, 2011)

Memorability, aphorisms and logos: The relevance to this argument can be expressed otherwise through the challenge of memorability in the face of engagement with complexity. Clearly the more complex the explanation -- whether in the case of the monster group or the crown chakra, for example -- the greater the challenge to memorability and communicability. This may apply both to the young in the earlier stages of learning and to older generations as their capacities diminish -- even though they may continue to have a role in articulating the challenges of governance.

Memorability could be understood as a response to the question: how are complex patterns to be re-membered? It is intriguing to note that one of the mathematicians most associated with the monster group is also responsible for developing both the Conway polyhedron notation and the Conway knot notation. The first could be seen as a means of ordering patterns of integration, with the latter an ordering of complexity -- epitomized by the Gordian knot for governance (Engaging globally with knots and riddles -- Gordian and otherwise. 2018). It is however even more intriguing to note that the issue of comprehensibility is a factor in neither -- although both may be understood as a matter of learning and insight. Do power laws offer clues to both memeorability and comprehensibility? Is biomimetics indeed a clue to memorability?

Curiously the polyhedron notation helps to frame the question of memorability otherwise -- through its use of a biological metaphor, namely a "seed" (as highlighted below left). In contrast to the indication of generation of forms by Conway operations on polyhedra, the speculative map on the right explores the numeric relations between the characteristics of a set of polyhedra, notably through the prime number factors associated with their product, as discussed separately (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). Of some relevance in relation to the primes evident in the monster group, of the complete set of the first 12 primes (2, 3, 5, 7, 11, 13, 17, 19, 23 29, 31 and 37), the first four are especially evident with respect to the tetrahedral pattern of Platonic polyhedra. Only 29 is not immediately evident, although 17 is seemingly relatively rare.

Creation of 12 forms from a cube by 3 operations
(indicative of Conway operations on a polyhedral "seed")
Relationships between spherically symmetrical polyhedra
(regular and semiregular; prime number factors in square brackets)
Creation of 12 forms from a cube by 3 Conway operations Map of relationships between spherically symmetrical polyhedra
Tomruen at English Wikipedia / Public domain; Preparation discussed separately

Arguably the fundamental challenge of memorability and communicability is one of designing the most efficient packaging -- as remarkably achieved by seeds. This is obvious in the case of a seed (pod). It has acquired importance for humanity in terms of seed banks and the need foreseen by science fiction for packing the totality of human culture into some form of container for the future, or to accompany migrants to distant planets. Packaging could be recognized as the skill so effectively employed by mnemonists, as explored by Frances Yates (The Art of Memory, 1966).

The concern with respect to global governance is whether insights can be memorably packaged, especially in a context of potential civilizational collapse (Societal Learning and the Erosion of Collective Memory, 1980). Given the emphasis on comprehension and memorability implied by the challenge of the crown chakra, of interest is the power law role in recent research into learning decay, memorability and forgetting (John T. Wixted and Shana K. Carpenter, The Wickelgren Power Law and the Ebbinghaus Savings Function, Association for Psychological Science, 18, 2007, 2; Aditya Khosla, et al, Memorability of Image Regions, Proceedings of the 25th International Conference on Neural Information Processing Systems, 1, 2012; Michael J. Kahana, et al, Note on the power law of forgetting). The issue is related to the power law of practice in learning.

Logos and emblems could be considered the epitome of integrative long-term memorability, especially in their heraldic form. They are the means by which followers are led to "battle" -- or remembrance of it. In the latter respect, especially curious is the considerable value attached to flowers (Ann Elias:, War and the Visual Language of Flowers: an antipodean perspectiveWar, Literature and the Arts, 20, 2008, 1-2). The role of flowers as a key to memorability in the face of complexity can be argued more generally (Flowering of Civilization -- Deflowering of Culture: flow as a necessarily complex experiential dynamic, 2014). Especially relevant with respect to this argument is the articulation offered by Keith Critchlow (The Hidden Geometry of Flowers: living rhythms, form and number, 2011).

Hence the case for exploring more engaging forms, or run the risk of their manipulative imposition, as in the case of the swastika (Eliciting Insight from Mandala-style Logos in 3D, 2020; Quantum Wampum Essential to Navigating Ragnarok, 2014; Swastika as Dynamic Pattern Underlying Psychosocial Power Processes, 2012). In this respect it is striking to recognize the potential relationship between logos, symbolically understood, and logos philosophically understood -- highlighted by the contrasting studies by the topologist René Thom (Structural Stability and Morphogenesis, 1972; Apologie du logos, 1990).

Music: It is remarkable to note the strange attraction of music in a global civilization torn both by disasters and by conflicting visions and insights. More curious are the numerous clues offered by music to memorability and integrative comprehension -- in a culture striving desperately for "harmony". Sonification is valued as a means of pattern recognition in complex data streams engendered by major science projects, as promoted by the International Community for Auditory Display (Sonification of Twitter Leadership at the G20, 2017). Considerable symbolic importance is associated with music in the formalities of governance through ritual  performance of national anthems. Strangely, although characteristically, this does not extend to any appreciation of music in enabling more effective engagement with the governance of complexity, as can be variously argued:

More remarkable are the insights potentially offered by musicologists into complexity and its comprehension, as suggested by the work of:

In relation to the monster group, it is intriguing to note McClain's use of the mathematics of lattice arrays, primes and their exponents to order tonal values in a detailed argument taking account of other number bases characteristic of other cultures (image on left). Does such exponentiation suggest a future possibility of comprehending the array of the sporadic simple groups, whether constituting the monster group or not, of which the image (below center) reflects current insight? Showing how they fit together, the diagram is based on that by Mark Ronan (Symmetry and the Monster, 2006).

Tonal values in hexagonal lattice array
Array of sporadic simple groups The Logic Alphabet Tesseract
37 Tonal values in hexagonal lattice array Array of sporadic simple groups The Logic Alphabet Tesseract by Shea Zellweger
Reproduced from Ernest McClain, Meditations Through the Quran: tonal images in an oral culture, 1981, p. 95) By en:User:Drschawrz - [File:Finitesubgroups.svg], CC BY-SA 3.0, Link Diagram by Warren Tschantz
(reproduced from the Institute of Figuring) .

Potentially of far greater simplicity, the dynamics of oppositional logic is indicated by the polyhedral array (above right) of the 16 possible binary Boolean operations -- the logic alphabet by Shea Zellweger in a four-dimensional cube (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018; Neglected recognition of logical patterns -- especially of opposition, 2017; Alessio Moretti, The Geometry of Logical Opposition, 2009).

In terms of the language of "monstrous moonshine", this could be considered a "baby monster" fundamental to the problematic nature of discourse in a global civilization currently vitiated by fake news. There is, for example, the delightful possibility that its 30 conjugacy classes offer a degree of correspondence to the 30 edges of the icosahedron determined from a magement cybernetics perspective to be a means of transcrending unproductive discourse (Stafford Beer, Beyond Dispute: the invention of team syntegrity, 1994; Joseph Truss, et al., The Coherent Architecture of Team Syntegrity: from small to mega forms, 2000). How might those dynamics be rendered comprehensible through music, given the four-dimensional implications of the logical tesseract?

There is a charming irony to the traditional tales of the manner in which archetypal monsters have been "tamed" or "pacified" by music, especially given the many references to it as a "monster" (Monster or Machine? A Profile of the Coronavirus at 6 Months, The New York Times, 2 June 2020). Speculation in that mode is to be recognized in the recent sonification of COVID-19 (Vineeth Venugopal, Scientists have turned the structure of the coronavirus into music, Science, 3 April 2020; Markus J. Buehler, Nanomechanical sonification of the 2019-nCoV coronavirus spike protein through a materiomusical approach, arxiv, 2003).

Understanding mathematics as a quest for meta-patterns, it can be readily imagined that their association with primes and groups will eventually engender a new understanding of order -- as with discovery of the supersingular primes of moonshine theory, through which the monster group was recognized (Marcus du Sautoy, The Music of the Primessearching to solve the greatest mystery in mathematics, 2003). This potentiality can be more speedily enabled by noting the potential correspondences among recognizably fundamental patterns across the disciplines. Given as yet unexplained preferences for sets of strategies numerically reflective of their characteristics, the dodecahedron and its dual are tantalizingly suggestive as comprehensible holding patterns for:

Arguably conflicts around the globe are exacerbated by misguided attachment to contrasting forms of cognitive geometry (Middle East Peace Potential through Dynamics in Spherical Geometry, 2012). This has been reinforced by theological negligence in failing to explore the mathematical theology on which preferred symbols are based (Mathematical Theology: Future Science of Confidence in Belief, 2011).

There is a considerable degree of irony to the traditional terminology of angelology regarding the distinctions to be made in assiduously numbered angelic hierarchies  -- to the extent that this terminology has been variously borrowed in relation to the current distinctions of global governance. Thrones, Principalities, Dominions and Powers are notably distinguished. In a contect tortured by "wicked problems", with "evil" as an explanation of last resort, a case can be made for exploring such numbering as patterns articulated in terms of powers (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016).

Playfullness and game-playing? Future recognition of a meta-pattern, and of any pattern language by which global goverance might be appropriately informed, would appear to call for human cultivation of pattern recognition beyond any dependence on artificial intelligence in that respect (Patterning Intuition with the Fifth Discipline, 2019).

The crown chakra is one such pattern. The crown chakra is especially valuable as a cognitive challenge in that it is associated with a contemplative discipline. Typical of such disciplines is the manner in which closure is called into question -- in contrast to the quest for a "solution", an "explanation" or a "description", conventionally held to be the most appropriate guarantor of human mastery of reality. To a remarkable degree this attitude resembles that developed in playfullness and game-playing, notably as celebrated in the classic by Hermann Hesse (The Glass Bead Game, 1943), as discussed separately (Evoking Castalia as Envisaged, Entoned and Embodied: the great game informed by the bertsolaritza cultural process? 2016; Humour and Play-Fullness: essential integrative processes in governance, religion and transdisciplinarity, 2005).

Rather than "containing" and "taming" complexity (and truth) as this implies, the more fruitful engagement may be quite otherwise -- through recognition of the premature nature of closure and the cognitive boundaries it imnplies (Engaging with Elusive Connectivity and Coherence: global comprehension as a mistaken quest for closure, 2018). Indicative in this respect to the Zen Buddhist attitude of shoshin, in which engagement is primarily characterized by lack of preconception (Christian Jarrett, How to foster "shoshin", Psyche, 17 May 2020).


References

Christopher Alexander:

Gregory Bateson. Mind and Nature: a necessary unity. Chandler, 1979

Stafford Beer. Beyond Dispute: the invention of team syntegrity. Wiley, 1994

Bryan Carr and Richard Dumbrill (Eds.). Music and Deep Memory. Iconea Publications, 2018

Keith Critchlow:

Victor S M De Guinzbourg. Wit and Wisdom of the United Nations: proverbs and apothegms on diplomacy. United Nations, 1961

Hubert Dreyfus and Sean Dorrance Kelly.  All Things Shining. Simon and Schuster, 2011

Marcus du Sautoy:

Barbara Ehrenreich:

Michael C. Finke. Metapoesis: the Russian tradition from Pushkin to Chekhov. Duke University Press, 2012

Susantha Goonatilake. Toward a Global Science: mining civilizational knowledge. Indiana University Press, 1999

Andrew Hui. A Theory of the Aphorism Princeton University Press, 2019

George Lakoff and Rafael Núñez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, 2000

Ernest McClain:

John Randolph Price. Nothing Is Too Good to Be True. Hay House, 2003

Mark Ronan. Symmetry and the Monster. Oxford University Press, 2006

René Thom:

Dmitri Tymoczko:

Arthur M. Young:

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