 / 
Introduction
Web resources
Toolkit for operacy?
Coherence, comprehensibility and credibility of a cognitive toolkit
Reconciling "positive" and "negative" operational insights
Cognitive operational sets recognized dynamically rather than statically
Operational insight sets as resonance hybrids?
Constraint of the 7fold on comprehension of the 20fold
Transformation of principles into "sides" through "truncation" of cognitive sets ?
Dynamics of forcedirected layout of concept sets beyond truncation of rigid polyhedra
Twenty as "God's number"?
References
Produced on the occasion of the 50th Anniversary celebrations of the Club of Rome
It is somewhat surprising to note the range of articulations of insights and methods specifically identified as numbering twenty. A loosely clustered checklist is provided below.
The interest follows from the presentation of a preceding exploration (Checklist of 12fold Principles, Plans, Symbols and Concepts: web resources, 2011). This was produced as an annex to a separate discussion (Eliciting a 12fold Pattern of Generic Operational Insights: recognition of memory constraints on collective strategic comprehension, 2011), presented in the following sections:.
As noted in the first section, the argument here follows from earlier initiatives (Representation, Comprehension and Communication of Sets: the Role of Number, 1978). This had resulted in analysis of a wide range of examples (Examples of Integrated, Multiset Concept Schemes, 1984; Patterns of Nfoldness: Comparison of integrated multiset concept schemes as forms of presentation, 1980). These initiatives were themselves presented within a set of related papers (Patterns of Conceptual Integration, 1984). This included an exercise in generalizing the qualitative distinctions between insights in sets of a given number  in sets of size from 1 to 20 elements (Distinguishing Levels of Declarations of Principles, 1980).
A major consideration was the importance to be attached to the muchcited study of George Miller (The Magical Number Seven, Plus or Minus Two, Psychological Review, 1956)  and subsequent research on human working memory capacity. A related concern was the challenge of the erosion of collective memory (Societal Learning and the Erosion of Collective Memory: a critique of the Club of Rome Report: No Limits to Learning, 1980; Pointers to the Pathology of Collective Memory, 1980). The argument was then developed in relation to new ways of articulating collective principles and the quest for mnemonic facilitation (In Quest of Mnemonic Catalysts  for comprehension of complex psychosocial dynamics, 2007; Structuring Mnemonic Encoding of Development Plans and Ethical Charters using Musical Leitmotivs, 2001; Structure of Declarations Challenging Traditional Patterns, 1993).
The question here is then "why 20"  in contrast to "why not 7" or "why not 12"? Is it purely a rhetorical convenience or a coincidence of no significance? However, given that 12 is already a stretch, when 7 (plus or minus 2) has proven to be so convenient, why the greater challenge to memory of 20? On the other hand is there any significant difference from 21 ("plus or minus 2"), as might by the articulation of Yuval Noah Harari (21 Lessons for the 21st Century, 2018).
Or is the choice of 20 simply a convenient doubling of the many uses of 10  most notably the 10 Commandments, as perhaps the ultimate articulation of human operacy? This is suggested by debate regarding 20 Commandments (Troy Lacey, Are There 20 Commandments? Answers in Genesis, 2 March 2015; The Other Ten Commandments, h2g2, 18 March 2008; Wallace Wenn, The Other Ten Commandments).
More intriguing is the possibility that 20 constitutes a subtle recognition of a form of completeness  as with the coherence implied by 20 Questions. The number 20 has particular properties which may contribute to this sense of completeness, notably with respect to the integrative pattern offered by the dodecahedron (20 vertices) and its dual the icosahedron (20 faces). It is also the number of proteinogenic amino acids that are encoded by the standard genetic code.
Potentially more intriguing still is its relationship to the vigesimal number systems. In many European languages, 20 is used as a base, at least with respect to the linguistic structure of the names of certain numbers. Vigesimal systems are common in Africa; twenty was a base in the Maya and Aztec number systems.
The question of why 20 could be asked otherwise by assuming there is a strange attraction to particular patterns  in the light of the argument of Jeremy Lent (The Patterning Instinct: a cultural history of humanity's search for meaning, 2017). A set of values could then be understood as a pattern of strange attractors (Human Values as Strange Attractors, 1993). As discussed here, this could be consistent with a perspective of cognitive psychology (George Lakoff and Rafael E. Nunez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).
A related question has been raised with respect to the seemingly arbitrary articulation of a recent strategic report to the Club of Rome into 40 segments (Ernst von Weizsaecker and Anders Wijkman, Come On! Capitalism, Shorttermism, Population and the Destruction of the Planet, 2018), as separately reviewed (Exhortation to We the Peoples from the Club of Rome, 2018). The latter includes a range of images and animations to emphasize the need for more complex articulations of strategic approaches  especially those of global relevance. Why 40? Is there some unexplored sense in which a 20fold understanding of operational insights is reflected in that 40fold strategic proposal?
The following checklist is based on simple search engine queries of the web. No significance is implied in the selection. The question is what significance, if any, the collection may imply. Issues with respect to the simple clustering have not been resolved. Clearly some lists presented as "rules" could be understood as "guidelines" or "principles", for example.
Thinking / Problem solving
Learning modalities:
Lessons:
Principles
Rules
Rules (Investment)
Insights
Strategies:
Methods / Ways
Guidelines
Proposals
Arguments / Reasons
Miscellaneous
Commentary: A more systematic approach would of course use the key words (principles, rules, strategies, etc) in searches for lists based on other numbers (18, 19, 21, 22, etc) to determine whether the proportion numbering 20 is statistically significant. The results are of course biased through the focus on English alone. It was however noted that French makes extensive use of "20 propositions"  not evident in English.
If the inference that a 20fold ordering is relevant to some form of operacy, this could be said to be subtly supported by the proportion of references relating to education, thinking, learning, and skill acquisition.
Somewhat unexpected is the large number of references to the 20 methods of Eastern martial arts, of which a sample is included:
It is perhaps most surprisng to note the even larger number of references to the 20 methods for the improvement of websites, of which only a sample are included here:
Martial arts could indeed be understood as a fundamental form of operacy. Somewhat amusing is the sense in which websites are effectively considered to be the systemic counterpart to the fortresses of centuries past, or the bunkers of the present day and recent past. Websites are now the fortresses of cyberspace, to be recognized as cyberfortresses, appropriately calling for operational insights.
An initial impulse for this exploration arose from an earlier study (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). This derived from the recognition, mentioned above, that 20 is the number of proteinogenic amino acids that are encoded by the standard genetic code  perhaps the most fundamental exemplification of "operacy".
As notably promoted in the many writings of Edward de Bono, operacy is the ability of an individual person to grow, to selfregulate and to lead other people to become successful. It includes the skills needed to become successful by doing whatever needs to be done correctly and consistently (Edward de Bono. Judgment, recognition and operacy, Extensor). Operacy carries with it the concepts of empowerment, safe and fruitful completion, and efficiency in action.
Having coined the term operacy, it is therefore an appropriate coincidence that of the 56 books written by Edward de Bono, 20 of those emerge from the selection offered (at the time of writing)  sorted by "thinking tools". Of particular and unusual mnemonic value, de Bono provides a distinctive iconic image for all 56 books, and therefore for the 20 considered to be thinking tools.
In an effort to augment the sense of a coherent toolkit, the 20 icons indicative of the 20 tools lend themselves to a display in three dimensions, rather than in the twodimensional form on the De Bono website. As indicated above, the display could be based either on the dodecahedron (20 vertices) or the icosahedron (20 faces), namely its dual.
It is immediately apparent that the 20 books do not represent distinctive skills. It is more appropriate to focus on the work as a whole of Edward de Bono over decades, especially his unusual interest in diagrammatic representation of skills, notably as featured in a book which does not feature in the list of 20 (Atlas of Management Thinking, 1981). Claimed to be the first book to be written deliberately for the right side of the reader's brain, the summary indicates:
Verbal descriptions of complex management situations are necessarily lodged in the left side of the brain. In order for us to use the right side of the brain we need a repertoire of nonverbal images. That is precisely what this book is to provide. The images provided by the drawings in this book enrich the perceptual map of the executive. The images allow him or her to add some rightbrain thinking to his or her usual leftbrain thinking. This makes it easier for the executive to recognize situations in a flash instead of having to build them up piecemeal. The book has been called an Atlas because it is a reference work of visual images.
Rather than focus on the 20 titles, it is therefore of interest to focus on the 20 icons as potentially indicative of distinctive styles of operacy as understood by Edward de Bono. The question is whether further insight is suggested by their configuration in three dimensions rather than in the two dimensions of the website or the Atlas.
Indicative mapping of thinking tool titles of Edward de Bono onto polyhedral animations  
Titles on dodecahedron 20 vertices  Titles on icosahedron 20 faces  Title icons on icosahedron (unfolding) 
Animations prepared with Stella Polyhedron Navigator 
An indication of the approach might be better made using some other 20fold set, given that the titles of the works of Edward de Bono are about his treatment of thinking and do not distinguish 20 thinking tools as such.
The argument here is that any toolkit is then necessarily best to be understood in systemic terms. Indeed the images in the original Atlas of Management Thinking merit some such treatment, as implied by the relation between the 2dimensional maps in any atlas and the 3D globe from which they are derived. Given their articulation, the argument applies to the other clusterings of thinking tools (Twenty Thinking Tools, 2006; Twenty Problem Solving Skills, 2011; Twenty Methods for Improving Problem Solving, 1958). Clearly these could be experimentally mapped onto polyhedra as indicated above or below  and would as such be more indicative of the cognitive content of any toolkit.
Sense of systemic coherence? Missing from the 20fold articulations is any sense of how the operational insights function together systemically, with necessary feedback loops and learning cycles. As triggers for further reflection the 3D configurations above can be "enriched" internally to imply such connectivity, in the light of arguments developed separately (Time for Provocative Mnemonic Aids to Systemic Connectivity? 2018). One approach is to suggests that the subjective sense of coherence and completion is somehow associated with the golden rectangles integral to the dodecahedron and icosahedron as shown below.
This implies that a set of 20, for example, is somehow held to be coherent through aesthetic integrity  much studied with respect to the golden rectangle. Many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing through its embodiment of the socalled golden ratio, perhaps to be understood as just proportionality  defined by the Greek letter phi.
Golden rectangles suggestive of internal coherence of a set of operational insights  
Dodecahedron (20 vertices)  Icosahedron (20 faces)  Icosahedron (15 golden rectangles) 
Internal coherence of 20fold set? The internal coherence can be explored further as implied by use of the great dodecahedron and the great icosahedron of 20 faces  and a morphing process between the latter and its dual, the great stellated dodecahedron of 20 vertices (right below). Elements of the argument are discussed separately (Representation of Creative Processes through Dynamics in Three Dimensions, 2014; From poster sessions to stellar futures via aesthetic visualizations, 2015). The latter includes images of the succession of 18 fully 'supported stellations' of the icosahedron.
Great dodecahedron (12 vertices)  Great icosahedron (20 faces)  Great icosahedron to Great stellated dodecahedron 
Functional distinctions? Of potentially greater interest is whether other items in the checklist above suggest more generic insights into the elements of a 20fold pattern  then necessarily more subtle and difficult to comprehend, or to label. What is the distinctive functional significance of each element of a 20fold set  if that is indeed an appropriate way in which to pose the question? The degree to which the martial arts use a 20fold pattern (as noted above) suggests that such distinctions can indeed be made clear in operational terms.
As previously indicated, a related question is raised in the light of the seemingly arbitrary articulation of a recent strategic report to the Club of Rome into 40 initiatives (Ernst von Weizsaecker and Anders Wijkman, Come On! Capitalism, Shorttermism, Population and the Destruction of the Planet, 2018), as separately reviewed (Exhortation to We the Peoples from the Club of Rome, 2018). This review includes a range of images and animations to emphasize the need for more complex articulations of strategic approaches  especially those of global relevance.
Why 40 in the report to the Club of Rome? Is there some unexplored sense in which a 20fold understanding of operational insights is reflected in that 40fold remedial global strategic proposal?
Obstacles to effective strategic implementation? Given the recognized resistance to such global calls for action  implied by "Come On!" in the title  is there some relevance to the 20fold pattern of Upakleshas of Buddhism, noted above, namely the 20 secondary "hindrances" binding people to illusion. These are:
20 Secondary hindrances according to Buddhism  



Reproduced from Chinese Buddhist Encyclopedia 
Also potentially relevant to any polyhedral mapping of psychological and behavioural "resistance" to 20fold operacy, Buddhism distinguishes a 5fold and a 10fold list of hindrances (Kleshas):
There is a case for interpreting each of these hindrances for individuals in terms of their collective implications.
Reconciling a 20fold and 6fold pattern: Irrespective of the coincidental clustering by Edward de Bono of 20 "thinking tools", his main focus has been on a 6fold articulation (Six Frames: For Thinking About Information, 2008). Curiously these make specific use of 6 mnemonic forms which are potentially related to coherent mappings in 3D (circle, triangle, square, diamond, rectangle, and heart). Some of the issues have been discussed in relation to the use of such forms on playing cards (Radical Localization in a Global Systemic Context: distinguishing normality using playing card suits as a pattern language, 2015).
To the extent that there is an intuitive recognition of coherence associated with regular polyhedra, it is therefore potentially interesting that Edward de Bono's longterm focus on 6 may be related to one of the patterns of great circles of which the spherical icosahedron has 6, 10 and 15 types, as detailed by Wikipedia (31 great circles of the spherical icosahedron). It was this recognition that enabled Buckminster Fuller to construct geodesic domes. Arguably, in the desperate quest for global strategic coherence, there is a collective need for comprehensible construction of global psychosocial analogues.
In the case of the icosahedron, the great circles are variously associated with its 12 vertices (6 circles), 20 faces (10 circles), and 30 edges (15 circles). It could be argued that there is presumably a form of subtle aesthetic appreciation of such patterns which engenders the sense of coherence giving rise to 12fold, 20fold and 30fold sets.
Coherent configurations of values: Is it a complete coincidence that the Universal Declaration of Human Rights (UNDHR) is articulated in 30 articles, or the European Commission's European Pillar of Social Rights in 20 principles? Given their acclaimed fundamental significance, especially the UNDHR, is the pattern as a whole of no particular significance in implying a degree of coherence, comprehensibility and credibility?
There is of course the possibility of mapping the UNDHR articles onto the 30 edges of the dodecahedron or of the icosahedron in quest of possibilities of eliciting greater coherence from the set. In an earlier exercise the 30 articles were mapped onto 30 faces of a rhombicosidodecahedron. The 18 articles of the European Convention on Human Rights were mapped onto faces of a rhombicuboctahedron, and the 53 articles of the Arab Charter on Human Rights were mapped onto faces of a rhombicosidodecahedron (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008).
Packing and morphing insights? Elaborating any set of insights could be said to call upon some sense of "packing" them together. It would be intriguing to explore any debate on extending a 20fold set to 21, or reducing it to 19  or extending a 30fold set to 31 or reducing it to 29. The issue has been evident in the case of the extension of the UN's Millennium Development Goals from 8 to the 16+1 of the Sustainable Development Goals. In the latter case the 17th Goal (Partnerships for the goals) is understood as coordination among the 16.
The issue of packing is the subject of special concern in terms of socalled sphere packing, namely an arrangement of nonoverlapping spheres within a containing space. The question here is whether there is a cognitive equivalent that merits attention. What then is the "containing space" of any kind of toolkit?
Dodecahedron (20 vertices)  Icosahedron (12 vertices)  
Spheres "unpacked"  Spheres "packed" / touching  Spheres "unpacked"  Spheres "packed" / touching 
Images prepared with Stella Polyhedron Navigator 
It has been established that the cuboctahedron is especially significant to the process of sphere packing. It can be used to provide an array of 12 Archimedean polyhedra in their most regular array around a truncated tetrahedron (omitted from the animation on the right). This approach has the merit of distinguishing visually the elements of a 12fold pattern, as is especially apparent from the animation on the right. The array is uniquely significant in terms of sphere packing in that the 12 are then all in contact with the 13th at the centre as extensively documented by Keith Critchlow (Order in Space: a design source book, 1969). Is there a corresponding need for "order in cognitive space" as reflected in 12fold, 20fold and 30fold sets of insights?
Dodecahedron/Icosahedron compound (32 vertices) 
Cuboctahedron (12 vertices) 
Cuboctahedral array of 12 Archimedean polyhedra 
Rotation of 12fold array of Archimedean polyhedra 
Of further relevance in the case of the cuboctahedron is that it can be transformed dynamically through a muchstudied "jitterbug" motion into other configurations, most notably the icosahedron (Robert W. Gray, Jitterbug Defined Polyhedra: the shape and dynamics of space, 2001; H. F. Verheyen, The Complete Set of Jitterbug Transformers and the Analysis of their Motion, Computers and Mathematics with Applications, 17, 1989, 13; Joe Clinton, R. Buckminster Fuller's Jitterbug: its fascination and some challenges, Synergetics Collab, 2011). Many videos of that movement have been produced (Buckminster Fuller's Jitterbug, 2007).
Such transformation are suggestive of a degree of cognitive continuity between various coherent Nfold patterns of insights, as discussed separately (Time for Provocative Mnemonic Aids to Systemic Connectivity? 2018).
Essential elusiveness of insights: Many sets of insights tend to be framed either in terms of what to do or what not to do. This leaves unclear what the insight is, especially if it involves striking some kind of dynamic balance between both extremes  a balance for which no singular label is adequate or particularly comprehensible. A further difficulty is that, as presented in checklists of insights, the terms use may be ambiguous or susceptible to various interpretations. This difficulty was addressed in the Human Values Project within the context of the Encyclopedia of World Problems and Human Potential.
There use was made of synonyms and antonyms of valuecharged words: 987 "constructive" and 1992 "destructive". These were then clustered in terms of 230 value polarities. The 20 Secondary hindrances according to Buddhism (as listed above) can be used as a point of departure in order to identify 20 implied cognitive attitudes or skills, given that the hindrances imply an attitude by which they can be transcended. The merit of that list is that it has a long tradition behind it and is relatively stable. The following is a very preliminary attempt to reconcile the Buddhist set with those value polarities, as was done with "destructive" concepts in the original project.
Buddhist secondary hindrances  Value polarities 
belligerence  AccordDisaccord / FriendshipEnmity / CongratulationEnvy 
resentment  CongratulationEnvy / ContentmentDiscontentment 
concealment  ModestyVanity / NaturalnessAffectation 
spite  LoveHate / FriendshipEnmity / KindnessUnkindness / GoodnessBadness 
jealousy  CongratulationEnvy / VirtueVice / ContentmentDiscontentment 
miserliness  EconomyProdigality / TemperanceIntemperance 
deceit  ProbityImprobity / SkilfulnessUnskilfullness / CommunicativenessUncommunicativeness 
dissimulation  ModestyVanity / NaturalnessAffectation 
haughtiness  ModestyVanity / PrideHumility / RespectDisrespect / SupportOpposition 
harmfulness  GoodnessBadness / KindnessUnkindness / HealthulnessUnhealthfulness 
nonshame  VirtueVice / ProbityImprobity / ModestyVanity / RightnessWrongness / ExultationLamentation / NaturalnessAffectation 
nonembarrassment  PrideHumility / WealthPoverty / FacilityDifficulty / AttentionInattention / CertainityUncertainty 
lethargy  ActionInaction / FeelingUnfeelingness / IntlligenceUnintelligence 
excitement  DesireAvoidance / MotivationDissuasion / EloquenceUneloquence / AttentionInattention / ExcitementInexcitability / FeelingUnfeelingness 
nonfaith  VirtueVice / BeliefUnbelief / ProbityImprobity / CertaintyUncertainty / PietyImpiety / DesireAvoidance / HopeHopelessness / ObedienceDisobedience 
laziness  ActionInaction / CarefulnessNeglect / TimelinessUntimeliness 
nonconscientiousness  TasteVulgarity / ProbityImprobity / DuenessUndueness / CarefulnessNeglect 
forgetfulness  KindnessUnkindness / CarefulnessNeglect / RemembranceForgetfulness 
nonintrospection  
distraction  SanityInsanity / AttentionInattention 
Framing ambiguity: The above exercise raised a variety of issues calling for further investigation. However it served to further highlight the underlying problem of the ambiguity of any terms with which a 20fold set of thinking tools is distinguished. It suggested the further possibility of variously framing that ambiguity in order to indicate the nature of a wellformed set of 20 subtle insights.
The possible approach is suggested by the following images based on the icosahedron (20 faces) and its dual the dodecahedron (20 vertices)  both with 30 edges. In the images below the edges are treated as polarities, namely a dynamic calling for a cognitive balance in practice between extremes. In the case of the icosahedron 3 such polarities frame a complex  the elements of the 20fold set. The vertices can themselves only be labelled ambiguously. It is this configuration as a whole which offers insights into the coherence of the set  despite any ambiguity in the labelling of its components.
Indicative schematic mappings framing dynamically cognitive insights of ambiguous subtlety  
Icosahedron  Dodecahedron 
Images prepared with Stella Polyhedron Navigator 
The challenge is then to explore the use of such a framing in the case of specific 20fold sets, and any correspondence with 12fold sets and 30fold sets.
Encoding distinctiveness in the light of regular polyhedra:
Distinctions between 5 Platonic and 13 Archimedean polyhedra Notation of (3, 4, 3, 4) means each vertex contains a triangle, a square, a triangle, and a square, in that cyclic order. 

235fold symmetry  234fold symmetry  
(3, 3, 3) tetrahedron  
(3, 3, 3, 3, 3) icosahedron  (3, 3, 3, 3) octahedron  
(5, 5, 5) dodecahedron  (4, 4, 4) cube  
(3, 6, 6) truncated tetrahedron  
(5, 6, 6) truncated icosahedron  (4, 6, 6) truncated octahedron  
(3, 5, 3, 5) icosidodecahedron  (3, 4, 3, 4) cuboctahedron  
(4, 6, 10) truncated icosidodecahedron  (4, 6, 8) truncated cuboctahedron  
(3, 3, 3, 3, 5) snub dodecahedron  (3, 3, 3, 3, 4) snub cube  
(3, 4, 5, 4) rhombicosidodecahedron  (3, 4, 4, 4) rhombicuboctahedron  
(3, 10, 10) truncated dodecahedron  (3, 8, 8) truncated cube  
Presentation following Keith Critchlow (Order in Space: a design source book, 1969). 
With edges understood as polarities, the vertices at which they intersect are necessarily indicative of a higher order of ambiguitysubtlety. In the relation between the icosahedron (12 vertices) and the dodecahedron (12 faces), there is perhaps a greater challenge to comprehension of the elements of a 12fold articulation than for a 20fold articulation.
"Flexibility" of insight sets? The ambiguity and elusiveness of operational insights  and of values more generally  suggests that the obvious "rigidity" of polyhedra, notably as typically portrayed, may well be more than misleading when used for mapping purposes. As is evident with respect to cognitive skills, their use implies a dynamic between different skills in the light of their respective advantages and disadvantages in a particular situation.
The form of a polyhedron reinforces implications (for cognitive mapping purposes) that it it can be treated as a socalled Newtonian material, namely one that exhibits a linear relationship between stress and strain rate from the perspective of materials science. More relevant to the dynamic ambiguity of the insights of concern here is the possibility that their relation to any polyhedron could be better understood in terms of nonNewtonian properties. Examples include:
These are suggestive of other ways of considering the nature of insight sets and the relationships between their elements. When the insights are values or principles, it then becomes clearer how they take solid, rigid or nonnegotiable form through their definition and use as slogans  but that they of a more fluid or liquid form as experienced in practice, or when put to the test. In this sense any mapping might be better made on a relatively malleable spherical ball, responsive to any temporary encounter with obstacles. The polyhedra noted above might then take the form of spherical polyhedra.
A striking example of this is the widely familiar association football (below left). The pattern is of the form of a spherical truncated icosahedron with 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. Given the argument above, there is a degree of irony to the fact that it is kicked and passed around in the quest for goals. It is necessarily flexible to a degree. It can be compared with the rigid Buckminsterfullerene a form of carbon with a cagelike fusedring structure (truncated icosahedron) that obviously resembles a soccer ball.
Spherical tensegrity: However it is the spherical tensegrity structure on the right which perhaps best illustrates the dynamics of a set of cognitive insights. A simpler form of direct relevance to this argument is the icosahedral tensegrity (variously illustrated in that link, and separately).
As the basic for the architectural integrity of geodesic domes, such a tensional integrity, or floating compression, is a structural principle based on the use of isolated components in compression inside a net of continuous tension, in such a way that the compressed members (usually bars or struts) do not touch each other and the prestressed tensioned members (usually cables or tendons) delineate the system spatially.
Suggestively comparable spherical forms with dynamic implications  
Football  Buckminsterfullerene  Tensegrity 
The question is whether a wellformed set of cognitive insights must necessarily be based on the structural principles of a tensegrity  as argued from the perspective of management cybernetics by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994). Given the dynamic ambiguity stressed above, it is perhaps appropriate to understand the rigid (noncompressible) and flexible (tension) elements alternating in a cognitive variant. The alternation could well be a matter of degree, rather than a binary switch between extremes.
Beer's work focused on the relevance of the icosahedron to discourse (Gunter Nittbaur, Stafford Beer's Syntegration as a Renascence of the Ancient Greek Agora in Presentday Organizations, Journal of Universal Knowledge Management, 0, 2006; J. Truss, C. Cullen, and A. Leonard, The Coherent Structure of Team Syntegity: from small to mega forms). The principle was used on the occasion of the UN Conference on Environment and Development (1992) in configuring strategic dilemmas in intersectoral dialogue, notably through mapping a Representation of Issue Arenas on Icosidodecahedral Net (1992).
In the light of the above argument for "flexibility" and a "nonNewtonian" approach to the configuration of insight sets  given their subtle ambiguity  further inspiration may be sought from the structure of the molecule fundmental to organic chemistry, biology and life  namely the iconic benezene molecule. The strange manner in which that structure was famously recognized in a dream by August Kekulé merits attention. In particular of interest is current understanding of the molecule as a resonance hybrid. To correctly interpret the molecular structure described by a resonance hybrid, all significant contributors of the resonance hybrid must be considered together, since the hybrid represents the actual molecule as their "average", Why indeed should it be assumed that fundamental cognitive sets should be of lesser complexity than the structure so fundamental to the viability of life?
Benzene molecule in KekulÉ's dream  Alternation in benzene molecule as a resonance hybrid 
By Haltopub  Own workbasÉ sur Benzene Structural diagram.svg et Ouroborossimple.svg, CC BYSA 3.0, Link;  Reproduced from Wikipedia 
The argument above suggested that Edward de Bono's Six Frames: For Thinking About Information (2008) could be usefully recognized as associated with 6 polyhedral great circles, most notably 6 of the 31 great circles of the spherical icosahedron. Of interest in the following images, the edges associated with great circles are not continuous in the two Platonic polyhedra on the left. They are however continuous in the two Archimedean polyhedra on the right. The 10 edges in each of the 6 great circles in the case of the icosidodecahedron (of 60 edges) can then be compared with the 6 edges in the 4 great circles of the cuboctahedron (24 edges).
Each great circle could then be understood as a form of cognitive resonance hybrid with its particular integrity and stability. As configured together they form a kind of cognitive cage of even greater integrity  for which, as chemical metaphors, clathrates and inclusion molecules raise valuable questions. How these various arrangements relate to the concerns above with regard to 12fold, 15fold, 20fold and 30fold sets is a matter for further consideration. Some indications are offered by LiYuan Zhang (Selfequilibrium and superstability of truncated regular polyhedral tensegrity structures: a unified analytical solution, Proceedings of the Royal Society A, 25 July 2012).
Edges of polyhedra distinctively coloured according to the great circles with which they are associated  
Platonic polyhedra  Archimedean polyhedra  
Dodecahedron  Icosahedron  Icosidodecahedron  Cuboctahedron 
Images prepared with Stella Polyhedron Navigator 
The introduction to this argument noted the importance attached to the muchcited study of George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review, 1956)  and subsequent research on human working memory capacity. Any such consrain is clearly of particular relevance to comprehension of the many 20fold cognitive toolkits cited above  whether of principles, strategies, rights, or values. Presumably it is both a constraint on comprehension of the 10 Commandments, or of the 30 rights in the Universal Declaration of Human Rights  "as a whole". In the latter case, it might be asked why the set of rights is effectively beyond human "working memory capacity"  without any consideration of the matter.
More generally it could be asked, if there is a 7fold constraint on use of a cognitive toolkit of requite variety, how is it appropriate to address the challenge of enabling people and society to "get their act together"  and to keep it together in the light of any aspirations to sustainability.
What is a required is a way of relating the 7fold to the 20fold. One of the polyhedral configurations above offers an approach to this. The cuboctahedron of 14 faces (above right) has 7 axes of symmetry through those opposing faces. As noted above, it also offers a muchstudied path of transformation between the Platonic polyhedra through the socalled jitterbug process explored through synergetics by Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979). Unfortunately, as discussed separately, Fuller does not interpret the promise of that title with respect to the cognitive issues explored here (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).
Mapping options: Those axes can be used as a means of mapping various 7fold sets for purposes of comparison. It is a means of giving an ordered focus to the confusion of decisionmaking and choice, whether in the momentary hereandnow for the individual, or on a larger scale for elaboration of global strategy. The examples mapped arbitrarily onto those axes below derive from:
Examples of 7fold sets mapped arbitrarily onto 7 axes of symmetry of a cuboctahedron  
7 WH Questions  7 Axes of Bias (Jones, 1961)  7 Pairs of Opposites (Robinson, 2018) 
Axes of symmetry generated by Stella Polyhedron Navigator. Axes through the vertexes  mauve  are not used 
Oppositional logic: Each example highlights the challenge of dilemmas in engaging with the elements of a 20fold set  especially those taking the form of opposites to be reconciled. Of great potential relevance are ongoing explorations of oppositional logic and oppositional geometry, as discussed separately (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of antiotherness, 2018; Neglected recognition of logical patterns  especially of opposition, 2017). The latter discusses explorations of logical geometry and Aristotelian diagrams, as most recently summarized in a very comprehensive paper by Lorenz Demey and Hans Smessaert (Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation, Symmetry, 2017). That paper develops the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, it focuses on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis hexahedron [dual of the truncated octahedron] the tetraicosahedron and the nested tetrahedron.
With respect to the above argument, the rhombic dodecahedron (12 faces, 14 vertices) is of interest as the dual of the cuboctahedron. Rather than mapping any 7fold pattern of dilemmas onto the cuboctahedron, it could then be more appropriately mapped onto a rhombic dodecahedron where the 7 axes through 14 faces (as above) are now presented as 14 vertices through which 7 axes pass (as shown below). The form is used to distinguish different extremes of the 7 WHquestions  contrasting the local hereandnow for the individual with the global challenges of collective strategy. The elusive compromise is then associated with the neologism glocal  essentially dynamic. It is understood as the simultaneous occurrence of both universalizing and particularizing tendencies in contemporary social, political, and economic systems. In the animation below it is usefully indicated by the central sphere  as a nexus of dynamic reconciliation of dilemmas.
Experimental 7fold mapping possibilities  
7 WHquestion dilemmas on cuboctahedron and its dual  Szilassi polyhedron  
Cuboctahedron (rotation) sides distinctively coloured 
Rhombic dodecahedron (rotation) edges coloured by great circle 
7sided Szilassi polyhedron (rotation) sides distinctively coloured 
Mapping of questionpairs (some sides transparent) 
Animations generated by Stella Polyhedron Navigator 
How distinctions could be usefully mapped in relation to one another onto such forms remains to be investigated. As noted above, one consideration in the case of the 7 WHquestions follows from exploration of their possible conformality to the set of 7 elementary catastrophes variously studied by René Thom (Structural Stability and Morphogenesis, 1972; Semio Physics: A Sketch, 1990).
Like the pairs of 8 triangular faces and 6 square faces of the cuboctahedron, these fall into two groups: 4 potential functions of one active variable (fold catastrophe, cusp catastrophe, swallowtail catastrophe, butterfly catastrophe) and 3 potential functions of two active variables (hyperbolic umbilic catastrophe, elliptic umbilic catastrophe, parabolic umbilic catastrophe). With the WHquestions understood as "cognitive catastrophes" of a kind  especially in the practice of global discourse  their possible relation to those catastrophes is discussed separately (Correspondence of WHquestions to elementary catastrophes, 2006).
Collapsing 7fold dilemmas? As axes through opposing vertexes, the dynamics of the associated set of cognitive challenges are usefully held in a global configuration. Of interest is the consequence of "collapsing" this challenge. This collapse may be usefully explored through a polyhedron of 7 sides  with "sides" then indicative of fixed positions in a dialogue. Valuable in this respect is the highly unusual Szilassi polyhedron (above centre) where the 7 sides (of 4 types) are each in contact with all the others. Like the rhombic dodecahedron, it has 14 vertices (but of 7 types). It has however 21 edges (of 12 types).
The elegant symmetry of the polyhedra discussed above is seemingly completely lost in a form which could be seen as a useful reflection of the ugliness of the "collapsed discourse" which is currently so characteristic of global debate. The Szilassi polyhedron is a curious reflection of mistaken assumptions regarding how to "get any act together". Having the "sides" all "in touch" with each other  upheld in the symbolism of many conferences  clearly avoids more fundamental issues of conceptual configuration (Visualization Enabling Integrative Conference Comprehension: global articulation of futureoriented 3D technology, 2018).
Whilst the sides of the Szilassi polyhedron can usefully hold the 7 WHquestions, potentially more interesting in the quest for coherence, is the use of the edges to hold pairs of WHquestions, as shown in the image on the right, and discussed separately (Mapping of WHquestions with questionpairs onto the Szilassi polyhedron, 2014). Somewhat ironically, given the above argument regarding the football, a related mapping can also be explored with respect to the "beautiful game" (Mapping of WHquestions with questionpairs onto a memorable polyhedron (a football), 2014).
The Szilassi polyhedron holds further surprises in its relation to the strange dynamics of cube inversion, as illustrated separately (Association of the Szilassi polyhedron with cube inversion, 2018). As a consequence, it offers insights into the Dynamics of discord anticipating the dynamics of concord (2018). Animations there show an unsuspected coherence to the dynamics of either 6 or 7 Szilassi polyhedra variously connected.
Truncation of cognitive sets: A key question is how a set of 30 insights, such as is embodied in the Universal Declaration of Human Rights, gets "reduced" to the 20 insights embodied in the European Commission's European Pillar of Social Rights in 20 principles, or to the UN's 15(+1) Sustainable Development Goals  itself "expanded" from the UN's Millennium Development Goals.
A valuable clue to this "cognitive operation" is provided by the wellstudied and illustrated geometry of truncation (Truncatering; Livio Zefiro, Vertex and edgetruncation of the Platonic and Archimedean solids leading to vertextransitive polyhedra). This is an operation in any dimension that cuts polytope vertices, creating a new facet (or side) in place of each vertex. The term originates from Kepler's names for the Archimedean solids. The Wikipedia entry illustrates the distinction between uniform truncation, edgetruncation, and alternation (or partial truncation), and generalized truncations. How are patterns of insights then to be understood as variously "truncated"?
It is then of interest to recognize how a set of principles mapped onto vertexes, at the extremes of axes of symmetry, get transformed by effectively turning each original vertex into a side  thereby increasing the number of vertexes. As can be seen from the animations below, the truncation is associated with the creation of a side closer to the centre of axial symmetry  the side expands in size as it moves away from the original vertex. This process could be understood as implying one of "taking sides" rather then recognizing what the original vertex "stood for".
In the animations below each polyhedron gets transformed through transformation into its dual. Thus the icosahedron of 12 vertex ("principles") gets truncated into a dodecahedron of 12 sides  itself to be understood as characterized by 20 (lesser?) principles. In the reverse process, the 20 principles are conflated ("integrated") to constitute 12 principles. A corresponding animation of the cuboctahedron of 12 vertexes is shown transforming into its rhombic cuboctahedral dual of 12 sides. In all the animations various intermediary configurations are evident.
Animations illustrating morphing of polyhedra by truncation around axes of symmetry (some faces transparent in animation on right) 

Icosahedron to/from Dodecahedron  Rhombic dodecahedron to/from Cuboctahedron  Dodecahedron to/from Icosahedron 
Animations generated by Stella Polyhedron Navigator 
The above animations fail to show the consequence of truncations of different degrees along different axes, or from the extremes of the same axis. These would typically generate polyhedra of greater or lesser degrees of regularity (understood as an approximation to a "global" configuration). This could then be seen as corresponding to a greater or lesser degree of adherence to a principle in some cases, in contrast with "taking sides" to a greater or lesser degree in others.
An example of a particular interest would result from truncation with respect to a single down to the centre of axial symmetry, as might correspond to the complete failure to recognize the principle associated with one extreme of that axis. The principle could then be said to have been transformed completely into a side, with all that that would imply for the nonglobality of discourse. The images below are then interesting illustrations of that condition.
Illustrations of consequences of truncation along various axes  
Decagonal prism  Dual of decagonal prism  Pentagonal rotunda  Pentagonal gyrobicupola  Dual of pentagonal gyrobicupola 
12 faces, 20 vertices, 30 edges  20 faces, 12 vertices, 30 edges  20 vertices, 17 faces  20 edges, 40 vertices, 22 faces  20 faces, 40 edges, 22 vertices 
Images generated by Stella Polyhedron Navigator 
Juxtaposition of the dual of a polyhedron frames the provocative possibility that if one serves as a recognizable mapping of a set of conscious cognitive insights (as in any declaration), the other may constitute an indicative mapping of the set of insights which is effectively denied or of unconscious significance. This would then be a mapping of the shadow in psychoanalytical terms, and as implied by the arguments of John Ralston Saul (The Unconscious Civilization, 1995). The variety of polyhedra would then offer a pattern language through the collective shadow could be discussed in the light of the cognitive biases implied by the truncations along various axes of bias
Transformational "route maps"? The relation between the articulations of insights suggestively mapped onto the polyhedra indicated above 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 (Pathway "route maps" of potential psychosocial transformation? 2015). As indicated there, this might be understood as a Polyhedral metapatterns of relationships? (2015).
Map highlighting distinctive relationships pathways between spherically symmetrical polyhedra
(regular and semiregular) F=faces, E=edges, V=vertices (Total of these in parenthesis) [Total reduced to prime number, other than 2, in square brackets] 
The emphasis above has been on rigid polyhedra and the possibility of their transformation through truncation. In practice emphasis could be placed at any moment on one or more concepts in a set of insights. Although these can be understood as configured in polyhedral form  as a 20vertex dodecahedron, for example  in particular circumstances this could be "distorted" out of its ideal regularity in response to circumstances. As mentioned above, the edges linking concepts could then be likened to rubber bonds, stretching or shortening within the constraints of the set as a whole.
A valuable approach to this is through the use of forcedirected graph drawing, characteristic of DataDriven Documents (d3.js), to elicit selforganizing convex polyhedra  without the conventional prerequisite for vertex coordinates (Elijah Meeks, D3.js in Action, 2015). The approach is discussed separately (Use of force directed layout to elicit memorable polyhedra, 2015). The following interactive examples are reproduced from that exercise. The polyhedra can be "distorted" by interactive use of the mouse. Some implications of such distortion are also discussed (Potential significance of memorable irregularity? 2015).
Of greater relevance to the above argument are the conditions in which the "force" distorting the polyhedra is a consequence of conscious or unconscious emphasis (or deprecation) on particular insights (or strategic modalities) in relation to others. This could be understood as sensitivity or insensitivity to particular axes, or to one or other extreme on that axis. It can be usefully reflected in exaggerating or diminishing those particular axes of symmetry  implicit, and therefore "hidden", with respect to visualization of the polyhedral articulation of the set.
Screen shots of selected results of forcedirected layout for selected polyhedra (not to scale relative to one another; subject to further tests; interactive animations currently only work in Internet Explorer) 

Dodecaahedron (20 vertexes) 
Cuboctahedron (forcelayout animation; gif; mp4) 
Truncated cuboctahedron (forcelayout animation) 
Drilled truncated cube (64 edges, 32 vertexes, 32 sides)  
(forcelayout animation)  (rotation)  
The images on the right illustrate the possibilities of mappings onto more complex polyhedra with their particular coherence and memorability. A proof of concept exercise for mapping on the drilled truncated cube is discussed separately (Changing Patterns using Transformation Pathways, 2015). The approach suggests the nature of the coherence offered by dynamic integrity  subject to a variety of distortions  beyond the rigid integrity of polyhedra as commonly understood.
There is worldwide familiarity with Rubik's Cube and the competitive approach to its solution. Its surfaces can be used as mapping devices for strategic preoccupations  notably the Sustainable Development Goals of the UN (Interplay of Sustainable Development Goals through Rubik Cube Variations, 2017). Those developing skills for its solution use algorithms and competencies which could be usefully considered comparable to those required for operacy in other domains.
Much has been made of discovering the minimum number of moves required to solve the cube. However it is only recently that it has been determined that the minimum number of moves in which it can be solved is 20  now variously acclaimed (Rubik's Cube (solved in 20 moves or less), DailyInfographic, 21 November 2013). The process has been described in the following terms by those involved (God's Number is 20):
With about 35 CPUyears of idle computer time donated by Google, a team of researchers has essentially solved every position of the Rubik's Cube™, and shown that no position requires more than twenty moves. We consider any twist of any face to be one move (this is known as the halfturn metric.)
Every solver of the Cube uses an algorithm, which is a sequence of steps for solving the Cube. One algorithm might use a sequence of moves to solve the top face, then another sequence of moves to position the middle edges, and so on. There are many different algorithms, varying in complexity and number of moves required, but those that can be memorized by a mortal typically require more than forty moves.
One may suppose God would use a much more efficient algorithm, one that always uses the shortest sequence of moves; this is known as God's Algorithm. The number of moves this algorithm would take in the worst case is called God's Number. At long last, God's Number has been shown to be 20.
It took fifteen years after the introduction of the Cube to find the first position that provably requires twenty moves to solve; it is appropriate that fifteen years after that, we prove that twenty moves suffice for all positions.
Aside from preoccupation with solutions of Rubik's Cube, it is also interesting to note the concern with twenty in other games, most notably chess:
Upheld as the essence of strategic thinking, this focus in chess merits careful consideration in the light of the generic question raised above.
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