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19 September 2022 | Draft

Cognitive Embodiment of Patterns of Governance of Higher Order

Memorable navigation of viable global pathways from 4-fold to 64-fold and beyond

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Introduction
Cognitive analogues of mechanical transmission systems?
Fundamental pattern of order in space-time
Configurations of polyhedra as transmission systems
Alternative configurations of polyhedral transmission systems
Vertical arrays of polyhedra implying higher degrees of order
Nested and cage-like polyhedral configurations implying degrees of implicate order
Implicate order through hypercube and drilled truncated cube?
Polyhedral representation of Sustainable Development Goals including "Own Goals"?
Dynamics of systemic connectivity as a challenge to invariance
Toroidal polyhedral ring arrays
Strategic implications of traditional angelic and demonic configurations?
Variable geometry of institutions as alternation between polyhedral patterns
References


Produced, coincidentally, on the occasion of the 7th World Congress on the Square of Opposition (Leuven, September 2022)


Introduction

Little needs to be said of the chaotic state of global governance and the crises with which it is faced at this time. It therefore continues to be useful to question the methods advocated and deployed to address the challenges as they are currently framed. It is necessarily perceived as presumptuous to assert the merit of alternatives, however vigorously their merits are described. The arguments against any alternatives are themselves vigorously articulated -- with little ability to transcend the dysfunctional consequences of this dynamic.

The challenges of the times are curiously exemplified by a strangely unquestioned relation to numbers (Comprehension of Numbers Challenging Global Civilization, 2014). History may see it as bizarre that a global civilization should be so preoccupied with 1.5 -- being, however coincidentally, both the key to the response to the pandemic through social distancing, and as the target cap on global warming (Humanity's Magic Number as 1.5? Dimensionless constant governing civilization and its potential collapse, 2020). Claire Lemercieris and Claire Zalc ask whether history is a matter of individual agency and action, or of finding and quantifying underpinning structures and patterns (History by Numbers, Aeon/Psyche, 2 September 2022). That argument cites the curious historical focus of economists on the productivity engendered in slave populations in the USA by an average of 0.7 whippings per hand per year.

With respect to assumptions of the vital need for economic growth, a limited set of statistics are a preoccupation for governance, including interest rates, inflation rates, tax rates, exchange rates -- and rates of growth, as indicated by GDP. The tendency is an invitation to satire, given the neglect of other indicators -- especially in the light of the current proxy war with Russia (Evaluating the Grossness of Gross Domestic Product: Refugees Per Kiloton (RPK) as a missing indicator? 2016). As illustrated by past military activity in the Afghanistan arena, the focus is indeed on performance with little attention to remedial capacity (Remedial Capacity Indicators versus Performance Indicators, 1981; Transforming the Unsustainable Cost of General Education: strategic insights from Afghanistan, 2009). There is general resignation to the extraordinary level of national debt of many leading countries, and its continuing increase -- a "black hole" of astronomical proportions.

Many global statistical reports are indeed produced offering insight into performance, but with little insight into remedial capacity over time (Dynamic Transformation of Static Reporting of Global Processes, 2013). The foreseeable consequences of increasing population growth are assiduously avoided -- incommunicable as a fundamentally inconvenient truth (Institutionalized Shunning of Overpopulation Challenge, 2008). Curiously there is a strange global dependence on obscure metrics of proven fallibility (Uncritical Strategic Dependence on Little-known Metrics: the Gaussian Copula, the Kaya Identity, and what else? 2009).

Potentially even more curious for the future is what might otherwise be framed as simple superstition, namely an unquestioned global preference for strategies articulated by particular numbers: 7, 8, 10, 14,  20 or 30, exemplified by the 12-fold (Checklist of 12-fold Principles, Plans, Symbols and Concepts, 2011). There is no systemic explanation for the 17-fold articulation of the Sustainable Development Goals of the United Nations (Systemic Coherence of the UN's 17 SDGs as a Global Dream, 2021).

In this context there is therefore a case for continuing to speculate on distinctive ways of ordering information about the current global condition. The aim in what follows is to elicit imaginative responses from a perspective which contrasts with the conventional articulation of proposals in bullet-pointed text. The weakness of such articulations is evident from the lack of relationship between the points made in any such checklist -- undermining any systemic implications and relevance. That pattern of connectivity is of a very low order. This ensures that the memorability of such a checklist is itself of a very low order and is effectively a guarantee that any acclaimed institutional "oversight" is associated with "blindspots" -- as ironically implied by the alternative understanding of "oversight".

For example, who is able to recall all 17 of the UN's Sustainable Development Goals (let alone its 169 tasks)? Who has any sense of the manner in which they interact with each other together to ensure their systemic viability? Is this equally true of the rights articulated in the UN's Universal Declaration of Human Rights?

In a world in which great emphasis is placed on the attractiveness and interest of visual representations and music -- with which people are indeed happy to engage -- why is so much reliance placed on text in the processes of governance? This occurs in a period in which governance is increasingly experienced as alienating, opaque, and untrustworthy -- and only too evidently incompetent. The contrast with engagement with song is remarkable, as has been appreciated by some autocrats (A Singable Earth Charter, EU Constitution or Global Ethic? 2006).

What follows is a speculative quest for distinctive patterns of order which have a possibility of eliciting imaginative responses -- at least for some. It follows from a case previously made (Time for Provocative Mnemonic Aids to Systemic Connectivity? 2018). The challenge is especially evident in the case of the problematic articulation of the core values of a civilization aspiring to democracy -- and faced with its obvious inadequacies (Values, Virtues and Sins of a Viable Democratic Civilization, 2022). The focus is on the related patterns of numbers which are widely upheld as of fundamental significance to the organization of memory, especially computer memory, following their earlier consideration (Cognitive organization by polyhedra of 16, 32 and 64 vertices, 2017; Polyhedral Pattern Language, 2008)

The approach explored is notably inspired by the visual representation by Keith Critchlow (Order in Space: a design source book, 1969). This offered a coherent visual insight into the closest packing of the 13 disparate, semi-regular Archimedean polyhedral forms -- in contrast with the 5 more familiar Platonic forms recognized for their spherical regularity, and widely valued for symbolic purposes. Critchlow later adapted this geometrical insight into the organization of flowers (The Hidden Geometry of Flowers: living rhythms form and number, 2011). Such insights focus the question of the "closest packing" of disparate information essential to its memorability.

The Archimedean forms have acquired particular significance in oppositional logic, namely the discipline which focuses on the organization of logical relations as they feature in discourse and in computerized search algorithms. Unfortunately the coherent representation of these logical operations with polyhedra has as yet translated only to the most limited degree into relevance to the challenges of governance, as indicated separately (Oppositional Logic as Comprehensible Key to Sustainable Democracy, 2018). Ironically, opposition to wider application of those insights has not evoked any form of self-reflective inquiry by that discipline.

The exploration in what follows is a quest for the unusual and the extraordinary in the configuration of information. It is not envisaged as a specific explanation or proposal but rather as a challenge to the imagination -- to the possibility and sense of "what if". It can be argued that we unconsciously inhabit what amounts to "cognitive glass houses". The question is the degree to which these are usefully recognized as "conceptual cages" within which we are effectively "incarcerated" -- whether as greenhouses or with glass ceilings. The edges of polyhedra are usefully suggestive of the bars of such cages, irrespective of their more valuable connotations.

If there is a need for "cages" of previously unimagined design -- as might well be easily enabled for intelligent primates in zoos (Primate Environmental Enrichment: automated reconfiguration of zoo enclosures, 2011). Changing metaphor, the argument is illustrated by the widespread familiarity with gear change mechanisms on bicycles and in automobile transmission systems. Under what circumstances do we need to "change gear" conceptually -- and how many gears might be appropriate to experience a higher degree of freedom? This frames the question of the need for conceptual "cage changing mechanisms" in 3D and 4D -- and their possibility in practice.

Symbolically it might be asked what relationship this might bear to Jacob's ladder as a linear inspiration for the Abrahamic religions. This suggests the possibility of a non-linear "cognitive ladder", as with the inspiration offered by the coiling of DNA, namely the pattern so fundamental to biological life (Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007). To the extent that it is widely appreciated that the house of divinity "has many mansions", this invites recognition of the possibility that it also has many "stairwells". The dynamic can be explored as the quest for alternative and more integrative paradigms (Navigating Alternative Conceptual Realities, 2002; Combining Clues to 'Ascent' and 'Escape', 2002).

The argument concludes with a presentation of the relationship between the hypercube (of significance to oppositional logic) and the 64-edged drilled truncated cube. The role of this polyhedron can be understood as a form of Rosetta Stone offering a means of reconciling a range of patterns variously upheld as of strategic significance: 4-fold, 6-fold, 7-fold, 8-fold, 9-fold, 12-fold, 14-fold, 16-fold, 20-fold, 32-fold and 64-fold. The argument is extended to a 72-fold pattern of traditional significance.

For a civilization faced with an unprecedented global migration crisis, this migration process could be understood as disguising the degree to which there is effectively a mass migration of humanity to a new cognitive frontier -- meriting a degree of recognition, as separately argued (Future Global Exodus to the Metasphere, 2022).

Cognitive analogues of mechanical transmission systems?

Inspiration is increasingly sought for technological innovation through the insights offered by biomimicry and biomimetics -- most obviously with respect to airplane design and flight. Inspiration can be similarly found for cognitive innovation from both biomimicry and technomimicry (Technomimicry as Analogous to Biomimicry, 2011), as illustrated from the process of helicopter development (Engendering a Psychopter through Biomimicry and Technomimicry, 2011). Potentially of greater relevance to governance are the insights to be drawn from the work of Nikola Tesla, most notably with respect to the alternation and rotation of magnetic fields (Reimagining Tesla's Creativity through Technomimicry, 2014).

There is widespread familiarity with the transmission systems vital to the operation of automobiles and bicyles. The process of changing gear -- and the need to do so -- calls for little commentary with regard to movement over terrain presenting different challenges. Far less evident is the cognitive equivalent requiring a shift from one mode of organization to another in response to "cognitive terrain" of different degrees of complexity.

Whilst familiar, the process of changing gear in an automobile is less evident in terms of the mechanical geometry of the invisible transmission system (or "gear box"). This can be surprisingly complex in the case of heavy duty trucks (see gift shift pattern, below left).

By contrast, the mechanism in the case of bicycles is especially visible (see below). As expressed by Wikipedia:

Bicycle gearing is the aspect of a bicycle drivetrain that determines the relation between the cadence, the rate at which the rider pedals, and the rate at which the drive wheel turns. On some bicycles there is only one gear and, therefore, the gear ratio is fixed, but most modern bicycles have multiple gears and thus multiple gear ratios. A shifting mechanism allows selection of the appropriate gear ratio for efficiency or comfort under the prevailing circumstances: for example, it may be comfortable to use a high gear when cycling downhill, a medium gear when cycling on a flat road, and a low gear when cycling uphill. Different gear ratios and gear ranges are appropriate for different people and styles of cycling....

On single-speed bicycles and multi-speed bicycles using derailleur gears, the gear ratio depends on the ratio of the number of teeth on the crankset to the number of teeth on the rear sprocket (cogset).... On a bicycle, the cassette or cluster is the set of multiple sprockets that attaches to the hub on the rear wheel. A cogset works with a rear derailleur to provide multiple gear ratios to the rider. 

Mechanical metaphors indicative of the nature and possibility of conceptual gear shifting
As suggested by a gear shift pattern in trucks A Shimano XT rear derailleur on a mountain bike 10-speed bicycle cassette
Example of gear shift pattern in trucks Shimano XT rear derailleur on a mountain bike 10-speed bicycle cassette
  C. Corleis, CC BY-SA 3.0, via Wikimedia Commons Sam Sailor, CC BY-SA 4.0, via Wikimedia Commons

Skilled cyclists and car drivers have little difficulty in appreciating that there is an art to changing gear -- and choosing into which gear to shift according to circumstances. As a metaphor, the ability to "change gear" is recognized in rhetoric and in the deployment of complementary strategies. Far less evident is the number of gears between which a choice can be made -- and how and when to "shift gear". Some consideration of the opportunity is evident in discussion of variable institutional geometry (see below).

The configuration of transmission systems merits a degree of comparison with the tradition of basket weaving and its symbolis (The Future of Comprehension: conceptual birdcages and functional basket-weaving, 1980). Ironically the relevance of such a metaphor could be seen in reference to "basket" with respect to the major Conference on Security and Cooperation in Europe as a key element of the détente process during the Cold War. This was organized into three "baskets" of issues (Findings Eleven Years After Helsinki -- Implementation of the Final Act: Basket I, 1987; Basket II, 1987; Basket III, 1987). Use of that insightful metaphor did not however extend to the manner in which the baskets were "woven". It is such relationships which are integral to insight into systemic viability (Interweaving Thematic Threads and Learning Pathways, 2010; E. Ronner, et al, Basket of Options: unpacking the concept, 7 July 2021).

Fundamental pattern of order in space-time

Framed by the potential of technomicry, the question is how to derive cognitive insights from understandings of order in space.

As noted above, the inspiration for the exploration below derives from the early work of Keith Critchlow presented in conventional text form with a static two-dimensional image (below left). This has invited adaptation of that representation into the dynamic three-dimensional visualization enabled by web technology (discussed further below). Critchlow's initiative has inspired other extensions (Robert C. Meurant, A New Order in Space: aspects of a threefold ordering of the fundamental symmetries of empirical space, as evidence in the Platonic and Archimedean polyhedra, International Journal of Space Structures, 6, 1991, 1)

Transformational "route maps"? The relation between the articulations of insights suggestively mapped onto  polyhedra offers the further sense that a "cognitive toolkit" could well vary in scope -- in the number of "tools" held to be relevant to a particular set of circumstances. This is suggested by a kind of "route map", discussed separately and reproduced below right (Pathway "route maps" of potential psychosocial transformation? 2015). As indicated there, this might be understood as Polyhedral meta-patterns of relationships? (2015).

Alternative schematic relationships between 12 Archimedean polyhedra
Closest packing configuration of polyhedra by Critchlow
(enhanced with arrow animation indicating transformations)
Conway relational chart
Showing 12 polyhedral forms created by 3 symmetry-preserving operations on the cube
Distinctive relationships pathways between spherically symmetrical polyhedra
Closest packing configuration of polyhedra Conway relational chart for polyhedra Route maps of psychosocial life suggested bysymmetrical  polyhedra
Engaging with Globality through Dynamic Complexity, 2009 Tomruen at English Wikipedia, Public domain, via Wikimedia Commons F=faces, E=edges, V=vertices
(total of these in parenthesis)

The emphasis of Critchlow's representation is on the geometry of the closest packing of 12 spheres -- clearly with implications for the packaging industry. Given that the semi-regular Archimedean polyhedra are recognized for a degree of spherical symmetry, Critchlow was also able to indicate how 12 of those forms could be arrayed together in a manner respectful of their geometrical relationship.

Critchlow's 12-fold geometrical array of disparate forms around a 13th, calls for recognition of the systemic connectivity implied by the geometry of that configuration, partially suggested by an animation of the connections in his original map (above, left). This can be explored in terms of the geometrical attributes which suggest ways -- pathways -- in which one geometrical form can be transformed into another. This is suggested by the two-dimensional map on the right above.

Method of loci and memorability of the succinct: Curiously "closest packing" invites recognition in terms of mnemonics and memorabilty, as suggested above. It could be argued that the art of memory lies in techniques whereby an array of disparate elements are held together. The principle has been extensively described as the traditional method of loci (Frances Yates, The Art of Memory, 1966). How are disparate elements "packed" compactly into memory -- enabling their succinct expression? As presented in what follows, the features of polyhedra offer "loci" with which "memories" can be variously associated.

Obvious examples include names of relatives and friends, celebrities, events, and songs. Geography poses a challenge with respect to towns, countries, rivers, mountains, and seas. Biology frames the challenge in terms of animal and plant species. Astronomy presents the challenge in terms of planets, stars and galaxies. The geograpic example clarifies a further issue, namely how to get from one place to another through s network of roadways. The challenge of remembering relationships is variously evident in the other cases.

A question that would seem to be avoided is the manner in which people remember the details within the domains with which they are especially preoccupied. Are significant proportions of the population effectively "list averse", namely challenged to remember details which take list form? What alternatives do they cultivate in preference to lists -- most notably in traditional oral cultures? How indeed do they organize and remember the relationships between relatives, celebrities, and the like?

Does any significant coherence of the complexities of civilization correspond in any way to the so-called islands of stability discovered by nuclear physicists among the vast array of isotopes of chemical elements?

Symmetry-preserving operations: Of potential relevance to development of this theme is the formal study of transformations between polyhedral forms, as succinctly indicated in the central image above regarding the Conway polyhedron notation, and discussed separately (Topological operations on polyhedra as indicative of cognitive operations, 2021).. This is understood as constrained by symmetry preservation, as separately discussed in considering the encoding of coherent topic transformation in global dialogue (2021):

"Ways of looking"? The current strategic challenge could be framed otherwise by asking how many "ways of looking" are potentially relevant to creative recognition of any remedial response. The phrase has achieved salience through the inspiration offered by the poem of  Wallace Stevens (Thirteen Ways of Looking at a Blackbird, 1917) and the resulting application in a variety of arenas -- beyond authoritative closure on the one right way, as in the case of the pandemic (Variety of "ways of looking" -- binary or otherwise, 2021).

The further challenge is how to interrelate such contrasting perspectives (Interrelating Multiple Ways of Looking at a Crisis, 2021). One approach is through configurations of axes of bias, as discussed separately with respect to values and virtues (Systems of Categories Distinguishing Cultural Biases, 1993; Global ethical nexus of disparate challenges, 2022).

Configurations of polyhedra as transmission systems

Cognitive engines and gear boxes? Technology has been able to develop beyond the 2-stroke engine -- without disregarding its role under particular circumstances. Is it possible to consider more complex "engines" for the democratic process? What insights might be inspired by development of four-stroke engines, six-stroke engines, or even an eight-stroke engine (cf How does an 8-stroke IC engine work? Quora)?

The question framed in what follows is the nature of the "gearbox" which would enable transition between 3-fold, 4-fold, and N-fold modalities, as previously framed (In quest of a "cognitive gearbox" or transmission system, 2021). The complexities of the associated transmission mechanics can be contrasted with simplistic assumptions regarding the communication "transmission" between governors and governed.

Mapping options? There is considerable familiarity with use of simple geometricsl forms to map relationships. Most obvious is use of a sphere to map the features of the planet. Less obvious is the extensive array of distinctive projections whereby this may be achieved -- if the map is to be reproduced in two-dimensions (List of Map Projections, Wikipedia). Some of these take the form of polyhedra which can be unfolded from 3D to 2D, as with the so-called Dymaxion Map.

Yet to be a focus of attention is the manner in which information, conventionally ordered by checklists, can be mapped onto polyhedra -- specifically in order to emphasize the systemic relationship between the elements of the checklist. A polyhedron is appropriately understood to be a system in its own right, as asserted by Buckminster Fuller asserted that: All systems are polyhedra: All polyhedra are systems. ((Synergetics: explorations in the geometry of thinking, 1975/1979, 400.56). However, despite the cognitive emphasis implied by the title,the relevance to strategic "thinking" is less than evident, as discussed separately  (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).

Judicious association of disparate elements on a polyhedral map therefore enables recognition of their systemic integrity. The form of the polyhedron, especially its symmetry, is then a valuable trigger for both the comprehension and memorability significantly absent from articulation in list form (Identifying Polyhedra Enabling emorable Strategic Mapping, 2020). With respect to visualization f organization and strategic coherence through 3D modelling, the latter discusses:

Commentary on patterns of N-foldness
Polyhedra of secondary value for memorable mapping?
Constructible polyhedra in the light of constructible polygons?
Memorability of 17 Sustainable Development Goals with 169 tasks
Relevance of polyhedral mapping of patterns of complexity with symbolic appeal
Strategic viability of global governance enabled by mappings on exotic polyhedra
Polyhedral "memory palaces": an ordering pattern for sustainable self-governance?
Memorability implied by Euler's isomorphism of musical and polyhedral order?
Reframing forms of connectivity through the logic of oppositional geometry

Relative simplicity of strategic articulations: As might be expected, many of the conventional articulations of strategic initiatives are potentially associated with more mappable (and memorable) polyhedra, as indicated in the following table

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

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

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

Cognitive implications: The cognitive implications and strategic relevance of such patterns invite variously explorations:

Alternative configurations of polyhedral transmission systems

The following configurations are presented as a trigger for imaginative speculation as being indicative of the predilection for hierarchical modes of organization. In contrast to the use by Keith Critchlow (above) of the truncated tetrahedron as a form of "keystone", here use is made of the simple tetrahedron. The numerical characterists of each configuration and its elements is appended in each case -- as an indication of mapping possibilities. In each case a progression from 4 (vertices) to 64 is indicated.

Configuration A Configuration B
Non-transparent faces Transparent faces Non-transparent faces Transparent faces
Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system
Component polyhedra Elements (vertices, edges, faces) Component polyhedra Elements (vertices, edges, faces)
V E F EV EF VF EVF V E F EV EF VF EVF
tetrahedron 4 6 4 10 10 10 14 tetrahedron 4 6 4 10 10 10 14
cube 8 12 6 20 18 14 26 cube 8 12 6 20 18 14 26
simplest torus 16 28 12 44 40 28 56 1 freq truncated tetrahedron 16 42 28 58 70 44 86
drilled truncated cube 32 64 32 96 96 64 128 augmented icosahedron 32 90 60 122 150 92 182
42-Rit proj 64 96 64 160 160 128 224   cubes-8 64 96 48 160 144 112 208

Use is made above of the so-called simplest torus as being of potential interest in ordering the Sustainable Development Goals of the UN (Current relevance of the "simplest torus"? 2019; Coherent mapping possibilities on the simplest torus? 2019).

Configuration C Configuration D
Non-transparent faces Transparent faces Non-transparent faces Transparent faces
Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system
Component polyhedra Elements (vertices, edges, faces) Component polyhedra Elements (vertices, edges, faces)
V E F EV EF VF EVF V E F EV EF VF EVF
tetrahedron 4 6 4 10 10 10 14 tetrahedron 4 6 4 10 10 10 14
triakistetrahedron 8 18 12 26 30 20 38 triakistetrahedron 8 18 12 26 30 20 38
5-5-4 acrohedron 16 30 16 46 46 32 62 5-5-4 acrohedron 16 30 16 46 46 32 62
rhombic triacontahedron 32 60 30 92 92 122 182 rhombic triacontahedron 32 60 30 92 92 122 182
19-Tat proj 64 112 48 176 160 112 224   cubes-8 64 96 48 160 144 112 208

In the animations below, the rotations are no longer made around a "vertical" axis of the tetrahedron.

Configuration E Configuration F
Non-transparent faces Transparent faces Non-transparent faces Transparent faces
Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system Configurations of polyhedral transmission system
Component polyhedra Elements (vertices, edges, faces) Component polyhedra Elements (vertices, edges, faces)
V E F EV EF VF EVF V E F EV EF VF EVF
tetrahedron 4 6 4 10 10 10 14 tetrahedron 4 6 4 10 10 10 14
gyrobifastigium 8 14 8 22 22 16 30 17-Thor (3) 8 24 32 32 56 40 64
diphenocingulum 16 38 24 54 62 40 78 diphenocingulum 16 38 24 54 62 40 78
44-Hinnit 32 96 120 128 216 152 158 augmented icosahedron 32 90 60 122 150 92 182
tetrahedra 8+6+2 64 96 64 160 160 256 352 19-Tat proj 64 112 48 176 160 112 224

Vertical arrays of polyhedra implying higher degrees of order

There is no lack of references implying the possibility of "higher" degrees of insight -- whether or not the metaphor is to be challenged. The recognition is obvious in the sequence of academic degrees and qualifications, and realted assumptions of IQ. It features in secret societies, as with the degrees of Freemasonry; it is a characteristic of initiations and the associated rituals (Varieties of Rebirth: distinguishing ways of being born again, 2004).

Far less evident is the application of this insight to governance, although wisdom in governance may be readily acknowledged, as with the occasional appointment of advisory bodies (Committee of Wise Persons of the Council of Europe; Committee of Wise Men on the Regulation of European Securities Markets), and solicitation of the wisdom of others deemed especially wise (The Elders). It is indeed unclear how many levels of skill in governance can be usefully distinguished, however these may culminate in the "wise governance" for which strategic sustainability presumably calls.

Arguably a degree of recognition of these distinctions is indicated by the traditional symbolic role of the sceptre (Embodying the essence of governance in ritual dynamics with mace, sceptre, fasces or vajra? 2021; Integrative "orbital" implications: Crown and Sceptre / Sahasrara and Axis Mundi, 2020).

Far more evident is the recourse to arrays of symbolic "pillars" in the articulation of strategies by intergovernmental institutions (Fundamental values and strategic pillars, 2008; Holders of value configurations -- and their "pillars", 2008; Coherent Value Frameworks: Pillar-ization, Polarization and Polyhedral frames of reference, 2008). For example, the European Union has developed various sets of "pillars" that might be understood as the implicit value architecture of a number of strategic initiatives. The EEC was renamed the European Community (EC) upon becoming integrated into the first pillar of the newly formed European Union in 1993. Classic examples include: the Pillars of Ashoka, the Five Pillars of Islam, and the Pillars of Hercules. Less evident however is attribution of distinct levels of significance by inscription at different heights of the pillar.

As an exercise, the following animations are formed by stacking polyhedra selected for the number of vertices associated with each -- rising from 4 at the base to 64 at the top, passing through 8, 16 and 32. These are the numbers which are so fundamental to the binary coding systems of computers -- based on 2n, where n ranges from 2 to 5. A distinct set of examples could be offered based on the number of edges, or the number of faces.

The numerical attributes of each polyhedra in a stack are presented below as an indication of how it might be used to map qualitative attributesr values, concepts or strategies, for example.

Stack A alternative renderings Stack B alternative renderings
Non-transparent Rainbow lighting Transparent Non-transparent Rainbow lighting Transparent
Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order
Component polyhedra Elements (vertices, edges, faces) Component polyhedra Elements (vertices, edges, faces)
V E F EV EF VF EVF V E F EV EF VF EVF
42-Rit proj 64 96 64 160 160 128 224 19-Tat proj 64 112 48 176 160 112 224
rhombic triacontahedron 32 60 30 92 92 122 182 pentakisdodecahedron 32 90 60 122 150 92 182
simplest torus 16 28 12 44 40 28 56 1-freq. trunc. tetra. geodesic sphere 16 42 28 58 70 44 86
cube 8 12 6 20 18 14 26 triakistetrahedron 8 18 12 26 30 20 38
tetrahedron 4 6 4 10 10 10 14 tetrahedron 4 6 4 10 10 10 14

There is an obvious case for including the truncated icosahedron in such a stack. As the stitching pattern for the standard association football, known world wide, it must necessarily be of a fundamental significance as yet to be fully appreciated, as discussed separately (Game ball design as holding insight of relevance to global governance? 2020). It is especially intriguing in that it is a 32-fold pattern which is so familiar, being much more complex than the 8-fold and 16-fold patterns preceding it in the stack. Potentially more intriguing is the pattern of 64 vertices in the polyhedron which follows it -- being so complex that it can only be represented as the 3D projection of a structure in 4D.

The pillar-like structure of the FIFA World Cup is presented below as holding the aspiration with which the 32-fold pattern is associated. It can perhaps be compared with a cup whose form is so fundamental to religious ceremonies, itself inviting structural insights (Complementary visual metaphors of "Chalice", 2011; In-forming the Chalice as an Integrative Cognitive Dynamic, 2011). Such designs can be explored as echoed in the forms selected to reflect the culmination of strategic endeavour through traditional iconographic use of the laurel wreath (Game-playing, bull-leaping and laurel wreaths, 2014; Winged logos, laurels and strategic uplift, 2020). The animation below left is an exercise in representing nested significance.

Stack C alternative renderings Compact visual analogues to vertical arrays
Non-transparent Rainbow lighting Transparent FIFA World Cup Chalice cup Laurel wreath
Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order Vertical array of polyhedra implying higher degrees of order FIFA World Cup Sacred cup - chalice Animation of progressive emergence from ball-passing movement
           
Component polyhedra Elements (vertices, edges, faces)    
V E F EV EF VF EVF xxx xxx
42-Rit proj 64 96 64 160 160 128 224
truncated icosahedron 60 90 32 150 122 92 182
cubes-2 16 24 12 36 36 36 52
triakistetrahedron 8 18 12 26 30 20 38
tetrahedron 4 6 4 10 10 10 14

Nested and cage-like polyhedral configurations implying degrees of implicate order

Traditional symbolic configurations: One provocation to further exploration derives from the mysterious function of the so-called Roman dodecahedron. This is a small hollow object made of bronze or stone (4-11 centimeters in diameter). Of dodecahedral form, it has twelve flat pentagonal faces, each face having a circular hole of varying diameter in the middle, the holes connecting to the hollow center. Over a hundred of these have been found across Europe, dating from the 2nd or 3rd centuries AD.

As discussed separately, they have evoked speculation of every kind as to their function -- with no conclusion (Roman dodecahedron, Chinese puzzle balls and Rubik's Cube? 2018). They can be usefully compared with carved stones balls (petrospheres) dating from the Neolithic period, and with the carved Chinese puzzle balls.

Roman dodecahedron Chinese ivory puzzle ball Neolithic carved stone ball Polyhedral model of solar system of Johannes Kepler
on Mysterium Cosmographicum(1596)
Roman dodecahedron Chinese ivory puzzle ball Neolithic carved stone ball Kepler solar systemnested polyhedra
By Lokilech [GFDL, CC-BY-SA-3.0 or CC BY-SA 2.5 ],
from Wikimedia Commons
British Museum [CC BY 2.0 ],
via Wikimedia Commons
National Museums of Scotland,
via Wikimedia Commons
Reproduced from Wikipedia entry

The animation of the array of 12 Archimedean polyhedra (in its collapsed form) suggested the further possibility of emulating the classical Chinese puzzle balls, or mystery balls (hsiang ya ch'iu or hsiang ya qiu). As a traditional gift to the Emperor, these were carved out of a single piece of ivory, but now from synthetic ivory, resin, wood, jade, and other materials (Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds", 2015).

They are called "puzzle balls" due to the mystery and puzzling explanation behind their making. There is the possibility of manipulating the inner balls so that their holes align with the outer balls, thereby "solving" the puzzle in a technical sense.

They consist of a number of concentric spheres -- typically from 3 to 7 -- which rotate freely with respect to one another. The sequence of balls is understood to represent the cosmos -- a symbolic reference to the sense of "worlds-within-worlds" as being the very nature of reality. Every sphere has distinctive symbolic carvings, usually of plants and animals. Most often, the outermost will either depict two dueling dragons, or hold a dragon (emperor/male), and a phoenix (empress/female), battling for hold upon the world and keeping it in balance, namely as representations of yin and yang. The most complex known is made of 42 spheres enclosing one another.

Representation of implicate order? As a theoretical physicist, David Bohm is concerned with the illusory nature of fragmentation and the manner in which distinct fragments emerge from wholeness in movement (Wholeness and the Implicate Order, 1980). As discussed separately, Bohm sees the perceptual problems with which he deals as being as relevant to a more healthy response to psychosocial fragmentation as to the problems of fundamental physics (Wholeness and the implicate order, 1983; Coexistence of variety: an implicate order? 2011).

Basing his investigations on insights from the current state of physics, Bohm:

In his challenge to prevailing views, Bohm notably articulated an understanding of holomovement as a key dynamic in quantum mechanics relatiing explicate and implicate order. (David Peat, David Bohm, Implicate Order and Holomovement, Science and Nonduality; The Cosmic Plenum: Bohm's Gnosis: The Implicate Order).

Any pointers suggestive of the relation between explicate and implicate order are therefore of value, as exemplified by the following indicative of patterns of change implied by by nesting, packing, and transforming symmetrical polyhedra (Psychosocial Implication in Polyhedral Animations in 3D , 2015)

Indications of cage-like and nesting polyhedral frameworks
Cuboctahedral array of 12 Archimedean polyhedra
(around an omitted 13th at the centre; totalling 984 edges, 558 vertices, 452 faces)
Nesting 5 Platonic polyhedra
octahedron, icosahedron, dodecahedron, tetrahedron, cube
with Rhombic triacontahedron (green) as a nesting framework
Rotation of cuboctahedron of Archimedean polyhedra Platonic polyhedra nested within Rhombic triacontahedron
Virtual reality variant (.wrl) virtual reality variants static: vrml or x3d;
mutual rotation: vrml or x3d; "pumping": vrml or x3d;
videos: "pumping" mp4; "rotation" mp4
Developed with X3D Edit and Stella Polyhedron Navigator
Screen shots of animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre
(without indication of the 13th at the centre, namely the truncated tetrahedron)
Cuboctahedral array of 12 Archimedean polyhedra collapsing into centre Cuboctahedral array of 12 Archimedean polyhedra collapsing into centre
Contextual cuboctahedron rendered partially transparent
Video (.mpg; .mov); virtual reality (.wrl; .x3d)
Wireframe version with all faces transparent
Video (.mp4; .mov); virtual reality (.wrl; .x3d)
Animations prepared with the aid of Stella Polyhedron Navigator

Particular coherence of 60-fold organization? The potentially fundamental implication for cognitive organization associated with the truncated icosahedron in the design of the association football (as noted above) merits further consideration in the light of the role of that polyhedron in the organization of a complex molecule only recently discovered. The molecule is that of C60, named after Buckminster Fuller as buckminsterfullerene.  It has a cage-like fused-ring structure made of twenty hexagons and twelve pentagons, therefore resembling a soccer ball.  It is the most common naturally occurring fullerene, also existing in space

Given the contrast with the organization of carbon in organic moledules so fundamental to life, there is a case for exploring tsycho-social implications of the polyhedral fullerenes for coherence, integrity and identity of a higher order (Sustainability through Global Patterns of 60-fold Organization:, 2022)

For example, 30 positive and 30 negative global trends can be identified (Convergence of 30 Disabling Global Trends, 2012). These can be suggestively mapped onto a 60-vertex truncated polyhedron, as shown in alternative renderings below. That on the right is a screen shot of the Interactive Mapping of 30 Problems with 30 Strategies onto Truncated Icosahedron using Force-directed Layout (2022). The blue nodes are challenging problems and the orange nodes are remedial strategies -- the titles becoming evident on mouseover. More skillful renderings are suggested in the visual representation of both alternatives.

Indicative mapping onto 60 vertices of a truncated icosahedron of 30 disabling trends with 30 remedial trends
(arbitrary mapping for illustrative purposes)
Mapping onto a truncated polyhedron (C60) of 60 disabling and enabling trends Mapping onto a truncated polyhedron (C60) of 60 disabling and enabling trends Mapping onto a force-directed truncated polyhedron (C60) of 60 disabling and enabling trends
Animations made using Stella Polyhedron Navigator Mapping with force-diected layout (interactive version)

Implicate order through hypercube and drilled truncated cube?

The justification for cage-like arrays can be clarified in the light of the recourse of logic, especially oppositional logic, to use of the hypercube (or tesseract) to represent the 16 Boolean connectives as operators of significance to computer operations (Reframing forms of connectivity through the logic of oppositional geometry, 2020).  These are variously illustrated in the following images.

As explained by Steven H. Cullinane (The Geometry of Logic: finite geometry and the 16 Boolean connectivesFinite Geometry Notes, 2007), a Hasse diagram of a Boolean lattice, may also be viewed as a tesseract (4-dimensional hypercube). The vertices represent the 16 traditional "binary connectives". The tesseract's 16 vertices may also be regarded as representing either the 16 subsets of a 4-set or the 16 elements of the affine 4-space A over the two-element Galois field. The pattern was originally depicted by Shea Zellweger, as a "logic alphabet", as shown below.

The images below show various attempts to facilitate comprehension of 4-dimensionality suggested by a central nexus. The animation of a hypercube (tesseract) on the right is very helpful in suggesting the necessarily paradoxical interplay of the alternation between an inner and an outer perspective.

Contrasting representations facilitating comprehension of 4-dimensionality and a central nexus
Cubic configuration of
BaGua trigram symbols
by Z. D. Sung

Logic Alphabet Tesseract
- a four-dimensional cube
by Shea Zellweger
(see coding).

Topological 4-statement Venn diagram
(graph of edges of a 4-dimensional cube as described by Tony Phillips)
Tesseract animation
simulating requisite
4-dimensionality?
Cubical representation  of BaGua pattern of I Ching The Logic Alphabet Tesseract by Shea Zellweger Topologically faithful 4-statement Venn diagram Tesseract animation
Reproduced from Z. D. Sung, The Symbols of Yi King or the Symbols of the Chinese Logic of Changes (1934, p. 12) Diagram by Warren Tschantz
(reproduced from the Institute of Figuring) .
A vertex is labeled by its coordinates (0 or 1) in the A, B, C and D directions; the 4-cube is drawn as projected into 3-space; edges going off in the 4th dimension are shown in green. by Jason Hise [CC0], via Wikimedia Commons

The literature of oppositional logic for the representation in truth tables of the 16 Boolean connectives (as indicated in the above hypercube representation in 4D) reduces them to 14, notably to enable their geometrical representation in 3D (Oppositional logic and its geometry -- 16 minus 2 connectives?, 2021;
From 16 to 14 connectives -- precluding a logical meta-perspective? 2021; Questionable confusion in configuring strategic frameworks: "fudging" self-reflexivity? 2019). A key polyhedron used to map the 16 (-2) Boolean logical connectives in that approach is the rhombic dodecahedron of 14 vertices and 12 faces.

The focus on a 14-fold configuration indeed offers significant insight (Pattern of 14-foldness as an Implicit Organizing Principle for Governance? 2021). However there is also potentially a case for exploring the value of 16-fold arrays (Deprecation of potential correspondences: 16-fold patterns? 2019).

In the light of Critchlow's arrangement of 12 semi-regular Archimedean polyhedra (as noted above), there is a case for exploring an arrangement of such polyhedra on a hypercube -- if only for mnemonic purposes. Noting that Critchlow's cloest packaging array is centered on the truncated tetranhedron (the 13th Archimedean solid), it can be assumed for the purposes of the exercise that the tetrahedron can be omitted from the set of 5 Platonic polyhedra. This then enables a total of 16 polyhedra to be arrayed as a hypercube. The 4 Platonic polyhedra can be positioned on vertices of the inner cube, together with 4 of the Archimedean polyhedra -- of which the remaining 8 can then be positioned on the outer cube (as variously shown below).

Of particular mnemonic interest is the possibility that the polyhedra can be judciously placed to reflect the symmetry-preserving operations of the Conway relational chart (reproduced above). As one initial effort in this respect, those polyhedra on vertices of the outer cube are primarily truncations of those on the inner cube.

Configuration of 16 Platonic and Archimedean polyhedra on a hypercube
(not to scale)
Solid faced Names (see variants below) Transparent faces
Configuration of Platonic and Archimedean polyhedra on a hypercube Configuration of Platonic and Archimedean polyhedra on a hypercube Configuration of Platonic and Archimedean polyhedra on a hypercube
Animations made using X3D- Edit and Stella Polyhedron Navigator

Drilled truncated cube: The pattern of 64 is nearly unique within the polyhedral context. However one interesting candidate is the toroidal drilled truncated cube with 64 edges -- with which any set of 64 elements could be associated (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). The issue is whether the manner in which they can be positioned on that framework constitutes a configuration which is meaningful in relation to particular cases, such as the 64 genetic codons or the 64 hexagrams of the I Ching.

It is possible that known constraints in the patterning in such particular cases can together offer guidance in the attribution of the distinct elements -- of relevance to each case, as tenatively explored (Changing Patterns using Transformation Pathways, 2015):

Use of drilled truncated cube as a mapping framework for 64 elements
Mapping attributions: preliminary assumptions from patterns of codons
Complementary mapping of I Ching hexagrams
Indicative mutual constraints between codon and hexagram patterns
Challenging cognitive business-as-usual: expecting the unexpected

As shown below, of particular interest is the manner in whuch a cube is effectively "nested" within the drilled truncated cube, as in the case of the hypercube above (Psychosocial Implication in Polyhedral Animations in 3D, 2017). The pattern invites a variety of experimental interactive animations (Dynamics of Parallel edges of Drilled truncated cubes in 3D; Experimental Interactive Display of Cube Edges in Movement in Virtual Reality.

Alternative views of hypercube (as above) nested within drilled truncated cube
Hypercube with polyhedral mappings nested within drilled truncated cube Hypercube with polyhedral mappings nested within drilled truncated cube Hypercube with polyhedral mappings nested within drilled truncated cube

An earlier experiments explored the use of that polyhedra for other mapping purposes (Configuring the 64 subjects of mathematics as a 64-edged drilled truncated cube, 2021). There its use was suggested for ordering the dynamics within a pantheon (Exploring potential dynamics within a pantheon?, 2021). The following examples are reproduced from a discussion in Enabling Wisdom Dynamically within Intertwined Tori: requisite resonance in global knowledge architecture (2012).

Experimental mappings with drilled truncated cube of 64 edges
Animation with faces non-transparent
(mapping 64 genetic codons)
Animation with faces transparent
(mapping 64 mathematical subjects)
Animation with faces transparent
(mapping 64 I Ching hexagrams)
Drilled truncated cube mapping of 64 mathematical disciplines onto 64-edged drilled truncated cube Drilled truncated cube of 64 edges with hexagram names
Animations prepared using Stella Polyhedron Navigator

Following the experimental configuration of 16 Archimedean and Platonic polyhedra on a hypercube (as illustrated above), and on the assumption that a hypercube can be mnemonically embedded within a drilled truncated cube, a further presentation of a 16-fold array can be explored. However, given that the drilled truncated cube has 32 faces, 16 can indeed be used for the Archimedean and Platonic polyhedra, but the remaining 16 faces can be used for the geometric duals of those polyhedra. In the case of the Archimedean polyhedra, these are the Catalan polyhedra. The duals of the Platonic polyhedra are themselves Platonic polyhedra (Duals of Platonic Solids, Wolfram Demonstration Project).

The images below position the 12 Archimedean polyhedra on 12 visible faces of the drilled truncated cube such as to correspond to the 12 Catalan polyhedra on the remaining 12 faces (on the reverse of the drilled truncated cube). Not shown in the images below, being only apparent in the animations which follow, are the 4 Platonic polyhedra (on the outer sides of the cube) and their duals on the corresponding inner surfaces of the cube).

Attribution of 12 Archimedean and 12 Catalan polyhedra on a drilled truncated cube
(4 Platonic polyhedra, with their 4 duals are not shown)
Attribution of 12 Archimedean and 12 Catalan polyhedra on a drilled truncated cube
Alternative animations of drilled truncated cube showing 16 Archimedean and Platonic polyhedra and their 16 duals
(names in image above, with variant names)
Animation of drilled truncated cube showing 16 Archimedean and Platonic polyhedra and their 16 duals Animation of drilled truncated cube showing 16 Archimedean and Platonic polyhedra and their 16 duals Animation of drilled truncated cube showing 16 Archimedean and Platonic polyhedra and their 16 duals
Animations prepared using Stella Polyhedron Navigator
Images suggestive of other design metaphors
Unfolding of drilled truncated cube
with polyhedra and duals (as above)
Wire frame of dual
of drilled truncated cube
UN Art Installation in Tblisi, Georgia
on Rubik's Cube
Unfolding of drilled truncated cube showing 16 Archimedean and Platonic polyhedra and their 16 duals UN art installation
Animations prepared using Stella Polyhedron Navigator Reproduced from UNIC@Work, October 2016 - Special UN Day Bulletin

Polyhedral representation of Sustainable Development Goals including "Own Goals"?

As indicated by the suggestive depiction of the UN Sustainable Development Goals on Rubik's Cube (above right), there is a case for imaginative exploration of alternative means of depicting the iconic images of the goals (Interplay of Sustainable Development Goals through Rubik Cube Variations: engaging otherwise with what people find meaningful, 2017). One approach is to use 16 of the goals (omitting the 17th as coordinative) and associate them with 16 faces of the drilled truncated cube as shown below.

This usefully evokes the judicious possibility of their placement to suggest systemic relationships between them. The edges between the faces are then indicative of some form of confrontation, feedback or challenge, with the vertices potentially indicative of trilemmas.

Correspondences between the faces also enables the polyhedron to be used as a means of indicating the tendency to fail in achievement of the goal -- perhaps to be understood globally as some form of civilizational "own goal" or "self-goal" (Variety of System Failures Engendered by Negligent Distinctions, 2016).. For this purpose, 16 of the icons have been coloured black and appropriately placed. [NB: The software used reverses images in certain positions, potentially suggestive of strategic misunderstanding -- until this is corrected.] It is appropriate to reflect on the inability to provide a coherent global overview of the set as displayed -- and hence the value of the animations on different axes below, in each case obscuring the icons at each extreme of the axis selected. Other design metaphors could of course be envisaged, including shading the "own goals" to varying degrees as an indication of remedial success.

Experimental polyhedral representation of Sustainable Development Goals of the UN (SDGs)
Use of 32-sided drilled truncated cube to represent 16 SDGs and 16 corresponding "own goals"
     
Use of 32 sides of drilled truncated cube to represents 16 SDGs and 16 "own goals" Use of 32 sides of drilled truncated cube to represents 16 SDGs and 16 "own goals" Use of 32 sides of drilled truncated cube to represents 16 SDGs and 16 "own goals"
Animations prepared using Stella Polyhedron Navigator

Dynamics of systemic connectivity as a challenge to invariance

The challenge of systemic coordination and coherence of the 16 SDG goals is indicated by the 17th: Strengthen the means of implementation and revitalize the global partnership for sustainable development. As explained, given its "keystone" role, this goal has been omitted from the 16 mappings above, as with the omission of the tetrahedron and the truncated tetrahedron in the polyhedral configurations.

An interesting design possibility for mnemonic purposes is to position the omitted polyhedra as animations within the inner cube of the drilled truncated cube (as shown below).

Angled animations of dynamics of tetrahedra and truncated tetrahedra nested within drilled truncated cube
(Screen shots of rotation on vertical axis with solid or transparent faces)
 Counter-rotating tetrahedron (red) and dual (blue)
Truncated tetrahedron Truncated. tetrahedron with dual
Angled animation of dynamics of tetrahedra nested within drilled truncated cube Angled animation of dynamics of tetrahedra nested within drilled truncated cube Rotation of truncated tetrahedron nested within drilled truncated cube Rotation of truncated tetrahedron with dual nested within drilled truncated cube

To the extent that the drilled truncated cube constitutes a potential cognitive Rosetta Stone, inclusion of animations within the form provides a valuable contrast to the stasis conventionally associated with invariant configurations -- and the nature of any "philosopher's stone". Following from the reference above to transmission systems, the rotation of the inclusions is also reminiscent of the fundamentlal importance of rotation in electromagnetic technology -- namely in motors and dynamos,  as developed by Nikola Tesla as noted above (Reimagining Tesla's Creativity through Technomimicry, 2014). Fundamental to his innovative discoveries was the rotation of a magnetic field, potentially suggestive of psychosocial analogues (Potential implications of alternation and rotation in psychosocial fields, 2014).

The manner in which the vertices of the rotating polyhedra successively contact the surrounding structure is suggestive of how processes of governance could be understood as triggered. More skillful animations would have edges of that structure successively highlighted following such contact, thereby highlighting particular "information pathways" through the structure of categories of categories it maps..

Categories "enstoned" and the myth of invariance? There is an unquestioned assumption that the most fundamental categories are metaphorically "set in stone" -- implying a degree of stasis which could be understood as contrary to their fundamental nature, especially when associated with a cube. Challenges to this assumption from the perspective of experiential reality can be variously explored (Fivefold Clustering of Ways of Being Stoned, 2012). The latter distinguishes between:

In the preceding discussion of the articulation of the traditional 7-fold set of values/virtues deemed fundamental to society, animations were presented using a cubic framework (Global ethical nexus of disparate challenges, 2002). This used a cube to offer a means of representing the dynamics by which 7 "axes of bias" could be distinctively explored. Clearly such a cube could be recognized as central to that framed by any hypercube-like structure, as discussed separately (Eliciting the dynamics of the cube: reframing discourse dynamics, 2018; Mark Ronan, The Rotations of a Cube; The Rotational Symmetries of the Cube, York University).

As with an indicative animation of a hypercube presented above, its paradoxical nature is further explored there through the seemingly impossible inversion of a cube. The drilled truncated cube also invites exploration of its potential dynamics and visualization (Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization? 2015; Dynamics of movement of parallel edges of drilled truncated cube).

Other dimensions to the mapping potential of the drilled truncated cube follow from the fact that its 32 faces are of 5 types, its 64 edges are of 9 types, and its 32 vertices are of 4 types. These are each associated with extensively explored patterns of significance to quite dispararate domains. The following section explores the use of other polyhedral configurations, especially with respect to the ring configuration associated with the 12-fold pattern, as noted above. As the animations above make clear, one particular perspective on the drilled truncated cube highlights a 12-fold pattern. framed by two distinctive 4-fold patterns thereby offering a 20-fold pettern (referenced above).

Intriguingly the ambiguity associated with "invariance" of fundamental categories has been explored by the musicologist Ernest McClain (Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato, 1976).

Beyond 64? The transformations from patterns of 4 to 64 are potentially taken further in numerical terms by the animations above.  In the case of the 4-edged tetrahedron (with a tetrahedron as its own dual), the 8 sides together extend the 64 of the drilled truncated cube to 72 -- thereby anticipating the traditional emphasis on the dynamics of that pattern, as discussed below in various Western traditions. Given the reference above to Jacob's Ladder, its degrees were to the number of 72 (according to the Zohar).

In the case of the animation of the 18-edged truncated tetrahedron (with its 18-edged dual), the pattern of 64 would be extended by 36 to a 100-fold pattern as variously appreciated. Combining both animations, the pattern of 64 would be extended to 108 -- of considerable significance in the traditions of Buddhism, Hinduism and Jainism.

Toroidal polyhedral ring array

The drilled truncated cube is recognized as approximating a torus. The argument can be developed further given recognition that fullerenes of carbon can take toroidal form (as mentioned above). This can be related to exploration of its psychosocial implications (Imagining Toroidal Life as a Sustainable Alternative: from globalization to toroidization or back to flatland? 2019).

A spreadsheet application has been developed by Sergey Bederov of Cortona3D to explore the possibility of a potentially more stable toroidal fullerene (Toroidal fullerenes as a complement to the global form, 2022). This introduces pentagons in accordance with research on toroidal carbon nanotubes (Florian Beuerle, et al, Optical and Vibrational Properties of Toroidal Carbon Nanotubes, Chemistry: a European Journal, 17, 2011, 14; Pakhapoom Sarapat, A Review of Geometry, Construction and Modelling for Carbon Nanotori, Applied Sciences. 9, 2019, 11). Such research recognizes that in order to maintain their Euler characteristic (zero for a torus), an equal number of heptagons (7-gons) needs to be added (coloured blue and red in the screen shots below to distinguish them by size). Whether pentagons or hexagons, these are in each case by hexagons; mathematically all vertices lie on a perfect torus and edge lengths differ by only 24%.

Screen shots of a toroidal fullerene with pentagons and heptagons surrounded by hexagons
Toroidal fullerene with pentagons and heptagons surrounded by hexagons (polar view) Toroidal fullerene with pentagons and heptagons surrounded by hexagons (side view)
Screen shots derived from a 3D model kindly developed by Sergey Bederov of Cortona3D. X3D model

Such experiments suggest the possibility of configuring the 12 Archimedean polyhedra together in the form of a ring as variously illustrated below -- in relation to a torus of otherwise. Alternatives could use the Catalan variants.

Configuration of 12 Archimedean polyhedra in relation to a torus
   
Animations prepared using X3D Edit and Stella Polyhedron Navigator

Such a configuration is consistent with earlier exercises focusing on the design metaphor of an archetypal Round Table -- together with its numeric relationship to the Last Supper and the communication issues implied by both:

Strategic implications of traditional angelic and demonic configurations

Stars and birds? Widespread use is made of stars on the flags of nations and intergovernmental institutions -- with little explanation of the associated symbolism and strategic implications, as discussed separately (Imagining the Flag of Europe otherwise? 2018). Given the importance of the flight metaphor in recognizing strategic achievement, such stars invite explanation in terms of the capacity of birds (Coordination of Wing Deployment and Folding in Politics, 2018). That framing developed the case for bird flight and landing as complementary metaphors of global strategic coherence under the followiong headings:

Star symbols as schematic birds?
Flying capacity implied by "wings" of a 5-pointed star?
Flying capacity implied by "wings" of a 6-pointed star?
Star rotation: achieving and sustaining controlled flight?
Symbolism of the Flag of Europe -- imagined in the light of bird flocking

Winged angels and demons? From a traditional perspective, it is somewhat extraordinary that "wings" should figure so frequently in the iconography of both demons and angels. For a secular society claiming to have transcended such framing, there is a case for exploring the transformation of the latter into problems and strategies respectively -- and in frequent reference to so-called political wings. In doing so, there is also a case for acknowledging the centuries of reflection on the manner of organization of the angelic and demonic realms -- especially given the continuing significance attributed to them by religious societies.

Extraordinary though it may appear from a secular perspective, it is noteworthy that King Charles III, in his message to the nation at his mother’s passing made specific reference to angels (Alison Milbank "May flights of angels sing thee to thy rest": Elizabeth II and the virtues of a Christian monarchy, ABC Religioon and Ethics, 13 Sep 2022). In the funeral itself, with its worldwide media coverage, reference was again made to angels and archangels, as in the commital service (Queen's funeral, The Sydney Morning Herald, 20 September 2022). Despite the attendance of world leaders, such language is not susceptible to challenge in practice (World leaders at Queen Elizabeth's funeral, Reuters, 20 September 2022).

From a secular philosophical perspective, it is intriguing to note that the "existence" in substantive terms of angels and demons is as tenuous and questionably founded as that of problems and strategies. More relevant is the effort invested in ordering such abstractions -- whether as artefacts of cultures or of collective awareness. There is also a case for respecting the engagement of the past with such matters -- whether readily deprecated with reference to scholastic debate regarding the "number of angels on the head of a pin".

Wicked problems and angelic strategies? It is of course the case that particular arrays of strategies, such as the 17 Sustainable Development Goals can be understood as having taken on the systemic role of distinct angels -- effectively held to embody what is deemed "sacred" in a secular society. They are arrayed aginst a set of problems -- effectively held to embody the wickedness by which the demonic has been traditionally recognized. Hence the current reference of the policy sciences to "wicked problems".

The manner in which values are upheld suggests that they are the "new sacred" for global civilization, especially democratic values, as discussed separately (Values, Virtues and Sins of a Viable Democratic Civilization, 2022). Similarly problems can be recognized as the "new demonic", appropriately confirmed by the extent to which "evil" continues frequently to be recognized and attributed to both phenomena and individuals (Existence of evil as authoritatively claimed to be an overriding strategic concern, 2016).

Any remedial contribution of the SDGs could of course be credibly described as "angelic" for public relations purposes, as noted by the United Nations with respect to the: “Angel for Sustainable Development” Global Awards, as offered by the World Organization for Development (WOD). The awards celebrate winners in seven categories that correlate with 11 SDGs. Award ceremonies will be held in cooperation with and under the auspices of UN bodies responsible for implementing the SDGs.

Memorable organization of the insubstantial? As discussed separately as an exercise in collective memory, there is therefore a case for exploring how the far vaster arrays of angels or demons might be organized in order to render them more memorable (Engaging with Hyperreality through Demonique and Angelique? 2016). This is partly justified by the extent to which the attributes of the many individual angels or demons have been explicitly identified -- potentially to a higher degree than is deemed credible in the case of strategies and problems.

The question here is then how such arrays might be organized, given that their number in each case is claimed to be 72 -- namely far in excess of what is readily indicated as memorable in the case of problems or strategies.

Following the polyhedral method used above, and in previous exercises, the pattern of 72 may be understood to factor as 8x9, 9x8, 6x12, or 12x6. The animation on the left below uses a 6-vertex octahedron to configure 6 icosahedra, each having 12 vertices. The central animations use an 8-vertex cube to configure 8 heptagrammic dipyramids, each having 9 vertices. That on the right uses a single heptagrammic dipyramid of 9 vertices to configure 9 8-vertex cubes.

Exploratory polyhedral configuration of 72 elements
Octahedron configuring 6 icosahedra Cube configuring 8 9-vertex heptagrammic dipyramids Heptagrammic dipyramid configuring 9 cubes
Octahedron configuring 6 icosahedra Cube configuring 8 9-vertex heptagrammic dipyramids Cube configuring 8 9-vertex heptagrammic dipyramids Heptagrammic dipyramid configuring 9 cubes
       

System diagrams for governance? The set of modes of systemic negligence could well be recognized as "demons" for which vigilance is required in the governance of any system, as discussed separately (Evil loops and sigils as a pattern language, 2016). Any understanding of the relationship between sigil and demon therefore merits careful attention, whatever mapping device is used. A sigil could be recognized as a traditional systems diagram -- especially since "sigil is now used in computer programming. Demons then offer an imaginative means of distinguishing patterns of nonviability in systemic terms -- effectively of "unwholiness" or "unholiness". In geopolitical terms, an unholy alliance refers to an alliance perceived as unnatural, unusual, or simply undesirable, sometimes between seemingly antagonistic parties.

As a first exercise, on the assumption that the 72 traditional demons usefully characterize distinctive styles of problematic wickedness, experimental use can be made of a truncated icosahedron to configure the sigils deemed to represent them.

A much-cited traditional list of sigils is in The Lesser Key of Solomon. This presents the sigils of the 72 "princes of the hierarchy of hell" to be employed with the skills and understanding of traditional magic of the Ars Goetia. Within that worldview, such sigils have been considered to be the equivalent of the "true name of a spirit" and thus granted the magician a measure of control over them. Examples of such sigils include the following. Wikipedia offers a detailed List of demons in the Ars Goetia and a summary list of Goetic demons in popular culture (notably in role-playing games). The essence of a wicked problem, as a challenge to governance, could indeed be described metaphorically by such terms.


Use of truncated icosahedron to display 2 sets of 32 demons
linked by an octahedron displaying a set of 8 demons (thus totalling 72)
Version A mapping of 32 Mapping of 8 Version B mapping of 32
Experimental use of truncated icosahedron to display a sets of 32 demons Experimental use of truncated icosahedron to display a sets of 32 demons
Generated using Stella Polyhedron Navigator

The elements of the above animations are presented more clearly in the following animations which help to highlight the distinctive designs that can be used to populate the global configurations presented subsequently.

Animation of sequence of 72 Angel names from the Shemhamphorasch
(in two contrasting representations)
Animation of sequence of 72 demonic sigils from the Ars Goetia
(with matching reversed images)
Animation of sequence of 72 Angel names from the Shemhamphorasch Animation of sequence of 72 Angel names from the Shemhamphorasch Animation of sequence of 72 demonic sigils from the Ars Goetia Animation of sequence of 72 demonic sigils from the Ars Goetia

Curiously it is however appropriate to note that there is currently no collection of systemically depicted problems which global governance is called upon to address. Ironically the nearest equivalent is the online Encylopedia of World Problems and Human Potential from which what amount to specific systems diagrams are generated dynamically, as described and illustrated by Tomáš Fülöpp (Loop Mining in the Encyclopedia of World Problems, Paper for the 17th International Futures Conference on Futures Studies Tackling Wicked Problems, 2015).

The pathetic institutional response to the challenges of global governance is curiously echoed by the manner in which it is claimed to be founded upon "core democratic values" -- but with little ability to indicate what they are in any memorable systemic manner, as discussed separately (Core democratic values? 2022). It is for the defence of that ill-defined set of values that it is held necessary to engage in costly warfare, as in Vietnam, Afghanistan, and Ukraine, for example.

Interrelating "anglelique" and "demonique"? Configuring the global problematique as a whole has been limited to the world modelling exercises with which the Club of Rome report on Limits to Growth (1972) was originally associated. For example, the World3 model is a system dynamics model for computer simulation of interactions between population, industrial growth, food production and limits in the ecosystems of the earth (Alexander Christakis, A Retrospective Structural Inquiry of the Predicament of Mankind Prospectus of the Club of Rome, 2000). Little effort is made to depict the global problematique in memorable systemic terms -- especially in relation to any global resolutique of strategic responses (Patterning the Resolutique, Global Strategies Project).

There is therefore a case for considering how the "angelique" and the "demonique" might be configured -- in relation to one another, as discussed separately (Engaging with Hyperreality through Demonique and Angelique? 2016). This explores the mnemonic clues to global governance from mathematical theology and hyperbolic tessellation. It discusses Traditional modes of cognitive engagement with hyperreality and Hyperbolic reframing of the Demonique and Angelique of tradition. The following animations are reproduced from the latter discussion. The  demonic metaphor can also be used to highlight the challenges to any remedial strategy (Mnemonic clues to 72 modes of viable system failure from a demonic pattern language, 2016).

Indication in 2D of the dynamic nature of a "hyperdimensional" interaction between radically distinctive forces
Alternative experimental configurations alternating between the 72 "angels" and 72 "demons"
Animation of 8 sets of 9
(enlargements for detail: angels / demons)
Animation of 9 sets of 8
(enlargements for detail: angels / demons)
The allocation of sets to the star "tables" in the schematics is based on the tabular form in which the 72 angel names (from the Shemhamphorasch) and the72 demonic sigils (from the Ars Goetia) are typically presented. See other presentations in the Wikipedia List of demons in the Ars Goetia (with comments on differences in variations between sources) and in The Demonic Paradise Wiki
Experimental configuration alternating between the 72 angels and demons Experimental configuration alternating between the 72 angels and demons
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 from a systemic functional perspective.

Implicit in any confrontation of 72 angels and demons is recognition of a capacity to make 144 distinctions (12x12) -- perhaps to be seen as a comparable challenge to memory to the 169 tasks (13x13) addressed by the 17 Sustainable Development Goals of the UN (Memorability of 17 Sustainable Development Goals with 169 tasks, 2020). Curiously there is one game which explores that degree of variety, namely mahjong with its use of 144 tiles, as discussed separately (Reframing the Righteousness Enabling Repetition of the Titanic Disaster: comprehension of 144 distinctions, 2020). Although from a culture with a distinctive iconography, this might then be seen as a game of "Angels" versus "Demons".

Experimental mappings of angels and demons onto polyhedra
72 Angels (as strategic values?) 72 Demons (as "wicked problems"?)
Version A Version B Version A Version B
       

Variable geometry of institutions as alternation between polyhedral patterns

There is continuing reflection on the possibility of a "variable geometry" of regional and institutional organization:

The possibility has been most developed with respect to a multi-speed Europe:

The relevance has been explored with respect to the United Nations (Alternation between Variable Geometries: a brokership style for the United Nations as a guarantee of its requisite variety, 1985).

The reference to transmission systems for gear-changing (as discussed above) can clearly be recognized as an application of the potential of variable geometry. The questions is how this can be related to the transformation between polyhedral patterns exemplifying alternation between contrasting patterns of movement of information.

The potential of variable geometry could well be especially relevant to the challenging relationships between divided countries (Korea, Israel-Palestine, China-Taiwan, and the like). With the passing of Elizabeth II, the possibility merits consideration with respect to the future role of monarchy, most notably in relation to the future organization of the Commonweaalth of Nations. More controversially is its relevance to the organization of NATO (Alastair Crooke, A Variable Geometry NATO? Really? Al Mayadeen 29 May 2022). As argued separately, the potentially hidden faces of global strategy might well be highlighted through polyhedra (Envisaging NATO Otherwise -- in 3D and 4D? 2017)


References

Keith Albar. The Language of Pattern: an enquiry inspired by Islamic decoration. Thames and Hudson, 1974

Stafford Beer. Beyond Dispute: the invention of team syntegrity. John Wiley, 1995

David Bohm. Wholeness and the Implicate Order. Routledge and Kegan Pual, 1980

Keith Critchlow:

Marcus du Sautoy:

R. Buckminster Fuller  with E. J. Applewhite:

Susan George. Variable Geometry to Design Positive Outcomes. Transnational Institute, 2005 [text]

Charles Grant:

Jeremy Lent. The Patterning Instinct: a cultural history of humanity's search for meaning. Prometheus, 2017 [contents]

Ernest McClain:

Anthony Pugh. Polyhedra; a visual approach. University of California Press, 1976.

Robert D. Romanyshyn. Technology as Symptom and Dream. Routledge, 1989

Robert Williams. Natural Structures: toward a form language. Eudaemon Press, 1972

Frances Yates:

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