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16 March 2020 | Draft

Reimagining Coronavirus in 3D as a Metaphor of Global Society in Distress

Crowning pattern that connects spiky organisms, satellite constellations, nuclear explosions, and egomania?

-- / --

Coronavirus representations compared with a global pattern of nuclear explosions
Designing a rotating global pattern of "erupting spikes" in 3D
Number of equidistant "spikes" in a global spherical array?
Options for representation in 3D of a "mushroom-spike"
Distribution of "mushroom-spikes" around a sphere
Biomimicry and global polyhedral organization
Polyhedral organization of coronavirus?
Global "spiky-prickly" indications of a "pattern that connects"
Global deployment of "proactive" technology
Psychosocial "global implication" of a "pattern that connects"?
Global spiking of knowledge and ignorance
Global connectivity as reinforced through symbolic configuration?


In a period of widespread concern about the coronavirus, and widespread media response, the concern here is with the possibility of another way of imagining its implication -- of relevance to responses of new kind. The approach follows from the presentation of aggressive-seeming depictions of the virus in 2D or 3D. The question these raise in aesthetic terms is what else might such a configuration suggest.

Furthermore, given its relatively great complexity, is there a need to comprehend the configuration as a whole in some new kind of way? More specifically, if humanity is confronted by a threat configured in that way, is it necessary to get to grips conceptually with complexity of that order? As such. does it constitute a pattern evident to some degree in other domains? The argument here is that there are other suggestive patterns of a similar nature with which people are more familiar for other reasons. Through such familiarity it may be possible to engage imaginatively with whatever the coronavirus represents as a challenge.

Given the tantalizing regularity of the extant depictions, if there is a pattern which is recognizable to some degree at other scales, the further argument is to what extent it is cognitively engendered. Does this follow from the arguments of George Lakoff (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). More provocatively and more speculatively, given the global form of the 3D representations, to what extent does this echo instances of the pattern as recognized in plant and animal species, for example -- and even in the global organization of society and various patterns of deployment of technology?

The coronavirus is clearly to be experienced as "aggressively invasive" -- especially at the present time. Is such aggressivity to be recognized in analogues at other scales? Provocatively again, are those aggressive functions matched by defensive functions in those other instances? What does that ambiguity imply? Given enthusiastic speculation regarding extraterrestrials, is the coronavirus to be considered as a form of invasion -- much as humanity might aspire to invading other planets capable of supporting life, as explored in the movie Avatar (2009)? Can humans function like viruses?

For Erik Davis as author of TechGnosis: Myth, Magic, and Mysticism in the Age of Information (1998):

I'm sure you have all read more than enough about the virus, so no big reflections here. I only want to say that, while I am not at all enjoying the hassle, or the extra anxiety layered into social planning, part of me is also totally fascinated watching the world system shudder as the biopolitical archons bare their fangs while my fellow citizens morph into memetized robot-crowds playing apocalyptic consumer scripts with unacknowledged jouissance. I am even fascinating in my own bouts of dread and paranoia. It's not that I am not taking precautions, but I am committed, at least for now, to prioritizing local human connections over the selfish isolationist survival drives that are everywhere on display. (March Blast, 2020)

The focus in what follows is firstly on the ability to represent variously the coronavirus in 3D and to compare it with other 3D structures in nature and on a global scale. Arguably it is the insights from such comparability which might then enable insight to move beyond the specific threats of the virus to physical health as they are variously hyped globally -- even to the point of engendering panic.

Are there vital learnings from the form of the virus -- named in terms of a crown -- that are valuable at the global level? This question follows from earlier speculation on the cognitive significance of a crown in psychosocial organization (Cognitive Crowns: all-encompassing, well-rounded experience, 2000; Implication of Toroidal Transformation of the Crown of Thorns: design challenge to enable integrative comprehension of global dynamics, 2011). The approach was extended even more speculatively, in a manner of relevance to the argument here (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? Experimental representation of crown chakra in virtual reality, 2020).

Any notion of a "crown" with respect to global governance implies a higher order of integration than is evident, or is yet to be understood. Ironically it could be said that there is more comprehension of the structure and operation of viruses than of global society. Could it be the case that the viability of virus organization offers greater insight into the viability of global organization (as yet to take form) than is currently considered worthy of discussion?

This speculative inquiry is therefore of some relevance to the vain (if not naive) pleas for urgent global integration in response to various global challenges, most obviously climate change and now the coronavirus pandemic (Mark Landler, A Fumbled Global Response to the Virus in a Leadership Void, The New York Times, 11 March 2020; EU top dogs slam coronavirus travel bans but can’t coordinate bloc’s response, as member states shut borders to defend themselves, RT, 12 March 2020; John Feffer, Will COVID-19 Kill Globalization? CounterPunch, 10 March 2020). The latter argues:

The global spread of a new pathogen has exposed the fragility of modern life. As it moves around the world, the coronavirus has compromised the circulatory system of globalization, dramatically reducing the international flow of money, goods, and people. The disease has done so rather economically, by infecting fewer than 100,000 people so far. Extrapolation and fear have done most of the work for it.

While world leaders are at last speaking out about it, why does the leadership in each case expect global consensus of such a simple form -- "follow my lead" -- when this too obviously serves particular interests and evokes increasing degrees of suspicion (The Consensus Delusion: mysterious attractor undermining global civilization as currently imagined, 2011).

Some commentators are already seeing the coronavirus pandemic as calling into question the nature of globalization as currently understood (Marshall Auerback, Coronavirus Reveals the Cracks in Globalization, Other News, 11 March 2020). Are there other ways of thinking of "global" organization to which the very form of the coronavirus may offer unsuspected clues?

The following approach follows from the perspective originally explored by the Society for General Systems Research and published over many years in the Yearbook of the Society for General Systems Research. That highlighted a degree of isomorphism between systems at quite different scales, despite their apparent lack of comparability and the opposition between disciplines focused on forms evident at one scale or another. So framed, "global" implies recognition of a higher order of integration than is characteristic of debate about the nature of "globalization" in relation to the planet as a "globe". With respect to the psychosocial implications (discussed in a concluding section), this neglected distinction was emphasized in an earlier paper for a conference of the World Futures Studies Federation (Future Generation through Global Conversation, 1997).

The framing of an elusive "pattern that connects" is a much-cited theme of Gregory Bateson as may be variously discussed (Hyperspace Clues to the Psychology of the Pattern that Connects, 2003; Walking Elven Pathways: enactivating the pattern that connects, 2006). 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 in a much-cited phrase: Break the pattern which connects the items of learning and you necessarily destroy all quality. The insight has been otherwise developed in terms of a "pattern language" by Christopher Alexander (A Pattern Language, 1977), itself treated as a template for complementary psychosocial patterns (5-fold Pattern Language, 1984).

Physicists proudly refer to the much-quoted statement by Niels Bohr in response to Wolfgang Pauli: We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough. To that Freeman Dyson added:

When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope! (Innovation in Physics, Scientific American, 199, 1958, 3)

Faced with global crises and social chaos, the question with regard to the much-sought "new thinking" with respect to "global governance", and the "governance of globalization", is whether any theory is "crazy enough" -- as may well be essential. In this light the newly announced UK initiative for high-risk innovative research, frames the question whether the requisite "craziness" will be inhibited by the same mindsets that have inhibited it previously (UK to launch £800m 'blue skies' research agency. The Guardian, 12 March 2020; Dominic Cummings calls for 'weirdos and misfits' for No 10 jobs, The Guardian, 3 January 2020).

The following speculative inquiry is in the spirit of the question asked by H. Bruce Franklin (What Is Covid-19 Trying to Teach Us? CounterPunch, 13 March 2020):

Neither Covid-19 nor a major recession poses a threat to our survival as a species. We do, however, face two existential threats, both created by our species, and each featuring our nation in the lead role. At the very moment when only global unity and cooperation can save us from threats of nuclear holocaust and environmental devastation, deadly nationalism is tearing our species apart. Can Covid-19 teach us that those two great menaces to our existence are also not zero-sum games? That our species either wins or we, as well as many other species, all lose?

Rather than a question of winning or losing, the challenge might be more appropriately framed in the transcendent spirit of Eastern martial arts as how to learn from one's opponent to avoid the enthusiasms for the restrictive cognitive conventions of zero-sum games, as variously imagined (Ensuring Strategic Resilience through Haiku Patterns: reframing the scope of the "martial arts" in response to strategic threats, 2006; James P. Carse, Finite and Infinite Games: a vision of life as play and possibility, 1986). Are there new possibilities, if we recognize that the form and operation of the coronavirus is a mysterious image of who we are, whether collectively or individually?

Coronavirus representations compared with a global pattern of nuclear explosions

Global dissemination of coronavirus depictions: There is no lack of depictions of various forms of the coronavirus as a spherical particle, otherwise termed a virion. It is the glycoprotein "spikes" protruding from each virion which make infection possible, enabling invasion of the host cells (Ana Shulla, et al, Role of Spike Protein Endodomains in Regulating Coronavirus Entry, The Journal of Biological Chemistry, 284, 2009; Scientists in the Netherlands reveal 3-D structure of coronavirus, CORDIS, 18 April 2003)

As widely indicated, the name coronavirus is derived from the Latin corona, meaning "crown" or "halo", which refers to the characteristic appearance reminiscent of a crown or a solar corona around the virions when viewed under two-dimensional transmission electron microscopy, due to the surface covering in club-shaped protein spikes.

Typically, in conformity with the constraints of global communication (even in times of crisis), these depictions are variously subject to copyright (and paywalls) and cannot be readily reproduced in this document. The following are reproduced thanks to the policy of Wikipedia, however it is appropriate to note the policy of some lading members of the International Publishers Association in enabling access exceptionally to documents on coronavirus research (but not on reproduction of their content).

Depictions of the coronavirus
Ultrastructural morphology exhibited by coronaviruses. Note the spikes that adorn the outer surface of the virus, which impart the look of a corona surrounding the virion, when viewed electron microscopically. Images combined from a 3D medical animation, depicting the shape of coronavirus
as well as the cross-sectional view. Image shows the major elements including the
Spike S protein, HE protein, viral envelope, and helical RNA
Representation of the coronavirus in 3D Cross-sectional representation in 2D of the coronavirus in 3D Cross-sectional representation in 2D of the coronavirus in 3D
CDC/ Alissa Eckert, MS; Dan Higgins, MAM
/ Public domain
Reproduced from Wikipedia / CC BY-SA

An animation of a coronavirus particle rotating to show structure with the characteristic halo of spikes is also accessible from Russell Kightley Premium Scientific Pictures.

Global pattern of nuclear explosions? The media and political focus on the coronavirus pandemic occurs in a period in which the famous Doomsday Clock, maintained since 1947 by the members of the Bulletin of the Atomic Scientists has been reset. Originally set at seven minutes to midnight in 1947, the clock was set at two minutes to midnight in January 2018, and left unchanged in 2019 due to the twin threats of nuclear weapons and the increasing effects of global warming. On 23 January 2020, it was moved forward to 100 seconds (1 minute 40 seconds) before midnight, based on the increased threats to global stability posed by "a nuclear blunder", exacerbated by the rate of climate change.

Of some relevance to the seemingly unusual comparison made here is the focus of an interview with Jeffrey Lewis (John Krzyzaniak, How the coronavirus outbreak is like a nuclear attack: An interview with Jeffrey Lewis, The Bulletin of Atomic Scientists, 20 March 2020).

As a visual metaphor, the argument developed here suggests a comparison between the "spikes" on the coronavirus (especially those in red on the left above) with the widely publicized depictions of the mushroom cloud of a nuclear explosion (as shown below). A nuclear "doomsday" would see a pattern of such explosions around the globe -- suggestively reminiscent of the pattern of spikes around the coronavirus.

Depictions of a thermonuclear explosion
Inside a rising mushroom cloud: denser air rapidly forces itself into the bottom center of the toroidal fireball, which turbulently mixes into the familiar cloud appearance. Animated atomic bomb explosion. Sequence showing mushroom cloud formation from a U.S. nuclear weapon test at the Nevada Test Site. Probably from Operation Tumbler-Snapper, 1952 Mushroom cloud from the atomic bombing of Nagasaki, Japan on August 9, 1945.
Schematic cross-section of a thermonuclar explosion mushroom cloud Animation of a thermonuclar explosion Photography of a thermonuclar explosion
Reproduced from Wikipedia Marcin n®, recreated from scratch by Quibik
Charles Levy / Public domain

Rather then any exclusive focus on the physical nature of nuclear explosions, their form could be understood as a metaphor for the "eruption" of socio-economic, environmental and psychosocial crises around the globe. Argued otherwise, in the light of increasing focus on information warfare and memetic warfare, the eruptions could be notably recognized in such terms.

Designing a rotating global pattern of "erupting spikes" in 3D

The challenge here, as an exercise in visual imagination, is to explore the possibilities of similarities between a pattern of "eruptions" or "explosions" at the global level of the planet and a pattern of invasively "aggressive spikes" at the molecular level. That the protein spikes are described as "club-like" is perhaps ironically appropriate in that various threats by one country to bomb other countries "back to the Stone Age" would result in a reversion to that mode of aggression (Nick Cullather, Bomb them Back to the Stone Age: an etymology, History News Network, 2006)

One inspiration is the interactive animation produced as one of the many exercises on Vertexshaderart. This permits the number of spikes on the rotating globe, and their form, to be varied. The spikes explode out and collapse back -- independently and randomly, as illustrated in screen shots below.

Screen shots of interactive animations
Screen shot of interactive animation Screen shot of interactive animation Screen shot of interactive animation
Reproduced from one of the many options on Vertexshaderart.

Number of equidistant "spikes" in a global spherical array?

With the coronavirus as the point of departure, an initial challenge was to detect any indication from the research literature on the number of protein "spikes" -- bearing in mind that the different forms of the virus may have a different number and any given form might well adopt different configurations. Any reference to "spike" may indeed need to be understood dynamically (Fang Li, Structure, Function, and Evolution of Coronavirus Spike Proteins, Annual Review of Virology, 3, 2016, 1).

Use of "spike" is also confusing since it features in media reference to "spikes" in the number of cases and fatalities reported from coronavirus, as well as in the "spiking" in dynamic systems more generally (Andrés Aragoneses, et al, Unveiling the complex organization of recurrent patterns in spiking dynamical systems, Scientific Reports, 15 April 2014). The latter notes: that: Complex systems displaying recurrent spike patterns are ubiquitous in nature. Understanding the organization of these patterns is a challenging task.

The flu virus is understood to have some 450 protein spikes, whereas the HIV virus has from 10 to 20 spikes -- usefully contrasted with images of the dense pattern in the case of dengue fever (John Schiller, et al, Why HIV Virions Have Low Numbers of Envelope Spikes, PLoS Pathogens, 10, 2014, 8). The contrasts can also be seen with access to "protein spikes", and especially "coronavirus virions" in images via Google.

There is very little information on the number of protein spikes around the sphere of the coronavirus particle. One estimate found was an average of 74 by Benjamin W. Neuman, et al, (A structural analysis of M protein in coronavirus assembly and morphology Journal of Structural Biology, 174, 2011, 1):

To test the quality of this model, we counted the number of spikes on three-dimensional reconstructions of MHV particles. These particles showed an average of 74 spikes per particle, which gives an approximate inter-spike spacing of 17 nm. Our model predicts ~90 spikes per particle.

Given that 74 is only an average, the focus in which follows will assume a pattern of 72 -- if only in a spirit of poetic licence. Again it should be emphasized that this is an exercise of the imagination, and the emphasis is metaphorical in a quest for memorable designs, as highlighted in a preceding discussion (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? 2020). That exercise drew attention to a configuration of 72 for geometrical reasons, having various traditional symbolic connotations. Nevertheless, it is appropriate to recognize that the number of protein spikes may vary on a coronavirus for several reasons. Of potential relevance from this perspective is the most recent initiative (Jonathan Amos, OneWeb increases mega-constellation to 74 satellites, BBC News, 22 March 2020). Why 74? The completed network aims to achieve an orbital configuration of approximately 650 satellites, 

The issue, as considered below, is then how to distribute "mushroom-spikes" evenly around a sphere -- with the primary focus on 72.

Options for representation in 3D of a "mushroom-spike"

The question is how a mushroom-like spike might be most imaginatively designed, especially if the design is to incorporate a dynamic -- in contrast to any simple static form implied by the term "spike".

The approach taken here exploited elements from a 3D model developed previously to demonstrate distinctive dynamics of a torus -- rotation as a ring and a "smoke-ring" dynamic (Interlocking tori: combining the two alternative representations, 2006). This featured in an argument for the Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics (2006). The interlocking was later used in discussion of the "fearful attraction of a hole" with respect to Visualization in 3D of Dynamics of Toroidal Helical Coils (2016). The latter concluded with an adaptation of the earlier models into the form of a horn torus to represent Cognitive heart dynamics framed by two tori in 3D.

Earlier dynamic models in 3D serving as the basis for construction of a mushroom cloud
Animation of horn torus in 3D Animation of interlocking tori Compound of 2 horn tori Stack of horn tori
Animation of two horn tori of major radius in proportion of phi
The X3D and VRML models in 3D were originally prepared by Sergey Bederov of Cortona3D.

The coding for the earlier 3D models and animations above offered several possibilities of immediate adaptation, illustrated by the animations below.

Exploratory "mushroom cloud" dynamic models in 3D
Cyclic ring dynamic with narrowing stem Smoke-ring dynamic with straight stem
Filled surface variant Wire frame variant Wire frame variant Filled surface variant

In response to these exercises Sergey Bederov used some of his earlier coding to produce the following -- which offers another design of the stem of the mushroom and is visually enhanced by the dynamic cloud effects.

Experimental dynamic mushroom-cloud models in 3D
Clouded covering
(mp4; x3d)
Wire frame variant
Experimental dynamic mushroom-cloud models in 3D Experimental dynamic mushroom-cloud models in 3D
Kindly prepared by Sergey Bederov of Cortona3D

Distribution of "mushroom-spikes" around a sphere

With respect to eliciting any pattern, the very specific challenge in spherical geometry is the determination of the even distribution of N points on a sphere, as variously discussed in a Stackoverflow forum. Two main options are highlighted:

The latter was used by Sergey Bederov in preparing the animations which follow. A third option considered was to use symmetrical polyhedral configurations of 72 or 74 vertices, as discussed below.

Animations of globe with array of 72 mushroom-cloud nuclear explosions
Solid variant (mp4; x3d) Wire frame variant (mp4) Solid variant colourd (mp4; x3d)
Animation of globe with array of 72 mushroom-cloud nuclear explosions Animation of globe with array of 72 mushroom-cloud nuclear explosions -- wire frame coloured Animation of globe with array of 72 mushroom-cloud nuclear explosions -- coloured
Kindly prepared by Sergey Bederov of Cortona3D, and partially adapted (see also 74-mushroom-cloud variant: image, x3d)

Biomimicry and global polyhedral organization

Wicked problems? In an earlier scenario-building exercise with an aesthetic emphasis, the process led (fortuitously) to recognition of an as yet unexplored degree of correspondence between the icosahedral ordering of both psychosocial mega-problems and of the micro-problems constituted by viruses (Encycling wickidity in the light of polyhedral viruses and their mutation, 2015). As infectious pathogens, viruses cause serious diseases and major threats for global public health (as in the case of influenza, hepatitis, and AIDS).

Inspired by biomimicry, this suggests the value of exploring virology as offering a "pattern language" with regard to antigens and antibodies as these might apply to the operation of possible "viral antigens" that could be developed to constrain so-called wicked problems. Biomimicry offers a valuable clue to the comprehension of problem wickidity through consideration of viruses and their potential mutations.

This raises the question as to whether the most hazardous viruses offer a valuable mapping template for exploring the wickedest global challenges. A wicked problem can then be usefully framed as a form of viral pandemic like influenza -- whose potential future emergence is an active concern of WHO. Pandemics include non-viral forms like the plague, epitomized in Europe by the dramatic consequences of the 100 million fatalities of the Black Death.

Icosahedral symmetry: As noted in that earlier discussion, a particularly significant class of viruses, polyhedral viruses consist of nucleic acid surrounded by a polyhedral (many-sided) shell or capsid, usually in the form of an icosahedron -- hence the use of "icosahedral virus" in virus classification (Roya Zandi, et al. Origin of Icosahedral Symmetry in Viruses, PNAS, 2004; UCLA Virus Research Group, The Origin of Icosahedral Symmetry in Viruses; D. P. Wilson, Protruding Features of Viral Capsids Are Clustered on Icosahedral Great Circles, PLoS ONE, 11, 2016, 4). Of potential relevance to problem wickidity, the latter demonstrates that the basis for icosahedral symmetry being so strongly preferred by viruses is that: allows for the lowest-energy configuration of particles interacting isotropically on the surface of a sphere. More explicitly, we find that the energy-per-particle is a minimum for configurations that involve 12 five-fold defects at the vertices of an icosahedron, and that these configurations are especially favored for "magic" numbers of particles corresponding to the "triangulation" ("T") numbers of Casper and Klug [1962]

The Wikipedia entry on capsids includes a tabular representation of icosahedral viral capsids in terms of T-numbers. With respect to the disparate characteristic of wickidity, a contrasting case for triangulation has been discussed separately (Triangulation of Incommensurable Concepts for Global Configuration, 2001). Similarly, with respect to "magic" numbers, fundamental to organization, there is a case for exploring the interplay of 5, 12, 20 and 30 (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).

Most animal viruses are icosahedral or near-spherical with chiral icosahedral symmetry. A regular icosahedron is the optimum way of forming a closed shell from identical sub-units. The minimum number of identical capsomeres required for each triangular face is 3, which gives 60 for the icosahedron. Many viruses, such as rotavirus, have more than 60 capsomers and appear spherical but they retain this symmetry. To achieve this, the capsomeres at the apices are surrounded by five other capsomeres and are called pentons. Capsomeres on the triangular faces are surrounded by six others and are called hexons. Hexons are in essence flat and pentons, which form the 12 vertices, are curved. The same protein may act as the subunit of both the pentamers and hexamers or they may be composed of different proteins.

Chirality: Many of the essential molecules for life on Earth can exist in two mirror-image forms, referred to as "left-handed" and "right-handed", but living organisms do not use both. Proteins are exclusively composed of left-handed amino acids; RNA and DNA contain only right-handed sugars. This phenomenon is known as homochirality. It is not known whether homochirality emerged before or after life, whether the building blocks of life must have this particular chirality, or indeed whether life needs to be homochiral. Protein chains built from amino acids of mixed chirality tend not to fold or function as catalysts, but mirror-image proteins have been constructed that work the same but on substrates of opposite handedness.

Hypothetically, it is possible to recreate an entire ecosystem from the bottom up, in chiral form. In this way, the creation of an Earth ecosystem without microbial diseases might be possible. In some distant future, mirror life could be employed to create robust, effective and disease-free ecosystems for use on other planets. Mirror life (also called mirror-image life, chiral life, or enantiomeric life) is a hypothetical form of life with mirror-reflected molecular building blocks. The possibility of mirror life was first discussed by Louis Pasteur. Although this alternative life form has not been discovered in nature, efforts to build a mirror-image version of biology's molecular machinery are already underway.

Mutations of viruses can be organized as truncations and chiral operations in pentagonal space (notably to form Archimedean polyhedra), as described and illustrated by Sten Andersson (General Polyhedra, Virus Structure and Mutation, Zeitschrift für Kristallographie, 225, 2010; The Structure of Virus Capsids, Zeitschrift für anorganische und allgemeine Chemie. 634, 2008). Further to the reference above to the stellations of the icosahedron, Andersson notes that the spike structures of viruses accurately can be described as stellations of polyhedra (Virus Structures, Stellations, Spikes, and Rods, Zeitschrift für anorganische und allgemeine Chemie, 635, 2009).

Pentagonal symmetry: As noted by Aditya Raguram, et al (A chiral pentagonal polyhedral framework for characterizing virus capsid structures, Trends in Microbiology, 25, 2017, 6):

Recent developments of rational strategies for the design of antiviral therapies, including monoclonal antibodies (mAbs), have naturally relied extensively on available viral structural information. As new strategies continue to be developed, it is equally important to continue to refine our understanding and interpretation of viral structural data. There are known limitations to the traditional (Caspar-Klug) theory for describing virus capsid structures that involves subdividing a capsid into triangular subunits. In this context, we describe a more general polyhedral framework for describing virus capsid structures that is able to account for many of these limitations, including a more thorough characterization of inter-subunit interfaces. Additionally, our use of pentagonal subunits instead of triangular ones accounts for the intrinsic chirality observed in all capsids. In conjunction with the existing theory, the framework presented here provides a more complete picture of a capsid's structure and therefore can help contribute to the development of more effective antiviral strategies.

Polyhedral organization of coronavirus?

Selected polyhedra with 72 vertices
(wire frame versions shown below for each case)
1-Frequency Snub Dodecahedral Geodesic Sphere 3-Frequency Geodesic Octahedron (dual) 3 Truncated cubes 8+1 Cubes (dual)
1-Frequency Snub Dodecahedral  Geodesic Sphere 3-Frequency Geodesic Octahedron (dual) 3 Truncated cubes 8+1 Cubes (dual)
1-Frequency Snub Dodecahedral  Geodesic Sphere (wire) 3-Frequency Geodesic Octahedron  (wire) 3 Truncated cubes  (wire) 8+1 Cubes (dual)
Images generated with Stella Polyhedron Navigator

Exceptional polyhedra with both 72 vertices and 72 faces (with wire frame versions)
Small Stellated Dodecahedra 5+1 Small Stellated Dodecahedra 5+1 (wire) Small Stellated Dodecahedra 5+1 (dual) Small Stellated Dodecahedra 5+1 (dual)
Small Stellated Dodecahedra 5+1 Small Stellated Dodecahedra 5+1 Small Stellated Dodecahedra 5+1 (dual) Small Stellated Dodecahedra 5+1 (dual)
Exceptional polyhedra with 74 vertices (with wire frame versions)
Stellated Cuboctahedron Stellated Cuboctahedron (wire) 6-Frequency Tetrahedral Geodesic Sphere 6-Frequency Tetrahedral Geodesic Sphere (wire)
Stellated Cuboctahedron 6-Frequency Tetrahedral Geodesic  Sphere 6-Frequency Tetrahedral Geodesic  Sphere
Images generated with Stella Polyhedron Navigator

The further step was then to explore the 3D variants of the polyhedra which variously corresponded to the assumed criteria of 72 vertices (or 74). As enumerated by Sergey Bederov, two were found possessing icosahedral symmetry. One of these, the pentakis snub dodecahedron, consists solely of triangles, but doesn't have mirror symmetry (namely it is chiral). The other is a polyhedron, seemingly without an official name (a "pentakis icosidodecahedron"?) has full icosahedral symmetry including mirror symmetry, but it also contains some squares in addition to triangles.

In the case of the 74-vertex polyhedra, none has icosahedral symmetry; the highest symmetry for these is cubic. Only one of them has approximately equal edges, and seemingly does not have an official name. In each case the polyhedral and mushroom-clouds models have been combined and variously adapted in the following which offers a sense of the distinctions between the vertices. Note that a further range of design options could be readily explored (colouring, motion rates, background, periodic peaking of mushroom-spikes, etc).

Global configuration of mushroom-clouds on vertices of polyhedra -- simulating a coronavirus
72 vertices -- icosahedral symmetry (mp4; x3d) 74 vertices -- cubic symmetry (mp4; x3d)
Animation of solid variant
Screen shot of wire frame variant Animation of solid variant Screen shot of wire frame variant
Animation of global configuration of mushroom-clouds on vertices of polyhedra -- simulating a coronavirus Animation of global configuration of mushroom-clouds on vertices of polyhedra -- simulating a coronavirus Animation of global configuration of mushroom-clouds on vertices of polyhedra -- simulating a coronavirus
Combination and adaptation of polyhedral and mushroom-cloud models separately prepared by Sergey Bederov of Cortona3D

Global "spiky-prickly" indications of a "pattern that connects"

Software enabling recognition of such pattern noted the role of polyhedra as highlighted above (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008).

The following examples suggest that nature has been constrained in its efforts to ensure the appropriate distribution of global protection of defensive/offensive devices on a sphere -- notably in the case of spheroidal virions. Many of those are specifically recognized as icosahedral viruses -- the capsid having 20 triangular faces, with regularly arranged units called  capsomeres, two to five or more along each side; with the nucleic acid densely coiled within. 

Ironically it might be said, in the case of the first example, that radiolaria "know" all about global configurations. As in that example, the others are variously constrained in geometrical terms in their deployment of "spiky structures", especially as a configuration for defensive purposes. The constraint in human social terms is evident in the constraints of "circling the wagons" (Circling the wagons, The Economist, 25 May 2013).

Radiozoa: Also called radiolaria, are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm. Radiolarians have many needle-like pseudopods supported by bundles of microtubules, which aid in the radiolarian's buoyancy. Some radiolarians are known for their resemblance to regular polyhedra, such as the icosahedron-shaped Circogonia icosahedra pictured.

Radiolaria (inversion of black and white coloration)
Circogonia icosahedra Haeckeliana porcellana Circostephanus coronarius
Circogonia icosahedra
Adapted from Wikipedia

Forms such as those above derive from the remarkable work of Ernst Haeckel (Kunstformen der Natur, 1904) with its emphasis on symmetry and levels of organization

Echinoidea: Sea urchins, are typically spiny, globular animals, echinoderms. About 950 species live on the seabed, inhabiting all oceans and depth zones. Their tests (hard shells) are round and spiny, typically from 3 to 10 cm across. Sea urchins move slowly, crawling with their tube feet, and sometimes pushing themselves with their spines. Like other echinoderms, urchins have fivefold symmetry as adults.

Examples of sea urchins
Phyllacanthus imperialis Echinus melo Paracentrotus lividus
Frédéric Ducarme / CC BY-SA Marco Busdraghi / CC BY Frédéric Ducarme / CC BY-SA

Of potential relevance to isomorphism in animals is the study by D'Arcy Wentworth Thompson (On Growth and Form, 1917)

Barrel cacti: These are various members of the two genera Echinocactus and Ferocactus of the cactus family, Cactaceae

Examples of barrel cacti
Echinocactus grusonii Ferocactus echidne Echinocactus parryi

Echinocactus grusonii Ferocactus echidne Echinocactus parryi
André Karwath aka Aka / CC BY-SA / CC BY-SA Michael Wolf / CC BY-SA

Of potential relevance to isomorphism in plants is the focus of  Keith Critchlow (The Hidden Geometry of Flowers: living rhythms, form and number, 2011). An equivalent study would be of value with respect to the spiny forms like those depicted above. (Gallery of Polyhedral Flower Arrangements: engendering sustainable psycho-social systems through metaphor, 2014)

Spiky seed pods (burs): Burs are seeds or dry fruit that have hooks or teeth. These catch on the fur of passing animals or the clothing of people. The hooks or teeth generally cause irritation, and some species commonly cause gross injury to animals, or expensive damage to clothing or to vehicle tires.

Examples of spiky fruits
Rambutan (unripened) Spiked melon Durian
Rambutan Spiked melon Durian
Kinglaw / Public domain Anagoria / CC BY مانفی / CC BY-SA

Burs serve the plants that bear them in two main ways. Burs tend to repel some herbivores, much as other spines and prickles do. Plants with burs rely largely on living agents to disperse their seeds; their burs are mechanisms of seed dispersal by epizoochory (dispersal by attaching to the outside of animals). Most epizoochorous burs attach to hair on the body or legs of the host animal,

Examples of spiky seed pods
Geum bur Bur of Arctium (Burdock) Xanthium bur Medicago bur
Geum bur Bur of Arctium (Burdock) Xanthium bur Medicago bur
Zephyris / CC BY-SA Prosthetic Head / CC BY-SA Enrico Blasutto / CC BY-SA  

Weapon designs: Examples of relevance to this inquiry include the contact naval mines of the Hertz horn variety (Sam LaGrone, A Terrible Thing That Waits Under the Ocean, Popular Science, 19 May 2014; John M. MacFarlane, Naval Anti-ship Mines, Nauticapedia, 2013). Of curiously similar form is the earlier club-like Morning star -- a spiked mace, whether with a rigid handle or a chain (known in German as Morgenstern). Given the origins of the latter, it is somewhat ironic to note the existence of the Morningstar global investment research company.

The Koon shot of Operation Castle was an early test of a thermonuclear device on (7 April 1954). The two-stage device was known as "Morgenstern", and had a highly innovative secondary stage. As indicated by Wikipedia, the name "Morgenstern" was chosen because of the shape of that secondary stage. It consisted of a central sphere from which spikes were radiating, resembling a morning star / mace. The spikes may have been an idea from Teller and colleagues to use implosive jets to compress the thermonuclear core. It would be well over two decades before weapons were designed that utilized a secondary concept similar to that first tested in the Koon shot.

It can be readily understood that efforts to imagine the design of the ultimate space weapon -- a "death star" -- would be faced with similar constraints. Of interest is the name attributed by Israel to its Spike Missile (Charlie Gao, Put a Spike in It: how Germany plans to kill enemy tanks in a war, The National Interest, 9 March 2019)

Global deployment of "proactive" technology

Of obvious relevance in the case of technology is the manner of its global deployment and the various constraints which become evident in terms of efficiency and efficacy. These are more obvious at the regional level in the deployment of "sites" -- whether in the form of supermarkets, wholesale depots, or manufacturing facilities. In terms of marketing and operational research, the mathematical optimization is determined with respect to catchment areas. In human geography, this is the area from which a city, service or institution attracts a population that uses its services. The matter is more complex when the "area" is the globe as a whole.

Attention has been devoted to the global deployment of resources by Buckminster Fuller in terms of a polyhedral configuration which has notably taken the form of a World Game. (Critical Path, 1981; Operating Manual For Spaceship Earth, 1969).

Of curious relevance to this argument is a form of complementarity between the highly secretive nature of some research on viruses and the highly secretive nature of global deployment of technology. Just as the first is a feature of biological warfare (of which the coronavirus may well be an example), the second is a feature of global militarisation.

Global deployment of nuclear missile launch sites: In the military quest for full-spectrum dominance to achieve global hegemony, considerable importance is clearly attached to the deployment of nuclear missile sites. The form of such missiles is clearly reminiscent of the defensive "spikes" of which various examples are given above. The efficient global deployment is necessarily further constrained by political arrangements.

As noted by Hans M. Kristensen and Robert S. Norris (Worldwide deployments of nuclear weapons, 2017, Bulletin of the Atomic Scientists, 73, 2017, 5):

As of mid-2017, we estimate that there are nearly 15,000 nuclear weapons located at some 107 sites in 14 countries. Roughly, 9400 of these weapons are in military arsenals; the remaining weapons are retired and awaiting dismantlement. Approximately 4150 are operationally available, and some 1800 are on high alert and ready for use on short notice. By far, the largest concentrations of nuclear weapons reside in Russia and the United States, which possess 93 percent of the total global inventory (Kristensen and Norris, 2013). In addition to the seven other countries with nuclear weapon stockpiles (Britain, France, China, Israel, India, Pakistan, and North Korea), five nonnuclear NATO allies (Belgium, Germany, Italy, the Netherlands, and Turkey) host about 150 US nuclear bombs at six air bases.

The long-articulated fears of simultaneous worldwide use of nuclear devices -- deliberately or by accident -- can therefore be related to the pattern articulated above.

Global constellations of communication and military satellites: Attention is currently aroused by the massive deployment of constellations of satellites, purportedly for communication purposes. There is obvious confusion with regard to the deployment of constellations of satellites for military purposes -- readily conflated with those claimed to be for communication purposes, given the dual use possibilities of the latter. Given the number of satellites involved -- tens of thousands -- the geometry of their configuration is clearly an important consideration, as discussed separately (Symbolic Disconnection from the Stars and the Universe? Surreptitious global implementation of full-spectrum dominance and shielding, 2019) and more speculatively (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? 2020).

Whether as a military or a communication configuration, it is readily comprehensible that a configuration of satellites serves both offensive and defensive purposes -- an analogue to the pattern of spikes discussed in the examples above.

In the above-mentioned spirit of "circling the wagons", the current implementation of a space force by various nations is a natural extension of satellites on their own. It may well be seamlessly integrated with any such configuration in 3D. It is recognized as one dimension of the militarisation of space.

In apparent contrast to the "militarisation" of space, the key role of communication satellites in enabling the global manipulation of public opinion -- whether overt or secretive -- is readily comprehensible. This can be explored in terms of panic containment, propaganda, "psychic numbing" and "dumbing down" -- as potential features of global information warfare, psychological warfare and memetic warfare. The latter is understood as involving the propagation of memes on social media through Platform Weaponization.

As has been argued in terms of precautionary strategies in anticipation of hypothetical extraterrestrials, and in science fiction, any such global configuration would potentially serve in the early detection of "space invaders" (as well as Earth crossing asteroids). The "shielding effect" has long been a feature of popular imagination with respect to any "star wars" scenario.

Global configuration of EMF towers -- Tesla towers and coils? The surface deployment of radio towers and arrays to ensure "global coverage" can be recognized as a pattern anticipating any configuration satellites, with communication functions increasingly integrated with such satellites. Arguably again, the form of those towers can be recognized in terms of the patterns of spikes noted above -- appropriately configured.

Potentially more intriguing are suggestions regarding the possibility of other functions associated with towers with some kind of EMF capacity. These range from the wireless distribution of energy to the functions attributed to High Frequency Active Auroral Research Program (HAARP), most notably by conspiracy theorists.

Especially significant in this regard are the inventions of Nikola Tesla, and the secrecy and conspiracy surrounding their later development. For example, as illustrated below, Wardenclyffe Tower (1901–1917), also known as the Tesla Tower, was an early experimental wireless transmission station designed and built by Nikola Tesla in New York in 1901–1902. Tesla intended to transmit messages, telephony and even facsimile images across the Atlantic to England and to ships at sea based on his theories of using the Earth to conduct the signals.

Could such a tower, and the principles on which it operated, be understood as a protoype of human "spike technology"?

Indication of potentially relevant innovations by Nikola Tesla
Tesla's Wardenclyffe Tower
Tesla's Magnifying "Apparatus for transmitting electrical energy" U.S. Patent 1,119,732 Tesla coil: Electrum sculpture (1998), the world's largest such coil. Builder Eric Orr is visible sitting inside the hollow spherical high voltage electrode.
Tesla's Wardenclyffe Tower Tesla's magnifying Apparatus for transmitting electrical energy Tesla coil: Electrum sculpture
Unknown author / Public domain Nikola Tesla / Public domain Joe Decker / CC BY-SA

The possible psychosocial implications are discussed separately (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011; Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014). In the latter it took the form of a separate section (Insight into global dynamics through Tesla's focus on the sphere, 2014). So framed, and as demonstrated by Tesla, it can then be asked whether the failure to manage positively and negatively charged conditions appropriately, results in the dangerously uncontrollable "sparks" so characteristic of society.

Psychosocial "global implication" of a "pattern that connects"?

In the spirit of the general systems and pattern language perspectives highlighted above, are there insights to be derived from the visual examples presented? In particular does the form of the coronavirus suggest a way of thinking about global psychosocial organization? More provocatively, could the emergence of the pandemic be recognized as a wake-up call evoked by a pattern of systemic negligence, as separately suggested (Vigorous Application of Derivative Thinking to Derivative Problems: transcending bewailing, hand-wringing and emotional blackmail, 2013; Systemic Crises as Keys to Systemic Remedies: a metaphorical Rosetta Stone for future strategy? 2008).

In framing this exercise in terms of the general systems approach (as noted above), it is appropriate to recall the long-standing exploration of the reflection of the microcosm in the macrocosm, and vice versa. This has been expressed succinctly in the aphorism as above, so below.

Speculatively, to what degree do the following psychosocial phenomena merit exploration as being characterized by a global "pattern of spikes" -- understood in their most generic terms, whether offensive or defensive? The above-mentioned confusion of terminology between static spikes and spiking dynamically is relevant to their consideration. Spikes may emerge and collapse, therefore to be understood as being in a latent or potential state.

Problems as spikes, globally configured as a "crisis of crises": The problems variously recognized by society could be usefully understood as spikes, notably as spiking dynamically -- and even in cycles. These are profiled online in the World Problems Project. A question is how these are best understood as configured globally rather than the subject of focus in isolation one from the other. How are the recognized problems connected -- and who recognizes that connectivity, other than in simplistic reference to "complexity" and "everything is connected to everything"

Strategies as spikes, potentially configured globally: Problems recognized as a threat, potential or otherwise, necessarily evoke strategies which could themselves be usefully understood as spikes, again as potentially activated or deactivated -- even in cycles of sensitivity. Strategies may well be understood as pre-emptive. Such strategies are profiled online in the Global Strategies Project. Again the question is how these are to be understood as configured globally -- exemplifying assumptions and expectations with regard to global governance. Obvious examples are the UN's set of Sustainable Development Goals, as successor to the Millennium Development Goals, and Agenda 21. These frame the obvious question as to what is omitted from such configurations -- indicative of zones of vulnerability in any global configuration.

The configurations envisaged by the United Nations -- spiking successively -- are a reminder that other agencies have envisaged other sets and configurations of strategies. Given the consequent lack of global coordination, is each such set to be understood as a "mega-spike" in its own right, competing with other such spikes?

Somewhat ironically, given a degree of strategic weaponisation, the matter can be explored further in the light of tank warfare. Readily perceived as taking the form of "mobile spikes", tanks offer a means of exploring the role of "think tanks", as discussed separately (Tank Warfare Challenges for Global Governance: extending the "think tank" metaphor to include other cognitive modalities, 2019).

Of some interest is the sense in which a strategic initiative is undermined by "putting a spike in it" (Marc H. Ellis, Put a spike in the wheel of injustice, Mondoweiss, 9 October 2014). A variant is to "put a spoke in one's wheel".

"Diplomatic minefields": The example above of the Hertz horn naval mine is usefully associated with common reference to a "diplomatic minefield" (Tom DiChristopher, Trump’s plan for the Iran nuclear deal runs straight through a diplomatic minefield, CNBC, 10 October 2017; David Maxwell and Mathew Ha, At the Latest Inter-Korean Summit, Kim Jong-un Created a Diplomatic Minefield for the United States, War on the Rocks, 21 October 2018).

From that perspective, any framing of the set of strategies as some kind of global "ocean of initiatives" is then clearly vulnerable to threats from such mines, in whatever patterns they may have been deployed -- or designed to drift randomly.

Human values as spikes, potentially configured globally: Individual values can be recognized as "spiking" dynamically in response to problems and in support of strategies, as experimentally order by the Human Values Project. They can also be understood as configured globally, as with the UN's Universal Declaration of Human Rights. However, as with the sets of strategies, this particular set of aspirational "spikes" can be understood as challenged or matched by others variously recognized (Universal Declaration of Human Responsibilities, 1997; Universal Declaration of Human Rights -- in the light of God's renewed Will, 2004; Universal Declaration of the Rights of Human Organization -- an experimental extension of the Universal Declaration of Human Rights, 1971; Universal Declaration of Responsibilities of Human Intercourse, 2007). Are such sets of spikes then to be recognized as spiking in response to one another -- as complements?

The question could then be asked as to how the spiking of values, individually or collectively can be explored in terms of "hope-mongering" in response to the value negativity recognized as doom-mongering, scare-mongering or fear-mongering? (Credibility Crunch engendered by Hope-mongering, 2008).

Principles, aphorisms, morals and fables as spikes configured globally? Given the argument with respect to values, this might be extended to principles, raising the question as to how many principles are fundamental to the coherence of a society or an individual -- and whether they can be understood as configured globally as "spikes" to safeguard that integrity. Sets of principles are variously articulated and recognized, as with sets of values.

The possibility is further clarified when principles are embodied in aphorisms and fables -- each with a moral. This is true to a degree in the case of sets of traditional fables, as with Aesop's Fables and the Jataka Tales. Arguably these each constitute a pattern of systemic understanding whereby the viability of a collectivity is sustained, as separately discussed (Fabulous Traditions of Managing Psychosocial Change, 2019). The approach has been used from a systems perspective by Russell Ackoff (The Art of Problem Solving: accompanied by Ackoff's Fables, 1978), later complemented by his articulation of a set of problematic f-laws (Management f-Laws: how organizations really work, 2006).

A similar approach, but deliberately paradoxical, is evident in the traditional collection of 48 Zen koans, known as the Mumonkan -- The Gateless Barrier -- and variously interpreted (Configuring a Set of Zen Koan as a Wisdom Container: formatting the Gateless Gate for Twitter, 2012).

Stakes, lines and points: The form of a spike can be readily recognized in reference to "stakes" and stakeholders (Addisu Lashitew, Stakeholder capitalism arrives at Davos, Brookings, 21 January 2020). Strategic agendas are typically articulated in terms of "points", most obviously in the bulleted points defining agreements and constitutions. The coherence of an argument may then be structured and ensured through "point-making". Extensive reference is made to "lines" in that respect (Enabling morphogenesis and transformation through catastrophic questioning, 2013; Engaging with Globality through cognitive lines, circlets, crowns or holes, 2009).

Especially intriguing is the sense in which "making a point" by someone can be experienced as a form of violence, even one which is potentially fatal to another worldview (Pricking the Bubble of Global Complacent Complicity: hyperdimensional insights from the physics of bubble blowing, bursting and collapse? 2017). The process and effect could be explored in terms of asking a "deadly question" undermining the integrity of a worldview (In quest of the most deadly question, 2013; Enabling morphogenesis and transformation through catastrophic questioning, 2013). The latter featured in a discussion with regard to engendering the future (Conferencing as putting identity to the question, 2013).

How might it be useful to recognize the configuration of an array of questions and points protecting the conventions safeguarding the identity of a collective or individual initiative? One possibility is offered by catastrophe theory (Conformality of 7 WH-questions to 7 Elementary Catastrophes: an exploration of potential psychosocial implications, 2006).

Human agencies as spikes: The emergence, deployment and collapse of organizations and corporations could be explored in terms of spiking. Especially interesting for any given country is the configuration of government departments or ministries. Even more intriguing in this respect is the global configuration of UN Specialized Agencies.

For any given government, how many "departments" (as "spikes") are currently held to be required to safeguard the integrity of the nation? Why do some countries have far more than others? How many are held to be required for global governance? By what parameters are the differences justified in systemic terms?

The question then is indeed whether and how they are to be understood as configured globally in response to an array of problems and for the deployment of a global array of strategies.

Human disciplinary specialization understood as a global configuration? Given the constrained manner in which disciplines and methodologies define their field of preoccupation, these too could be considered as spikes. From a historical perspective, or given shifting budgetary priorities, these are given or lose significance -- then to be recognized as spiking.

It is typical for the topics with which disciplines are preoccupied to be ordered in nested hierarchies, as evident in the library organization of knowledge. Efforts have been made to organize the array of subjects as a matrix, potentially inspired by the periodic table of elements, itself the focus of many attempts at non-tabular ordering (Functional Classification in an Integrative Matrix of Human Preoccupations, 1982; Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order, 2009).

In the quest for a higher order of interdisciplinary integration, the possibility of a spherical ordering of topics and disciplines merits consideration -- a "global" organization of topics. So framed, the preoccupation (or relative lack of it) with any given discipline can then be understood as a distinct spike. The emergence or collapse of any one mode of knowing, as a topic or discipline, can then be understood as spiking. This corresponds to a degree to widespread reference to "trending" in social media.

"Spiky / Prickly" personalities: The psychosocial phenomena indicated above can be understood as variously associated with particular personalities, preoccupied with particular sets of problems, strategies, values, and the like. Depending on their attachment to their preferred preoccupation, this can lead to their recognition as "prickly" personalities. This may be deprecated in slang terms as "being a prick" -- readily extended to recognition of a "world of pricks". A fundamental challenge is why a "prick" framed by one person may be perceived by another as heroic -- and exemplary.

This emphasis can be explored otherwise through the sense in which people may be recognized as "upright" (morally or otherwise) -- or may so perceive themselves. This extends into recognition of egotism, if not egomania. This phenomenon is frequently evoked as undermining strategic and other initiatives -- reinforcing a variety of problems.

Of particular relevance is the sense in which the array of people considering themselves "upright" could be fruitfully recognized as configured variously around a globe. From this perspective their different orientations then offers an insight into why their consensus is problematic if not impossible. The global form, as with people on the surface of the Earth, is then an indication of why there is disagreement as to the position of the Sun in the sky. This contrasts with the common assumption that people should "get with the program" or "sing from the same hymn sheet" -- both indicative of a 2D understanding in a context requiring a 3D understanding (at least).

As argued, the prickly response of any individual can be explored through the sense in which each person is potentially "globular" in form, as may well be understood imaginatively (Personal Globalization, 2001).

Such speculation can be taken further, inspired to a degree by the inventive inspiration of Nikola Tesla with respect to positive and negative charges and how they might be fruitfully interrelated with respect to the generation of electrical charges and other effects (as partially illustrated above). Through the manner in which he separated positive and negative, is there a case for imagining those technologies as offering metaphors for the manner in which an "upright" individual identifies with the "positive" and repressive the "negative"?

Especially fruitful to any such speculation is Tesla's remarkable insights into the rotation of electromagnetic fields. In the spirit of technomimicry, the possibilities with respect to a psychosocial analogue merit consideration as separately discussed (Potential implications of alternation and rotation in psychosocial fields. 2014).

Act of God and extraterrestrials: From a legal perspective, there is currently considerable discussion as to how the pandemic may be understood contractually in the light of provisions for an "Act of God" or "force majeure", especially by the insurance industry (Coronavirus Contract Considerations Beyond ‘Act of God’ Issues, Bloomberg, 13 March 2020; Force Majeure in the time of Coronavirus, Chief Executive, 13 March 2020).

Understood as an invasion, variously reframed as a "war", responses to the coronavirus could however also be seen as anticipating the long-envisaged possibility of an invasion by extraterrestrials. Science fiction would have little difficulty in portraying the coronavirus as instigated by extraterrestrials. More intriguing, in the light of the enthusiastic quest of science for extraterrestrial life, is the possibility that such life might well take the form of a virus. Humans returning from Mars might indeed prove to be carriers. Such life might be quite otherwise than optimistically foreseen by science (Sensing Epiterrestrial Intelligence (SETI): Embedding of "extraterrestrials" in episystemic dynamics? 2013).

This suggests that such "unforeseen" events should be more carefully written into contracts, aside from the precautionary and emergency measures that it is only now recognized as requiring attention. Whilst "coronavirus" has currently displaced "climate change" in media and policy discourse, there is a case for exploring the implications of such discourse and coverage if "climate change" were to be substituted for "coronavirus" in policy measures now considered "urgent" -- especially since many of the consequences of climate change will be framed as "Acts of God". Do contractual provisions adequately anticipate the disruptive effect of the arrival of extraterrestrials?

Global spiking of knowledge and ignorance

"Spikes"? The argument frames the question as to whether the patterns of knowledge and/or ignorance can be usefully understood in terms of arrays of spikes -- potentially configured globally in three or more dimensions. Whether ignorance is understood in its problematic sense, or as the "unknowing" highlighted by mystics (as in The Cloud of Unknowing), the possibility would appear to merit particular attention -- as indicated by the recognition increasingly accorded to its significance (Stuart Firestein, Ignorance: how it drives science, 2012; Nicholas Rescher, Ignorance: on the wider implications of deficient knowledge, 2009).

Understanding the "noosphere" as the "universe" of a global-knowledge-based civilization, suggests the provocative possibility of setting any exploration of it within a "university of ignorance", as separately argued (University of Ignorance: engaging with nothing, the unknown, the incomprehensible, and the unsaid, 2013). As one means of gaining understanding of the "shape" of that universe, speculative use was made there of some of the simplest sustainable micro-organisms -- radiolaria (as above) -- given the general systems inspiration regarding isomorphism. The following imagery, with its associated animation, is derived from that exploration.

Indicative representation of a "University of Ignorance"
as a pattern of resonance between cognitive extremes
Using the radiolarian Aulonia hexagona, whose morphology was extensively studied by Ernst Haeckel
(image derived from a study by Christina Brodie, prior to modification for illustrative purposes)
areas as domains of ignorance (dark holes)
connecting network as knowledge (thin light links)
ignorance as a background to a configuration of knowledge
areas as fields of knowledge (light holes)
connecting network as barriers (thin dark links)
knowledge as a background to a configuration of ignorance
University of Ignorance as a pattern of resonance between cognitive extremes University of Ignorance as a pattern of resonance between cognitive extremes
Animation indicating the intermediary conditions in the alternation between the above extremes (tentative)
(click for separate animation [2MB gif] with 64 contrasting images, including the following)
Specific condition of ignorance Specific condition of ignorance Specific condition of ignorance Specific condition of ignorance
Images for the animation were generated using filter effect options of Photoshop to modify the 2 extreme images,
thereby suggesting an arrray of contrasting relationships between knowledge and ignorance
(NB: Such an animation could benefit from greater aesthetic skills with respect to composition and pace)

"Holes"? The geometry of radiolarians has been an inspiration to digital artists, such as those associated with the journal Vague Terrain, including the following image, for which a commentary is provided by the author.

Digitally generated "radiolarian"
Digitally generated "radiolarian"
Image published by Robert Hodgin, Radiolaria Studies, Vague Terrain, 31 July 2009)

The complementary contrast to "spikes" can be seen as the presence of "holes", notably recognized in terms of the gaps in any pattern of knowledge. Of relevance is the Cognitive mystery of holes, lacunae and incompleteness, as discussed separately (Is the World View of a Holy Father Necessarily Full of Holes? Mysterious theological black holes engendering global crises, 2014). In referring to necessary incompleteness, that discussion highlighted the insights of deliberately omitting, or unconsciously missing, a dimension essential to systemic viability in the light of the work of the biological anthropologist Terrence Deacon (Incomplete Nature: how mind emerged from matter, 2012).

A fruitful question relates to the capacity of radiolaria in engendering such forms -- and their necessary sustainability, even at that scale. Could the organization and "shape" of global civilization benefit from such systemically sensitive design? More provocative is the sense in which such "unseen" organisms (of which few have heard) imply a configuration of silences. In terms of the mystical tradition of Buddhism, this offers an image of the "emptiness of form".

With respect to coronavirus, Farhad Manjoo focuses on Coronavirus Is What You Get When You Ignore Science (The New York Times, 4 March 2020). Amy Lauren Fairchild focuses on Science Can’t Save Us From Coronavirus Panic -- But Trump’s Information Crackdown Can Certainly Make Things Worse (Foreign Affairs, 10 March 2020). The difficulty is that neither "panic" nor "Trump" are topics on which "science" claims to offer any insight -- or is seemingly qualified to do so as its mandate and disciplines as conventionally defined. With respect to what is "missing" (in the light of Deacon), the point can be argued in greater detail (Knowledge Processes Neglected by Science: insights from the crisis of science and belief, 2012; Are Environmentalists and Climate Scientists in Denial? Climate change recognized as primarily a psychological challenge, 2019)

Metaphor? As noted by Richard Ennals and Philip Molyneux (Managing with Information Technology, 2012), the metaphor of sun and corona was coined by Howard Rosenbrock (see Engineering as Art in Artificial Intelligence, Culture and Language: On Education and Work, 1990). In relation to creativity and tacit knowledge, Rosenbrock used it to to explain the idea of human-machine symbiosis:

As objective knowledge is used in practice, it is seen as adding to the tacit dimension of human knowledge. Similarly, as tacit knowledge is practised, it becomes accessible and explicable and thus adds to the objective part of the knowledge. In other words, both objective and tacit knowledge increase through practice. This means that tacit knowledge will always exist so long as the objective knowledge exists; in other words, tacit knowledge can never be fully explicated. The sun metaphor signifies objective knowledge, and the corona metaphor signifies tacit knowledge. Just as the sun cannot exist without its corona, so objective knowledge cannot exist without its complementary tacit knowledge. (Managing with Information Technology, 2012)

Given reference above to a 3D variant of the sunflower as a mode of organization of spikes based on a Fibonacci spiral, of some relevance is detailed discussion of it separately (Adaptive Hypercycle of Sustainable Psychosocial Self-organization: designing a mapping of a Chinese metaphorical pattern language, 2010). This includes sections on:

Global connectivity as reinforced through symbolic configuration?

Comprehension of order: This argument could be reframed by the question as to where the perception of patterns of order can be understood as "coming from". Mention was made of the study by George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). The question is central to the study Jeremy Lent (The Patterning Instinct: a cultural history of man's search for meaning, 2017), as critically reviewed (Patterning Intuition with the Fifth Discipline, 2019).

Subsequent to his work on pattern language, the question was explored otherwise in a 4-volume study by Christopher Alexander (The Nature of Order: An Essay on the Art of Building and the Nature of the Universe, 2003-2004). This framed conclusions on New Concepts in Complexity Theory (2003) and on Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole (International Journal for Unconventional Computing (IJUC), 5, 2009). These evoked a review of the implications, as separately discussed (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).

Especially intriguing is the widespread use of circular symbols as integrative of the integration of governance which is proving so elusive, whether at the global or local levels. Thus the UN and its agencies, as well as other international bodies -- governmental and nongovernmental -- have recourse to logos in 2D based on the circle. It could be asked, however naively, how global integration is to be imagined and reinforced through a 2D symbol when it is readily comprehensible that a symbol in 3D (at least) would be more appropriate. The disconnect between 2D imagination and 3D is arguably at the root of the disconnect between a Flat Earth (local) view of the world and one which acknowledges its (global) 3-dimensionality (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008).

Can the coronavirus indeed be fruitfully understood as a symbol? However, beyond any notion of a symbol as a "conceptual bauble", how is it to be understood as cognitively embodied -- whether collectively or individually? The irony of that question is that as a virus, now in a mode of pandemic propagation, it is indeed "embodied" physically by individuals and society.

Understood otherwise, as argued here, the question is how is it embodied cognitively and how might that question be most appropriately framed? One pointer is offered by George Lakoff and Mark Johnson (Philosophy In The Flesh: the embodied mind and its challenge to Western thought, 1999). Johnson has explored the possibilities dynamically, namely as embodied in movement (The Meaning of the Body: Aesthetics of Human Understanding, 2007), as with Maxine Sheets-Johnstone (The Primacy of Movement, 1999), as discussed separately (Reintegration of a Remaindered World: cognitive recycling of objects of systemic neglect, 2011).

Crowning symbols of governance? With respect to the symbolism of governance, at least locally and nationally -- as well as from an imperial perspective -- it is intriguing to note the importance traditionally attached to the sceptre, orb, and crown. It is less than clear, in the language of today, to which cognitive and operational functions of governance these correspond.

Nevertheless, and intriguingly, widespread use is made of the sceptre or mace as an indication of authority (Embodying the essence of governance in ritual dynamics with mace, sceptre, fasces or vajra? 2019). A crown continues to be used in monarchies and by some religious hierarchies -- although variants are clearly valued in crowning the champions of sport and other disciplines.

The value of the orb, which might otherwise be understood to correspond most closely to any sense of globe, is less obvious -- other than through papal declarations on solemn occasions (Urbi et Orbi). Suspiciously however, it is useful to explore how each of these symbols has been effectively weaponised, with the orb then to be recognized as a bomb -- a weapon of mass destruction?

Corona in religious iconography: One embodiment of the corona is widely evident in depictions of a halo (nimbus or aureole) around the head of a person deemed holy, namely a crown of light rays, circle or disk of light. This recognition may be extended into the sense of an aura -- a human energy field -- more probably described in terms of charisma.

New acknowledgement of such a phenomenon is evident in the increasing recognition of the reality distortion field of certain leaders and entrepreneurs. Similarly unrepresented (except sarcastically) as a halo, this may be more conventionally articulated as charisma or presence. Much diluted variants are evident in recognition and cultivation of "image" and "look". Understood quite otherwise, for many the personal experience of a corona might be in terms of existential confusion, panic and secrecy.

Democratic implications of "corona"? It could be recognized as both ironic (especially at this time) and symbolic in its own right that what is considered the most famous judicial oration, On the Crown, was delivered by the prominent Athenian statesman and orator Demosthenes in 330 BC. His remarkable rhetorical skill has been the focus of admiration down the centuries, and continues to be the subject of commentary and analysis. Recognized as one of the most splendid political pleas ever written, it is upheld as one of the finest achievements of Greek prose. The speech made an immediate impression on contemporary Greeks and for centuries served the writers and speakers of antiquity as the primary model of forceful argument and vigorous style (Harvey Yunis, Demosthenes: De Corona, 2001; Thomas Leland's comments and translation of the oration On the Crown).

Of relevance to this argument, having been presented in the final days of liberty in classical Greece, as noted by George Cawkwell (The Crowning of Demosthenes, The Classical World, 19, 1969, 1):

In the course of Demosthenes' lifetime, indeed within a mere decade, the whole balance of power in the Greek world was destroyed. By 338 the city states were completely overshadowed by the national state of Macedon, and it is the concern of all students of Demosthenes to analyse this dramatic change.... It is on the Athenian orators that we have to rely, the very men most concerned in the politics of Athens, in the act of glossing over and denying their own share in the disaster and of misrepresenting that of their opponents.

Of particular relevance to the current crisis of global governance is the capacity of rhetoric to organize persuasive argument appealing to a diverse audience. As noted by Cecil Wooten (The Nature of Form in Demosthenes' "de Corona", The Classical World, 72, 1979, 6)

One of the problems with which classical rhetorical theory does not deal in much detail is the question of the various ways in which an orator can hold a speech together. In a short speech, this is not usually a crucial problem since the parts of the model speech are a clear indication of where the speech is going. In a long speech, however, this informing principle is not sufficient by itself since the parts become too large to be manageable and are not always clearly recognizable by the audience.

It might be further argued that the most persuasive orators of today tend to be those with least to say, whereas those with most insight to contribute lack the skills with which to present their case effectively (whatever that may be held to mean). Wooten argues that the much-cited insights of Kenneth Burke into rhetorical form can be used to explain why the discourse of Demosthenes is so effective in giving coherence to the whole.

As clarified by Dave Rick (Burke’s Counter-Statement, The World's Last Mysteries, 10 February 2015), in Counter-Statement (1931/1968) Burke offers up an extensive discussion of artistic form, striking key distinctions between information and the form in which that information is produced. This focus on form recalls both the seminal approach of Christopher Alexander (Notes on the Synthesis of Form, 1964) and the case made for the United Nations University by Johan Galtung (Forms of Presentation: a forgotten aspect of social science epistemology, Goals, Processes and Indicators of Development, 1980). The latter gave rise to further consideration of the matter (Forms of Presentation and the Future of Comprehension, 1984).

Given the early focus of Demosthenes in the cradle of democracy, it is especially intriguing to note that "corona" came to be associated in imperial Rome with the "ring of onlookers" or bystanders around a debating arena -- as well as with the garland accorded to any champion:

Trials took place in the Forum in the open air and could easily attract the attention of the crowd; the ring of casual spectators (quaintly called the corona or "garland") was a regular feature of Roman trials, and advocates found it as important to gain their favour as much as to gain that of the jury (Jonathan Powell and Jeremy Paterson, Cicero the Advocate)

There is therefore a highly suggestive association between conventional understandings of crown and that engaging an observing population -- one which has seemingly not been explored, despite its significance for democracy and the disconnect of any democratic deficit. The etymology of corona does not include that fruitful early ambiguity. The commentary above on Cicero of Rome, with its multiple mentions of "corona", does however include reference to the De Corona of Demosthenes -- offering a degree of llinkage between the two understandings.

Does the neglected duality of the connotation of the crown -- so conventionally restricted otherwise to the highest authority -- emphasize the sense in which the extremes of the corona-as-crown and corona-as-encircling-onlookers is better understood in terms of a field effect -- of which a "reality distortion field" offers one insight? As concentric rings, the two even recall the insights of Nikola Tesla regarding the power of rotating electromagnetic fields.

Pentadic analysis: The desirable engagement of any ring of auditors -- the audience -- is an explicit focus of Kenneth Burke, although seemingly unrelated (except by implication) to the theme of On the Crown by Demosthenes, however much it was a feature of the latter's skill (as implied by Cecil Wooten). Burke's insight merits consideration in relation to the variety of figures of speech which enable that engagement (Questionable Classification of Figures of Speech -- as fundamental to the need for powerful rhetoric in governance, 2016).

For Rick:

In defining form, Burke writes, Form is the creation of an appetite in the mind of the auditor, and the adequate satisfying of that appetite (31). He further argues that form orients to the experience of the audience, not the subject (32). Thus, he defines “artistic felicity” as the correct use of form upon the audience, and eloquence is when the form used elevates the information conveyed (37)....

Burke offers the definition that Form in literature is an arousing and fulfillment of desires (124). He then divides form into five aspects, including syllogistic and qualitative progression, repetitive form, conventional forms, and minor or incidental forms (124). Syllogistic progression refers to progression directly along a logical form, such that A to E entails steps B, C, and D (124). Qualitative progression is more subtle, where a quality present makes another quality “appropriately follow” (125). Repetitive form is the consistent maintaining of a principle under new guises (125). Conventional form is perhaps the most familiar, a staple of most genre expectations, and refers to “when a form appeals as form” (126). As to minor or incidental forms, they encompass various literary and verbal devices (127). This breakdown of form offers a useful method to consider the ways that form manifests in narrative.

Later in his lexicon, Burke turns to discussing “patterns of experience,” which he breaks down into: universal experience, or those emotions and states of being that are common to human beings (149); modes of experience, or the circumstances that give rise to universal experiences (150); and patterns of experience, when modes of experience are repeated and give rise to symbols (152). Here Burke enters into a thorough discussion of the appeals of symbols, fitting very neatly into the “interests” presented in The Rhetoric of Fiction presented by Wayne Booth.

Burke's framework has resulted in various commentaries on pentadic analysis, notably as offering a means of transcending a number of binary discussions, thus providing interesting perspectives for research and education (e.g. as a reflection tool):

The work of Burke is noted as responding to the developing insights of cybernetics (Jeff Pruchnic, Rhetoric, Cybernetics, and the Work of the Body in Burke's Body of Work, Rhetoric Review, 25, 2006, 3). Seemingly as yet to be explored are its later relevance to oppositional geometry and the associated logical implications in discourse (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018).

Confluence of geometries: As noted above, there is some indication that the protein spikes on a coronavirus may average at 74. For the purpose of this speculative exercise this was assumed to be 72, given the traditional importance attached to patterns of coherence associated with that number. The relevance of such a focus derives from a variety of constraints on human comprehension and memorability, as discussed separately (Comprehension of Numbers Challenging Global Civilization, 2014). Further exploration is justified by the evident failure of conventional approaches to global organization of connectivity, as separately argued (Time for Provocative Mnemonic Aids to Systemic Connectivity? Possibilities of reconciling the "headless hearts" to the "heartless heads", 2018).

In this spirit there is a curious confluence between the considerations of sacred geometry and the orbital geometry of a constellation of satellites -- given the focus on the 72 orbital planes of the StarLink configuration (each with 22 satellites?). Religious connotations have been exploited through reference to "ANGELS" (David Shiga, ANGELS to watch over US air force satellites, New Scientist, 4 August 2006). The following was presented in an earlier discussion of Unexamined symbolic implications of disrupted star light? (2019)

A provocative checklist by Wikileaks on the number 72, offers 22 references to its relevance in religious and other frameworks including:

Any such pattern of coherence lends itself to further speculative exploration (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016). The following was developed from an argument there regarding an experiment in Hyperbolic reframing of the Demonique and Angelique of tradition. This has the merit of reflecting the qualitative distinctions in a pattern of 72, whereas the global configurations of spikes above only do so in terms of their geometrical orientation around the sphere.

Indication in 2D of the dynamic nature of a "hyperdimensional" crown-corona
Alternative experimental configurations alternating between the 72 angels and demons
Animation of 8 sets of 9
(enlargements for detail: angels / demons)
Animation of 9 sets of 8
(enlargements for detail: angels / demons)
Experimental configuration alternating between the 72 angels and demons Experimental configuration alternating between the 72 angels and demons
The allocation of sets to the star "tables" in the above schematics is based on the tabular form in which the 72 angels and demons are typically presented. The rows are presented "around the tables" in one schematic, and the columns are presented "around the tables" in the other. The sequence around the tables is questionable, demanding further consideration.

Such patterns can be understood less controversially if the "demons" are recognized in terms of the protein spikes of a coronavirus -- and the "angels" as the set of proactive responses to them -- possibly as a global set of values or strategies. The dynamic in 2D, is then suggestive of how any integrative crown is necessarily to be understood as other than in the statically as favoured by convention (Implication of Toroidal Transformation of the Crown of Thorns: design challenge to enable integrative comprehension of global dynamics, 2011). As potentially to be understood in 3D, the following animations are indicative, reproduced from Modifying the cyclic symmetry of the star torus (2019).

Illustrative use of geometry of star torus for mapping purposes
(use browser facilities to enlarge animations and labelling)
5-fold 8-fold 12-fold 16-fold
Pattern of 5-fold cyclic symmetry of star torus with WuXing labels Pattern of 8-fold cyclic symmetry of star torus with BaGua labels Pattern of 12-fold cyclic symmetry of star torus with information functions Pattern of 16-fold cyclic symmetry of star torus with UN SDG goals
Animations generated with Stella Polyhedron Navigator

The above experiments were evoked by the preoccupation with 16(+1) Sustainable Development Goals of the United Nations. Why 16(+1), especially given the lack of indications both as to how these are interrelated in systemic terms and how the wider world -- the "corona" (?) -- is to be expected to comprehend them as the memorable keys to global governance at this time? The question can be articulated in other terms (Interplay of Sustainable Development Goals through Rubik Cube Variations: engaging otherwise with what people find meaningful, 2017).

As with the spiked global forms depicted above from many domains, of particular interest is why the number of spikes of very different orders has emerged through evolutionary processes. What parameters define the adequacy in a given case -- the inadequacy of a lower number and the unnecessary dependence on any higher number? Why 74 in the case of the coronavirus, if that is indeed the case? Why do the Sustainable Development Goals not number 72, for example -- as discussed more generally (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980)?

A quite distinctive key to such memorability, and the associated sense of global coherence, is suggested by music, as argued separately (A Singable Earth Charter, EU Constitution or Global Ethic? 2006). Any further exploration of patterns in terms of meaning in the light of such configurations would benefit from consideration of the work of musicologist Ernest McClain (The Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato, 1978; Meditations Through the Quran: tonal images in an oral culture, 1981).

Despite the complexity of a 72-fold pattern, McClain's approach to musicality (through the simplicity of multiple exponents of prime numbers) offers one clarification of the mnemonic power of such a 72-fold configuration as 23 x 32 -- and with 360 degrees as 23 x 32 x 5.

Especially intriguing is the possibility that new forms of relationship may be found between complex metaphorical patterns, such as those of the I Ching and the Kama Sutra. The discovery of fullerenes ("bucky balls") as a new form of carbon, based on the spherical configuration of 60 carbon atoms, is indicative of such possibility, as separately discussed (Understanding Sustainable Dialogue: the Secret within Bucky's Ball? 1996).

The configuration below (left) of the 64 hexagrams of the I Ching by József Drasny in his Yi-globe is suggestive in this respect, as argued separately (Triangulation of Incommensurable Concepts for Global Configuration, 2011; In Quest of a Dynamic Pattern of Transformations, 2012). In contrast with the 72-fold pattern of 8 x 9, this is a pattern of 8 x 8, namely 26. That on the right (below) is derived from a separate discussion (Symbolizing Collective Remembering Otherwise: encompassing the "headless hearts" and "heartless heads" through their dynamic entanglement, 2018).

How is the coherence of the 8 x 8 pattern, so esteemed by Chinese culture, to be compared with that of 8 x 9, or one of 9 x 9, as explored by Chinese culture (9-fold Magic Square Pattern of Tao Te Ching Insights -- experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006)? In contrast with any geometrical configuration, irrespective of its conformity to sacred geometry, the traditional 64-fold, 72-fold and 81-fold patterns distinguish the systemic function of their elements in some detail enabling a degree of engagement which is far from evident in other cases.

A potential potential exception is offered by the periodic table (D. H. Rouvray and R. Bruce King, The Mathematics of the Periodic Table, 2005; Wilmer Leal and Guillermo Restrepo, Formal Structure of Periodic System of Elements, Max-Planck-Institut für Mathematikin den Naturwissenschaften, 2019). The latter generalizes the periodic system as a set endowed with a system of similarity classes, whose elements hold an order relation; this structure corresponds to an ordered hypergraph, where similarity classes are hyperedges. The insight can be crudely explored in relation to traditional patterns (Hypergraph Experiment with I Ching, 2006).

Selected images of the Yi-globe of József Drasny

Animation of single and complementary Lissajous curves on horn torus
Yi-globe of József Drasny Yi-globe of József Drasny Animation of Lissajous curve on horn torus Animation of two complementary Lissajous curves on horn torus
Reproduced with permission from József Drasny
(The Image of the Cosmos in the I Ching: the Yi-globe, 2007)
Animation on left reproduced, with permission, from Wolfgang Daeumler (Horn Torus);
that on the right is a simplistic adaptation thereof

These images point to the possibility of a correspondence between the spherical organization of conditions of change and the more familiar understanding of how the Earth, as a globe, is exposed to light and darkness


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