Introduction
Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension?
Dynamic bonding patterns in ntuple helices engendering nfold rotating symbols
Embedding the triple helix in a spherical octahedron
Embedding the quadruple helix in a spherical cube
Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum
Embedding sixfold, eightfold and tenfold helices in appropriately encircled polyhedra
Embedding twelvefold, elevenfold, ninefold and sevenfold helices in appropriately encircled polyhedra
Neglected recognition of logical patterns  especially of opposition
Dynamic relationship between polyhedra engendered by circles  variously implying forms of unity
Symbol rotation as dynamic essential to engaging with valueinversion
References
The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (15641642). His muchcited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse  known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other muchvalued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. Recognizing this movement may be vital to understanding how the groups associated with them are interrelated  rather than being opposed to one another, if not violently so as is commonly the case.
The argument here assumes that much more is based on circles than is conventionally held to be the case. They can be explored as having a fundamental function in engendering order. On the one hand circles are readily understood to frame and identify social circles, their bounded nature, and an implied dynamic  acquiring one manifestation in the arenas of many games and the holelike structure of goals in many team sports. This extends to a degree of understanding of rings and cycles  exemplified by circlets of beads. On the other hand much is understood about the geometry of circles, most notably for this argument through the manner in which as great circles (with a common centre) they variously frame the spherically symmetrical polyhedra (5 Platonic and 13 Archimedean)  as approximations to a sphere. In this sense those polyhedra might then be better understood in dynamic terms as patterns of cycles, if not as standing wave forms. The contrasting preoccupations of such social circles suggests that they can be usefully recognized as variously "oriented" to one another in a geometric sense.
The interlocking of great circles frames polygons on a polyhedral surface  typically triangles, squares, pentagons, hexagons, etc. These are matched by corresponding polygons on the opposite side of the polyhedron  but typically inverted. Between such opposing polygons, it is potentially significant to explore the manner in which spirals link them through the polyhedral centre. The spirals may be triple, quadruple, quintuple, etc  effectively "engendered" by the sides of the polygon (whether three, four, five, etc), Those sides are themselves defined by the great circles variously oriented to one another. A triangular polygon thus "engenders" a triple helix, a pentagon a quintuple helix, as previously described and illustrated (Psychosocial Learnings from the Spiral Form of Hurricanes: implications of the triple helix and the 3fold triskelion as "cognitive cyclones"? 2017).
Of further relevance, polyhedra have acquired new significance as mapping devices in oppositional logic for ordering patterns of connectivity in Boolean logic and truth tables. Of concern is the extent to which current academic preoccupations with polyhedra are fundamentally impersonal and "unsociable"  and proud to be so. They are dissociated from psychosocial experience to the point of being essentially meaningless and irrelevant to a civilization in crisis. New forms of cognitive "bridging" are therefore required to the cultural symbols with which people have identified over centuries. The point can be argued with respect to the remarkable work of Buckminster Fuller, so specifically upheld as the "geometry of thinking" (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).
Given the inspiration via biomimicry of the twostranded DNA helix, the question then arises as to the contrasting patterns of bonding between the strands of a triple helix, a quadruple helix, or quintuple helix, for example. There is the curious possibility that this pattern of bonding may be very suggestive of common cultural symbols with which different groups variously identify  notably in star form (whether triangular, square, pentagonal, hexagonal, etc.). Those bonding symbols might then also be understood as "emerging" from the helical patterns, or "sinking" into them through a "cognitive wormhole" at the centre (and inverted through it). The academic modality is seemingly unable to engage effectively with the psychosocial function and holding power of symmetry, readily comparable in its beauty with the hypnotic power of a whirlpool. How indeed do symbols become psychoactive?
In moving and transforming the symbols may rotate in ways suggestive of new kinds of insight, especially in 3D visualization (Envisaging NATO Otherwise  in 3D and 4D? Potentially hidden faces of global strategy highlighted through polyhedra, 2017). The movement in relation to the spiral "strings" might also be usefully associated with patterns of sound of mnemonic value, as explored through sonification.
An earlier experimental exploration of this kind  of far simpler structure  is presented separately (Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation, 2013; see video). An application of current relevance is separately described (Middle East Peace Potential through Dynamics in Spherical Geometry: engendering connectivity from incommensurable 5fold and 6fold conceptual frameworks, 2012).
Many symmetrical symbols appear to serve as inadequately acknowledged "holding patterns" for significance. With flags bearing symbols of unity accorded overriding significance on every official occasion, those symbols would seem to merit "upgrading" as devices for heralding governance of change in an informationbased society. Is the assumption of the silent static nature of logos and cultural symbols essentially misleading? Is there indeed a valuable possibility to Eppur si muove?
The following argument can be explored through various threads whose relationship could be mapped as follows  offering a variety of directions of reading.
Symbolic stars: The psychosocial challenge of the times is partly indicated by the extensive range of symbolic stars, whether associated with unity understood in a religious or political sense. They are most obvious on national flags. Notable also is the use of such stars in awards for valour, as with the 5pointed Africa Star, the Bronze Star and Silver Star of the USA, and the various gold star medals of other countries. What indeed might be the implications of the 5pointed star on the reverse face of the medal of the Nobel Prize in Economics  as marking the highest order of economic insight? Why are the celebrities of many profession so commonly described as "stars"  with careers framed as an aspiration to "stardom"?
Strategic pillars: It is surprising to note the number of different sets of strategic pillars elaborated by international institutions and groups. There is the strange possibility that the enthusiasm of many institutions for a specific number of strategic "pillars" could be understood as a more concrete manifestation of the values they associate with the symbolic stars of their iconography serving architecturally as cognitive keystones. The pillars, although not meaningfully configured radially, are then effectively the points of an invisible symbolic star upheld as the elusive integration of the values embodied in the strategic pillars.
Any articulation in terms of valuepillars begs the question as to how they are understood to be systemically and functionally interrelated, other than as a checklist  an issue considered by the Chief Executives Board for Coordination of the United Nations System (Coordination among pillars of the United Nations System). Within the UN, such integration is understood as most necessary with respect to sustainability (Action on crosspillar linkages). However what are the "overarching" forms which link the tops of the "pillars" to the "stars" which reconcile them  and provide support for the "roof" of the cognitive shelter?
Polyhedra: However they may be depicted, the symbolic stars are then emblematic of what is occasionally termed "joinedup thinking"  in contrast with the typically disparate nature of questionably related pillars (Coherent Value Frameworks: pillarization, polarization and polyhedral frames of reference, 2008). The latter notes the possibility of pillars in a polyhedral configuration  echoing the significance of polyhedra in this argument.
Helices: The approach taken here is a development and generalization of that introduced in earlier papers:
The focus there was on the contrast between the longstanding significance associated with variants of a threefold traditional symbol (the triskelion), and that associated with emergent strategic interest in the Triple Helix model. The question was how their cognitive and psychosocial dimensions might be more effectively related  especially given the enduring quality of the traditional symbol. Reference was also made to tentative proposals for a Quadruple Helix model and a Quintuple Helix model (Elias Carayannis and David F. J. Campbell, Triple Helix, Quadruple Helix and Quintuple Helix and How Do Knowledge, Innovation and the Environment Relate To Each Other? International Journal of Social Ecology and Sustainable Development, 1, 2012). A variety of animations was provided in support of such explorations using 3D and virtual reality technology. Implicit reference to the biological role of the double stranded DNA in mutation and morphogenesis reinforces the potential relevance of any analogues to psychosocial analogues, notably as highlighted by René Thom (Structural Stability and Morphogenesis: an outline of a general theory of models, 1972).
The question here is how to take this argument further using a more systematic approach to take more effective account of the variety of implications explored by related disciplines. The primary concern is how to embody more widely meaningful coherence to a greater degree  in a coherent set of interrelated cognitive models.
Logic: There has been considerable development of logic, most notably that of oppositional logic and the concern with truth tables. This clarification is potentially vital to a society confronted with opposing strategic forces and perspectives as well as by fake news. Of particular relevance to this argument is the manner in which these distinctions are configured on starlike polygons and particular polyhedra.
Comprehension (through movement): The threads above can then be understood as contrasting framings of ways of knowing, best recognized as potentially complementary. There is clearly concern as to how strategic issues and possibilities are to be comprehended, especially if their articulation is necessarily more complex than would otherwise be preferred  given the cmplexities of society and the challenges of its governance. The need for movement to encompass greater complexity meaningfully is presented as the justification for the rotation of symbols rather than their static depiction. Their rotation in practice invites speculation (Moving Symbols: radical change in psychosocial energy possibilities? 2008).
Development of the argument:
In contrast to each of the above modalities is the curious role variously attributed to multiple circles and rings, whether as a pillar or line (edge on) or as some form of cognitive container (implying a sphere). Examples include: Claus Leggewie, Seven circles of European memory (Eurozine, 20 December 2010) and the Six Circles of an Effective Coalition (Georgetown Law). Mathematical problems are notably associated with specific numbers of circles. By contrast, a widely appreciated fantasy is built on the imaginative myth of a set of "rings of power" with "One Ring to bring them all and in the darkness bind them" (Cognitive Cycles Vital to Sustainable SelfGovernance: The Lord of the Rings as an emergent integrative dynamic, 2009; The "Dark Riders" of Social Change: a challenge for any Fellowship of the Ring, 2002).
Missing dimensions? Whether as medals or on flags, the unthinking appreciation of stars suggests intuitively valued implications which are missing in ways which might be usefully called into question:
Together these reinforce the arguments of Terrence Deacon as a cognitive anthropologist, that what is missing is what is missing (What's Missing from Theories of Information? 2010). Expressed otherwise, if values can be fruitfully understood as "strange attractors", any configuration of them is then liable to be even "stranger" (Human Values as Strange Attractors, 1993).
The question here is how a symbol emerging from an ntuple helical pattern might variously interrelate the separate helical pathways as it rotates (and as it might descend into the central cognitive wormhole). The examples below are purely suggestive with each vertex to be understood as in contact with a helical strand. Each line in the symbol may alternative between various conditions  as alternate types of bond. The perspective on the symbols presented is looking "down" or "up" the axis of the ntuple helix, as indicated in the screen shots above.
Understood generally, various approaches to such bonding patterns might be recognized:
The Second Coming by W. B. Yeats (first verse) 
Turning and turning in the widening gyre The falcon cannot hear the falconer; Things fall apart; the centre cannot hold; Mere anarchy is loosed upon the world, The blooddimmed tide is loosed, and everywhere The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity. 
Triangular bonding? The following animation is suggestive of the pattern of bonding in the case of a triple helix, notably with respect to its initial application to academia (universities), business and government. Of related interest is its relevance to the three pillars of sustainability  economic, social, environmental  as most recently recognized by the 2005 World Summit on Social Development. This contrasts with the three pillars of the European Union, defined in instititutional terms. The Three Pillars of the United Nations are human rights, peace and security, and development.
Animation of 3fold bonding pattern 
Quadrangular bonding? Various possibilities merit consideration in relation to the pattern of bonding in the case of a quadruple helix. Especially intriguing is the challenge of any relationship to the Christian cross and to the extremely charged symbol of the swastika in its different orientations. The animations on the right were developed in a separate argument which notably referred to the pattern of Knight's moves in chess across a matrix of 3x3 cells (Swastika as Dynamic Pattern Underlying Psychosocial Power Processes: immplicate order of Knight's move gameplaying sustaining creativity, exploitation and impunity, 2012).
Animations of 4fold bonding pattern  
Quadrangular  Leftfacing Swastika  Rightfacing Swastika 
Inspired by the possibility of representing the basic pattern of 8 distinctions of the I Ching trigrams in the triangle above, the animation on the left is a tentative exploration of an approach to configuring the tetragrams of the associated Chinese classic, the Taixuanjing (Canon of Supreme Mystery), as discussed separately (9fold Magic Square Pattern of Tao Te Ching Insights  experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006). Although unsuccessful it focuses the question of how to develop complex patterns of notation, as discussed further below. Of further potential relevance is a consideration of 4dimensionality (Fourdimensional requisite for a timebound global civilization? 2015).
As indicated above, strategic concerns may be articulated in terms of four pillars. For example the United Nations identifies four foundational pillars, four pillars of a global counterterrorism, as well as four pillars of transitional justice (truth, justice, reparation, guarantees of nonrecurrence), with UNESCO identitying four pillars of learning, and the World Bank identifying the four pillars of the knowledge economy, with four pillars of human rights distinguished by UNDHR (personal rights, relationships with a social group, civil liberties and political rights, and economic rights). The World Economic Forum identifies four pillars of economic corruption. Proposals have been made to extend to four the three pillars of sustainability (discussed above), namely with the inclusion of culture. OECD distinguishes four central pillars of inclusive growth.
Pentangular bonding? In arguing for recognition of a 5fold dynamic, it is somewhat ironic to note the correspondence between fundamental traditional understandings of East and West, as exemplified by the pentagrams of Hygeia and Wu Xing discussed separately (Fivefold cognitive dynamics of relevance to governance? 2015). The following images were reproduced from an earlier discussion (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012). The animation is suggestive of the pattern of bonding in the case of any quintuple helix. The image on the right derives from System Dynamics, Hypercycles and Psychosocial Selforganization: exploration of Chinese correlative understanding (2010).
Hygeia (Hugieia) Pentagram of Pythagoreans 
Animation of 5fold bonding pattern  Chinese 5phase Wu Xing cycle  Hypercycle representation 
Reproduced from Hygiea entry in Wikipedia (G. J. Allman Greek Geometry From Thales to Euclid, 1889, p.26) with labels added 
Adapted from Wu Xing entry in Wikipedia Interaction arrows: black=generating; white= overcoming 
Reproduced from entry in Principia Cybernetica 
Of relevance to bonding explorations, a more rigorous interactive approach to the 5 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order5 Magic Stars, 2003).
As indicated above, concens with strategy and principle may be articulated in terms of five pillars, as with the Five Pillars of Islam or the Five Pillars of US cyber security. The fundamental principles of Wikipedia are summarized in terms of "five pillars". OECD distinguishes five pillars of effective development action. UNFPA distinguishes five pillars of population and development. Of relevance is the widely appreciated distinction by Peter Senge (The Fifth Discipline: the art and practice of the learning organization, 1990). The strategic significance attributed to The Book of Five Rings is noted below.
Hexangular bonding? The following animation, as presented in the previous paper, is suggestive of the pattern of bonding in the case of a sixfold helix.
Animation of 6fold bonding pattern 
Of relevance to bonding explorations, a more rigorous interactive approach to the 6 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order6 Magic Stars, 2003). Understood in terms of 6 pillars, examples include: The Six Pillars of SelfEsteem (1995) articulatd by Nathaniel Branden; The Six Pillars of Character; Six Pillars of Knowledge Economics; Six Pillars of Peace. As argued by David Grebow, missing from the abovementioned "fifth discipline" articulation is a learning technology required to teach and continuously support the other five disciplines (The Sixth Discipline, Brandon Hall Group, 2013).
Heptangular bonding? Importance is conventionally attached to the effective size of decisionmaking groups, notably in relation to information processing capacity (George A. Miller, The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing informationt and size, Psychological Review, 63, 1956). Patterns of bonding in such contexts therefore merit consideration. As noted above, this raises distinctive problems in the case of the great circle approach advocated here and the related absence of symmetrically regular spherical polyhedra. Of interest is the recent elaboration of a "ritual of the heptagram" in the neopagan tradition  specifically noting forms of relationship within the pattern which can be undestood as distinctive forms pf bonding.
Images of 7pointed star, implying a bonding pattern  
Reproduced from Wikipedia  Copyright Shutterstock 
A detailed mathematical description is provided by Harvey Heinz (Order7 Magic Stars, 2003). It is potentially appropriate to note that the wrapping of DNA around the nucleosome core results in 7fold compaction of DNA (N. Ramaswamy, et al, Structure of DDNA: 8fold or 7fold helix? The EMBO Journal, 1983). With respect to recognition of any set of "seven pillars", many examples exist. Most memorable is perhaps the Seven Pillars of Wisdom (1926), as articulated by T. E. Lawrence, but featuring in Proverbs 9:1 (Henry M. Morris. The Seven Pillars of Wisdom, Institute for Creation Research). Other examples include the The Seven Pillars of Life (Science, March 2002) articulated by Daniel E. Koshland, the Seven Pillar of Ecosystem Management (Landscape and Urban Planning, 1998) articulated by R. Lackey, and the Seven Pillars of Democratic Governance (Synergy Associates, 2009) by Mel Gill.
Octangular bonding? The following animation is suggestive of the pattern of bonding in the case of an eightfold helix.
8fold bonding pattern  
Animation  Image 
Of relevance to bonding explorations, a more rigorous interactive approach to the 8 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order10 Magic Stars, 2003). Expressed in terms of pillars, the Dali Lama and Archbishop Desmond Tutu have articulated Eight Pillars of Joy. Google allegedly operates in terms of Eight Pillars of Innovation. Extensive reference is made to the articulation of Eight Pillars of Prosperity (2011) by John Allen.
Enneagram bonding? As noted above, the enneagram is a notable feature of a particular school of thought which associates modes of thought and personality characteristics with the vertices of a variant of the 9pointed star. A detailed mathematical description is provided by Harvey Heinz (Order8 Magic Stars, 2003). Of relevance to bonding explorations, a more rigorous interactive approach to the 9 pointed star is offered under a pseudonym in the GeoGebra environment. Understood in terms of pillars, examples include: Stephen Sideroff (Nine Pillars of Resilience and Success, 2015), Gunnar Sevelius (Nine Pillars of History, 2010), and the Nine Pillars of Conservatism of the European Young Conservatives.
Especially relevant to the argument here for increasing the mnemonic characteristics of symbols, musical notes are associated with those vertices  as presented in an animation in the enneagram entry in Wikipedia. That image is adapted in the following simplistic animation which points to the possibility of associating bonding relationships with complex patterns of memorable chordal relationships through sonification.
Animation of 9fold bonding pattern 
Adapted from Wikipedia 
In the mathematical disciplines of topology, geometry, and geometric group theory, the Heawood graph is a basic feature in discussion of the subtleties of orbifolds (for "orbitmanifold"). An orbifold is a generalization of a manifold. It is a topological space (an "underlying space") with an orbifold structure. Seemingly incomprehensible to most, orbifolds have been applied to music theory. As discussed separately (Musical implications of orbifolds for comprehension of questioning dynamics, 2014), there is the possibility that the distinctive cognitive feel for logical distinctions and connectivity might be associated with chords  in the light of the work of Dmitri Tymoczko (The Geometry of Musical Chords, Science, 2006; A Geometry of Music, 2011):
A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the nonEuclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different nearsymmetries and suggest different musical uses.
Decagonal bonding? A detailed mathematical description is provided by Harvey Heinz (Order10 Magic Stars, 2003). Of relevance to bonding explorations, a more rigorous interactive approach to the 10 pointed star is offered under a pseudonym in the GeoGebra environment, this is associated with an overlapping variant of which screen shots are presented in the animation below.
Especially interesting is the manner in which the pattern is generated using 7 circles and 4 lines.
Animation of 10fold bonding pattern 
Adapted from GeoGebra 
There are numerous examples of articulations in terms of "ten pillars": Ten Pillars of Economic Freedom, Ten Pillars of Buddhism, Ten Pillars of Financial Independence, Ten Pillars of Successful Strategic Planning, Ten Pillars of Successful Technology Implementation, The Ten Principles of the UN Global Compact.
Hendecagram bonding? A detailed mathematical description is provided by Harvey Heinz (Order11 Magic Stars, 2003). Their symbolic use is exceedingly rare, although one detailed description of its various symbolic articulations (as an endekagram) is noted by the Hermetic Order of the Golden Dawn (Polygrams and Polygons).
Dodecagram bonding? Considerable importance is conventionally associated with 12fold configurations, most notably juries and the round tables of the wise (Checklist of 12fold Principles, Plans, Symbols and Concepts: web resources, 2011). It could therefore be assumed that the challenge of shifting patterns of bonding would invite intensive study. Of particular relevance is the approach of Arthur Young (The Geometry of Meaning, 1976) as separately discussed (Typology of 12 complementary dialogue modes essential to sustainable dialogue, 1998).
Of relevance to bonding explorations, a more rigorous interactive approach to the 12 pointed star is offered under a pseudonym in the GeoGebra environment. As might be expected, there are many articulations in terms of "twelve pillars" including: the 12 pillars of competitiveness of the World Economic Forum, 12 pillars of wisdom as the ultimate intelligence test (New Scientist, 26 October 2010), 12 pillars of wellbeing articulated by Rick Hanson, 12 pillars of trust, articulated by Jane Anderson.
This model is a development of that in the earlier paper, most notably through use of constraining great circles and the octahedron they imply  as suggested by their use in framing the Pentagramma Mirificum (discussed there in relation to a Quintuple Helix model).
Screen shots of triple helix embedded in octahedral great circles  
Octahedral great circles highlighted  Octahedron highlighted  Triple helix highlighted 
Interactive 3D versions: x3d; wrl/vrml. Video: mp4 
Experimental design constraints applied to all models: The basic approach taken is to develop the 3D models explored in the previous documents in a more systematic manner. To this end emphasis was placed on the following design criteria:
The models presented below could be readily improved from a design perspective  in addition to the interactive facilities which could be added. They have already benefitted to some degree from the guidance of Sergey Bederov of Cortona3D.
Screen shots of quadruple helix embedded in octahedral great circles  
Cube great circles highlighted  Cube highlighted  Quadruple helix highlighted 
Interactive 3D versions: x3d; wrl/vrml. Video: mp4 
In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equalarea mapping and binning scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere).
In the light of the spherical geometry of the Pentagramma Mirificum, and its seminal role in global navigation, the relevance of both a great circle and polyhedral framing for a dynamic approach to symbols was argued in the previous paper (Suggestive representation of dynamics of a "cognitive wormhole" associated with a quintuple helix, 2017). This followed from earlier discussion of its significance (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).
Screen shots of quintuple helix embedded within great circles framing complementary forms of Pentagramma Mirificum  
Given the role of "Napier's Rules" for navigation, the question raised is whether there was a set of "Napier's Rules" to be articulated for governance  of mnemonic significance or otherwise? The image on the right shows a late phase in the dynamic of emergence and reabsorption of a star pentagon from the "cognitive womhole" at the centre of the model. No polyhedron is shown in this case. 

Interactive 3D versions: x3d; wrl/vrml. Video: mp4 
Variants of the 5pointed star are extremely common in a range of cultures, figuring prominently on flags (USA, Venezuela, Honduras, Philippines, Cuba, Panama, Jordan, Italy). They have been etensively associated with communism (USSR/Russia, Vietnam, North Korea, People's Republic of China). The "Flag of Europe", has a circle of twelve 5pointed stars. Constructed with five straight strokes, the pentagram was used symbolically in ancient Greece and Babylonia, and is currently valued through its magical associations by many Wiccans. Christians once more commonly used the pentagram to represent the five wounds of Jesus. The pentagram has associations with Freemasonry and is also used as a symbol by other belief systems. A 5pointed star, the Haykal, is representative of s the Bahá'í Faith.
Screen shots of quintuple helix embedded within great circles framing dodecahedron (some great circles omitted) 

Dodecahedron great circles highlighted  Dodecahedron highlighted  Quadruple helix highlighted 
Interactive 3D versions: x3d; wrl/vrml. Video: mp4 
Especially interesting in this case, both with respect to the configuration of 5 great circles in these models and the bonding argument above, is the considerable strategic interest in the Zen martial arts classic The Book of Five Rings.
There are interesting constraints on the use of the above approach with respect to the following polyhedra for Ntuple helices of even number.
Sixfold helix: The remaining Platonic polyhedron, the icosahedron, does not lend itself to embedding a sixfold helix. More useful are several of the semiregular Archimedean polyhedra:
This form is potentially valuable because of the great significance attached to the 6pointed star hexagram (as discussed in the previous paper). Most obviously used as the Star of David, the symbol is commonly associated with Judaism; it is also used in Christianity, Islam and Hinduism, but on a less frequent basis. The Great Seal of the United States contains a sixpointed star made up of 13 stars (representing the original 13 colonies).
The 4 hexagonal faces of the truncated tetrahedron suggests a distinctive approach, as discussed further below in relation to the Platonic tetrahedron.
Eightfold helix: Possibilities among the Archimedean polyhedra, to embed an eightfold helix with matching octagonal faces, include
The 8pointed star octagram is a frequent motif in Mughal Islamic art and Islamic architecture and is widely associated with beautification in Arab cultures. It notably figures on the flags of Iraq and Azerbaijan. A variant is the Star of Lakshmi of Hinduism.
Tenfold helix: The Archimedean polyhedra offer two possibilities with matching decagonal faces:
The 10pointed star decagram is a frequent motif in Mughal Islamic art and Islamic architecture and is widely associated with beautification in Arab cultures. It notably figures on the flags of Iraq and Azerbaijan.
Given the successful framing of helices in the previous section, the question is by what might this other set of ntuple helices be comprehensibly framed? To what ways and manner of thinking does their asymmetry correspond? Do they imply and embody a peculiarly unsuspected form of "cognitive twist"? (Cognitive "twist" 2007, Enantiodromia: cycling through the "cognitive twist" 2007).
Twelvefold helix: The Archimedean polyhedra do not include any with dodecagonal faces by which application of the great circle process could be further explored. The following animations of unusual polyhedra, derived by further truncation from the truncated cube, were discovered. However, in order to reproduce that configuration so as to explore the great circle process, it proved necessary to construct in 3D a cubic arrangement of dodecagonal faces (righthand image below).Animations of variants of truncated cube with dodecagonal faces  Framework of dodecagonal faces  
Reproduced with permission from The Truncated Cube, with Two Variations Featuring Regular Dodecagons (RobertLovesPi's blog, 2016)  Constructed by use of Stella Polyhedron Navigator and X3DEdit 
Given the importance conventionally accorded to a 12fold patterns of dialogue, most notably in round tables of the wise and in juries, the question explored by the great circle process was the potentially implied pattern of interactions. Three sets of 12 great circles were therefore applied to the dodecagonal framework as a possible prelude to introducing a 12fold helical pattern.
As indicated below, the 36 great circles create a complex interweaving pattern in their own right, possibly precluding addition of helical patterns (or implying them in some way). As to any emergent symbol, this might be better understood as taking a 3D form (rather than 2D, as in the cases above). Given that any of the KeplerPoinsot star polyhedra could be considerd too complex, a better symbol might be the 8vertex compound of two tetrahedra (otherwise known as Stella Octangula), and discussed separately with respect to the Merkabah as a 3D variant of the Star of David (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017).
Successive addition of 36 great circles to dodecagonalfaced cubic framework (aboveright)  
Application of 1st set of 12 great circles  Application of 2nd set of 12 great circles  Application of 3rd set of 12 great circles 
Patterns dynamiclly combining red / green / blue circles are shown in the animation. Interactive 3D versions: x3d; wrl/vrml. Video: mp4 (7mb) 
Use of a dodecagonalfaced truncated cube pattern is especially interesting for mapping purposes in that 72 edges are subtended by the 36 great circles. However 8 of these edges are associated with two great circles, offering 64 edges for distinctive mapping. A further 24 edges are excluded from this encirclement. The pattern of 72 edges recalls the traditional symbols articulated as the contrasting qualities of the angelic order on the one hand, and the demonic order on the other, as discussed separately (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016; Variety of System Failures Engendered by Negligent Distinctions: mnemonic clues to 72 modes of viable system failure from a demonic pattern language, 2016).
Structurally consistent with the 3D structure of the dodecagonal configuration, as based on the truncated cube, is that of the drilled truncated cube, unique in its pattern of 64 edges (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). As discussed there, this offers a 3D mapping surface for the 64 distinctions made by the I Ching encoding or the genetic codon combinations .
Drilled truncated cube of 64 edges  
Screen shot (faces nontransparent) 
Animation with faces transparent (random attribution of codon combinations) 
the 12pointed dodecagram ***
Elevenfold helix? Examples of use of the 11pointed star hendecagram are rare. They include the Topkapi Scroll with an 11pointed star Girih form used in Islamic art. Potentially far more intriguing are implication of the 11dimensional form of string theory. Curiously the extra dimensions resulting from compactification are rendered comprehensible by suggesting that the common recognized dimensions can be understood as a line which, if observed close up, takes the form of a tube with whose circularity those dimensions are associated. The "great circles" of this argument might then be fruitfully understood as helical toroids.
Ninefold helix?: The 9pointed star ennegram features most notably in the enneagram of personality, a model of human personality which uses an enneagram figure, and in the Fourth Way enneagram, a diagram used in the teachings of G. I. Gurdjieff and others (A. G. E. Blake, The Intelligent Enneagram, 1996). The 9pointed star is a common symbol of the Baha'i Faith representing unity and Bahá'í. As indicated separately (Imagining the nature of cognitive "flight" in terms of the enneagram, 2014), the polyhedral implications have been noted by management cybernetician Stafford Beer through recognition of the association between the enneagram and the icosahedron. This he describes as emerging from collaboration with Joseph Truss  in a chapter on The Dynamics of Icosahedral Space (Stafford Beer, Beyond Dispute: the invention of team syntegrity , 1994, pp. 196209). For Beer:
But it is a matter of great interest that in the whole of the literature... the enneagram occurs as a plane figure. Nowhere had there been the slightest hint that a threedimensional manifestation existed... No wonder the search took so long, given that the diagram was discovered spread across four vertical planes... The icosahedron is the actual origin of the enneagram... (p. 206)
Various efforts have been made to depict the enneagram in 3D  readily available on You Tube (Francisco Meana, Enneagram from 3d perspective, 2007; Francisco Meana, Sufi Enneagram, 2009; Chuck Middaugh, 3D Enneagram MOD 9, 2013), The degree of relationship to the icosahedron is not especially evident. Beer provided no depiction, but this is offered in subsequent documents (J. Truss, et al, The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003; J. Baldwin, BuckyWorks: Buckminster Fuller's Ideas for Today, 1996, p. 220). The Beer/Truss argument is also discussed by Andrew Pickering (The Cybernetic Brain: sketches of another future, 2010).
Since those promoting syntegrity are especially sensitive to copyright issues, a different depiction is offered in the following generated with virtual reality software..
9fold enneagram embedded within an icosahedron 

View of enneagram associated with only one pattern of vertices of icosahedron (view in 3D with virtual reality plugin) 
View of enneagram from left image with the icosahedron framework hidden (view in 3D with virtual reality plugin) 
Note the colour coding and positioning of the icosahedral vertices  which offer guidance when rotating the above models in virtual reality in order to render visible the enneagram pattern. GreenMagenta links of the enneagram are the only links embodied within icosahedral edges (on the left), where they are invisible. Three of the 12 vertices, positioned on the vertical axis (of the image on the right), do not form part of the 9fold enneagram pattern (Red, Cyan and Black). The various possibilities for rotating the models in three dimensions affect the proportions of the enneagram as portrayed and the relative visibility of the Cyan and Black vertices. 
Of some relevance is the experimental configuration of helical toroids, as described separately
(Concordian Mandala as a Symbolic Nexus Insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016; Visualization in 3D of Dynamics of Toroidal Helical Coils  in quest of optimum designs for a Concordian Mandala, 2016) ***
Sevenfold helix? The 7pointed star has been valued for the complex, magical interpretation of its vertices and by alchemists to represent the 7 elements of the world and the seven planets originally known. Christians have used that star to represent the 7 days of creation, as a symbol of perfection by some sects, and as a means for warding off evil. In Islam, the heptagram is used to represent the first seven verses in the Quran; the Cherokee people use a 7 pointed star to symbolizes peace. It is also termed the elven star or fairy star in neopagan traditions and druidism. The flags of Australia and Jordan incorporate such a star. It features in a variety of insignia of police forces.
Despite its importance, or because of it, the sevenfold signals a break in pattern. In endeavouring to apply the great circle process to frame emergence, and the limitations of any heptahedron, one possibiity is use of a heptagrammic pyramid as shown below. In such cases any great circle subtends lines of the heptagram as well as the apex of the pyramid  indicative of a unique approach to symmetry.
Variants of heptagrammic pyramid  
If great circles are considered to be primarily associated with the axes of a polyhedron, considerable significance is attributed by R. Buckminster Fuller to the seven great circles generated by the axes of the fourteen faces of the cuboctahedron:
Synergetics also discloses the foldability of each of these seven great circles into local bowtielike patterns, which act as localcircuit shunts and are reassemblable into wholesphere integrities. Totally assembled, they reconstitute the whole greatcircles patterning of the completed spheres. They demonstrate thereby that these great circles may act as local informationshunting and holding circuits. (Cosmography: A Posthumous Scenario for the Future of Humanity, The Estate of R. Buckminster Fuller, 1992 p. 67)
Such reference to the cuboctahedron is especially intriguing in that its geometric dual, the rhombic dodecahedron has been used (as noted below) as a pattern to order Boolean connectives. This raises the question of how the framing by great circles might be applied from that perspective  as each would then pass through the apex of the pyramid.
The break by the sevenfold in the sequence of patterning introduced by the spherically symmetrical polyhedra (above) can be explored otherwise through the strange form of the Szilassi polyhedron with the highly unusual property that each face shares an edge with every other face. Its dual is the Császár polyhedron which has no diagonals, every pair of vertices being connected by an edge. Both polyhedra, having a central hole, bear a strange relationship to a torus. They invite consideration of their potential as mapping surfaces (Mapping of WHquestions with questionpairs onto the Szilassi polyhedron, 2014; Potential insights into the Szilassi configuration of WHquestions from 4D, 2014).
Szilassi polyhedron  
Rotation (reproduced from Wikipedia) 
12 types of the 21 edges (highlighted by colour) 
7 Pairs of vertices with indication of questionpairs (coloured by vertex pair, as with 12 edge types; edge lengths not to scale in this rotated perspective of the polyhedron, with faces transparent) 
Just as the enneagram is curiously embedded within the icosahedron, there is the strange possibility that the twisted cognitive transition associated with the 7fold may be embodied within the cuboctahedron/rhombic dodecahedron dynamic through the Szilassi/Császár dynamic  and the hole that the latter frames. The relationship between their characteristics in variously embodying sevenness are suggestive of this, as indicated below.
Symbolic embodiment of sevenness?  
Polyhedra  Corresponding duals  
Cuboctahedron  Szilassi polyhedron  Rhombic dodecahedron  Császár polyhedron  
Faces  14  7  12  14 
Edges  24  21  24  21 
Vertices  12  14  14  7 
Morphing animation between cuboctahedron and rhombic dodecahedron 
Produced with Stella Polyhedron Navigator 
Of some symbolic relevance:
In contrast to the approach tentatively articulated above, there is continuing interest in polyhedral patterns as a means of clarifying distinctions in logic. What could be considered amazing at this time is the manner in which this focus is restricted to an extremely limited range of polyhedra. No questions seem to be asked as to why related polyhedra are not of significance  even when extensively studied by other disciplines, This is especially striking in the case of the rhombic dodecahedron, favoured as a pattern in the study of logical connectivity. This polyhedron is a geometric dual of the cuboctahedron whose particular characteristics with respect to transformation between polyhedral forms have been highlighted by Buckminster Fuller and notably proved fundamental to enabling him to design geodesic domes.
As with the "fixation" on the static nature of symbolic stars, as highlighted above, it would appear that the world of logical reflection is equally "fixated" on static logical configurations  "logical stars". This fixation contrasts with the issues raised by George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). Poentially even more striking is the manner in which the conceptual frameworks of logic conform to the exigencies of academic journal publication on 2D surfaces becoming in effect the product of a "cognitive flatland". The metaphor has of course been repeatedly used by mathematics to highlight other potential modes of reflection (Ian Stewart, Flatterland, 2001; Dionys Burger, Sphereland: a fantasy about curved spaces and an expanding universe, 1965).
The issue merits particular consideration given the recent advances in artificial intelligence in surpassing human capacities exemplified by the game of go as the most complex game known  despite its origins in a nonwestern culture beyond serious consideration by conventional logical expertise (AlphaGo Zero: Google DeepMind supercomputer learns 3,000 years of human knowledge in 40 days, The Telegraph, 18 October 2017). Especially significant has proven to be the ability to discover moves previously unrecognized by he experts in the game.
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):
Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B_{4}, viz. the rhombic dodecahedron, the tetrakis hexahedron [dual of the truncated octahedron] the tetraicosahedron and the nested tetrahedron. After an indepth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance, the tetraicosahedron performs worse, and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.
If polyhedra are to be used as distinctive patterns of ways of knowing, through what transforms might it be possible to "translate" from one such pattern of thinking to another? Use of "distance" in the abovecited title recalls the intriguing studies of six degrees of separation  with the implication that the seemingly "distant" ways of knowing, by which society is so divided, might be separated by no more than six transformations of some kind. Is the challenge such that the 16 Boolean connectivities could be considered comparable to the many dimensions with which string theory is preoccupied  all but very few being "curled up" by compactification in ways which render them incomprehensible?
Despite the crisis of governance in a civilization in crisis, experts in logic seem however to have been unable to develop their insights to enable more appropriate modes of discourse  and seem not to have the slightest motivation for doing so. One of the abovecited authors may however be a valuable exception (Lorenz Demey, Structures of Oppositions for Public Announcement Logic, 2012).
Given the application of Fuller's polyhedral insights, is the design of "cognitive geodesic domes" for global governance to be usefully anticipated, as can be speculatively discussed (Transforming Static Websites into Mobile "Wizdomes": enabling change through intertwining dynamic and configurative metaphors, 2007).
Oppositional logic: With obvious implication for the conlictual challenges of governance, useful summaries are provided in the various contributions to the blog on Oppositional Logic: mathematics (and philosophy) of opposition, notably pages on active researchers and related references (some of which ae indicated below). Introduced as:
Stemming from works on the geometry of logical negation (rooted in a discussion of the foundations of paraconsistent logics), "oppositional geometry" (OG, for short) is the name of what seems to be a new branch of mathematics, similar to, but different from theories like diagrammatic logic, graph theory or knot theory. Previously known as "Nopposition theory" (i.e. NOT), OG gives one of the most general frameworks explaining what are things like the "logical square (or "square of opposition") (200 a.D.), the "logical hexagon" (1950), the "logical cube" (2004), the "logical tetrahexahedron" (1968) and so on (all seen, inside OG, as constituted of "oppositional bisimplexes of dim. m").
The main structures of OG are the alpha, beta and ?structures. They are mutually related by a rule of OGtranslation, which allows, for instance, to translate modal logic or generalised quantifiers or conceptual networks (expressed via ?structures) into "oppositional hypergeometry" (expressed via betanstructures). The astructures and betastructures are made of "logical bisimplexes of dimension m" The betanstructures yield a new notion of mathematical equivalence, the betanequivalence (between ?structures). For going from the theory of the oppositional bisimplexes to the theory of the oppositional polysimplexes go to the general structures.
By unfolding the mathematics of opposition in form of a new kind of geometry, OG gives to mathematicians, logicians, philosophers, linguists, ontologists, computer scientists and many others a general framework for modelling and handling through "oppositional structures" any "opposition phenomena" (first static, but then also dynamic). The concept of "opposition" being a very fundamental, elementary and pervasive one (recall that, for instance, "negation"  a highly important concept for logics and mathematics  is just one particular kind of opposition), OG is already finding applications in linguistics, modal logic, manyvalued logic, formal ontology, artificial intelligence, philosophy and in the humanities (semiotics, anthropology, sociology, psychoanalysis, gender studies, ...).
Four basic "conceptual actors" of oppositional logic and stages in their elaboration Reproduced from Oppositional Logic . Clarification: the term 'tetraicosahedron' only seems to occur in the logicoriented research of Moretti, Pellissier and Luzeaux et al. Furthermore, this term has not been used entirely consistently in the literature; for example, Pellissier [2008] and Luzeaux et al. [2008] draw a tetrakis hexahedron, but call it a 'tetraicosahedron', while Moretti [2009], conversely, draws a tetraicosahedron, but calls it a 'tetrahexahedron'.
"Logical garnet" vs "Vector equilibirum"? A comment on the work of Shea Zellweger (Untapped potential in Peirce's iconic notation for the sixteen binary connectives, 1997) in a blog (Opposition Geometry: mathematics (and philosophy) of opposition, 30 Setember 2015) notes:
The American psychologist Shea Zellweger (...) seems to be the first person to have remarked (in 1997?) that the 14 nontrivial binary connectives (i.e. the 16 binary connectives minus the "tautology" and the "contradiction" connectives) can be embedded into a 3D rhombic dodecahedron (which he called "logical garnet"). However, he does not seem to have been aware of the fact more or less the same structure (that is: the same structure but expressed in a different way, so to exhibit 6 logical hexagons in it) had been proposed by Sauriol in 1968.
As noted by Louis Kauffman:
Zellweger did an extensive study of the sixteen binary connectives in Boolean logic ( "and", "or" and their relatives  all the Boolean functions of two variables), starting from Peirce's own study of these patterns. He discovered a host of iconic notations for the connectives and a way to map them and their symmetries to the vertices of a four dimensional cube and to a three dimensional projection of that cube in the form of a rhombic dodecahedron. Symmetries of the connectives become, for Zellweger, mirror symmetries in planes perpendicular to the axes of the rhombic dodecahedron... Zellweger uses his own iconic notations for the connectives to label the rhombic dodecahedron, which he calls the "Logical Garnet". This is a remarkable connection of polyhedral geometry with basic logic. The meaning and application of this connection is yet to be fully appreciated. It is a significant linkage of domains. On the one hand, we have logic embedded in everyday speech. One does not expect to find direct connections of the structure of logical speech with the symmetries of Euclidean Geometry. It is the surprise of this connection that appeals to the intuition. Logic and reasoning are properties of language/mind in action. Geometry and symmetry are part of the mindset that would discover eternal forms and grasp the world as a whole. To find, by going to the source of logic, that we build simultaneously a world of reason and a world of geometry incites a vision of the full combination of the temporal and the eternal, a unification of action and contemplation. The relationship of logic and geometry demands a deep investigation. This investigation is in its infancy (The Mathematics of Charles Sanders Peirce, Cybernetics & Human Knowing, 8, 2001)
However, as indicated above, it is perhaps more surprising that the relation between the "logical garnet" and the cuboctahedron should have been neglected with respect to the quest for the organization of knowledge. Buckminster Fuller referred to the latter as both the vector equilibirum and (more informally) as the "jitterbug", due to it s transformational properties (Vector Equilibrium and its Transformation Pathways: comment, 1980). In a work coedited by Louis Kauffman, it is seemingly only recently that the two forms have been presented together with respect to issues of logic and the Laws of Form (Lynnclaire Dennis, et al., Building on the Known: a quintessential jitterbug, In: The Mereon Matrix: unity, perspective and paradox, 2013).
Cubical representation of BaGua pattern of I Ching 
The Logic Alphabet Tesseract 
Topologically faithful 4statement Venn diagram is the graph of edges of a 4dimensional cube as described by Tony Phillips 
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 4cube is drawn as projected into 3space; edges going off in the 4th dimension are shown in green. 
With respect to the image on the right above, as noted by Tony Phillips (Topology of Venn Diagrams, AMS), in a topologically faithful Venn diagram for the 4 statements A, B, C, D, the dual graph will have 16 vertices, labelled from not A and not B and not C and not D up to A and B and C and D. Each region must share an edge with exactly four other regions, since there are 4 places where "not" can be inserted or eliminated. Correspondingly, the dual graph would have to show 4 possible directions at each vertex; this gives the graph of edges of a 4dimensional cube.
The animations below are reproduced from a discussion of Succinct mapping of multidimensional psychosocial dynamics? (in: Destabilizing Multipolar Society through Binary Decisionmaking: alternatives to "2stroke democracy" suggested by 4sided ball games, 2016). With respect to any focus on the unit cube, the matter can be discussed otherwise (Fivefold ordering of strategic engagement with time, 2015).
Interactive virtual reality variant in 3D  Screenshots of noninteractive video renderings (offering limited speed control) 

Virtual reality variants: vrml/wrl; x3d.  Videos (.mov, 0.5mb; .mov, 11mb)  Wireframe video (.mov, 0.8mb) 
Cognitive engagement with hyperreaity through embodying higher dimensionality in hypercubes: There is a strange sense in which logic and mathematics part company with increase in the complexity of higher dimensionality  and especially with the nature of the meaning to be associated with its comprehension. The preceding images include reference to the tesseract as indicative of a fourdimensional cube. This form is but one of the ndimensional analogues to the cube and the square  known as hypercubes.
The question here is whether the cognitive challenge of the times can be adequately framed by such a tesseract or whether the corresponding the forms in 5dimension (the penteract) and 6dimensions (the hexeract) are relevant  if not those to which the symbolic patterns of bonding correspond, through the starlike graphs by which they are represented (hepteract, octeract, enneract, dekeract, and more). In terms of the notation systems variously indicated, the question might be framed as follows.
Juxtaposition of logical, notational, depiction and operational dimensions (tentative) 
Arguably the failure of logic to integrate consideration of the 32fold and 64fold degrees of connectivity is an indication that "something is missing" in the sense identified by Terrence Deacon (What's Missing from Theories of Information? 2010). This may relate to both the challenge of selfreference within physics and to insensitivity to higher orders of feedback in the cybernetic terms required for a viable system model  exemplified by highly constrained capacity for selfcriticism. It could also be explored in terms of culture and "spirit"  especially given the ironic extent to which populations are now dependent on psychoactive substances.
If the challenge of Boolean connectives is embodied in the tesseract, it is then appropriate to consider the implications of the penteract and hexeract  and their comprehension. With respect to the subtheme of this argument, it is significant that their representation calls for the use of movement. With respect to both comprehension and practical application, logic is also faced with the challenge of eppur si muove.
Tesseract animation 
Penteract (screen shot of phase in animation) 
Hexeract 
by Jason Hise [CC0], via Wikimedia Commons  by Jgmoxness, via Wikimedia Commons  by Jgmoxness, via Wikimedia Commons 
As is perhaps not surprising, the 2D orthographic projections of the hypercubes above can take a starlike form, as shown below.
Selection of 2D orthographic projections of hypercubes  
Tesseract 
Penteract 
Hexeract 
Hepteract  Octeract  Enneract  Dekeract  Hendekeract  Dodekeract 
Reproduced from Wikipedia entry on hypercubes 
Psychosocial change framed by polyhedral transformation? Given the respect accorded to the game of go in the realm of artificial intelligence, the restricted framework of current logical preoccupations is remarkably illustrated by the neglected encoding systems elaborated in the culture from which the game originates, most notably that which inspired Leibniz, namely the I Ching and its trigrams. Extensive commentary on the cubic configuration (left above) is provided in the study by Z. D. Sung (Symbols of Yi King  or symbols of the Chinese logic of changes, 1934) as noted in that of PiengLam Kho (YiJing (IChing) Matrices, 2004). The corresponding structure on the right features, with animations, in a discussion of a Succinct mapping of multidimensional psychosocial dynamics? (2016). That on the left offers various resemblances to the polyhedral structures elaborated with respect to logic. The tesseract in the centre derives from the work on the the Logic Alphabet by Shea Zellweger, namely an iconic set of symbols that systematically represents the sixteen possible binary truth functions of logic.
Buckminster Fuller has extensively explored the possibilities of transformation implied by the vector equilibirum and illustrated by jitterbug models. The logical implications of the various possibilities of morphing between duals, as illustrated by a feature of Stella Polyhedron Navigator, seem similarly to have been neglected. The software offers eight types of smooth morphing between a polyhedron (or compound) and its dual. Two of these are supported in 4D. The types are by: sizing, truncation, augmentation, expansion tilting quads, tilting triangles, tilting to compound, and tilting to rectify.
As one of the earliest encoding patterns, predating Aristotle's hexagon, the original binary encoding system of the I Ching pattern of course includes 8, 16, 32 and 64 degrees of distinction. How the 16 Boolean connectivities might be "enriched" by 32 and 64 indeed invites further reflection  especially it it is considered irrelevant by disciplines seemingly unable to enhance the capacitiy of governance at this time. Does the increasingly expressed need for "joinedup thinking" call for connectivities beyond the Boolean 16?
Selfreference and the "sign of itself": Given the fundamental importance attached to "great circles" in this argument, the commentary by Louis Kauffman usefully reframes the geometrical foc us:
Let us not forget the circle. As we saw in descending from Peirce's existential graphs to the calculus of indications (by allowing a variable to take the unmarked (true) state), the circle lives in a language where it is a sign of itself. That language, the calculus of indications, unfolds the patterns of the existential graphs and marks a larger unfolding of language, mathematics and logic as a patterning of possible distinctions. This point has been discussed at greater length earlier in this paper. We bring it up again here to remind the reader of the essential reflexivity of the basic language in which each circle, seen as a distinction, refers to any other circle seen as a distinction. (The Mathematics of Charles Sanders Peirce, Cybernetics & Human Knowing, 8, 2001)
How indeed might degrees of selfreflexivity be related to the framing by great circles of polyhedra and the helical pathways within them, as presented above? The question merits consideration in terms of the insights of cybernetics into higher orders of selfreference, as discussed by (Maurice Yolles, Knowledge Cybernetics: a new metaphor for social collectives, Journal of Organisational Transformation and Social Change, 3, 2006, 1). From that perspective, feedback processes of first, second, third and possibly higher order are envisaged, engaging any observer to an ever higher degree, as discussed with respect to Cybernetics of cybernetics: complex adaptive systems? (Consciously Selfreflexive Global Initiatives, 2007). Kauffman only refers to second order cybernetics in his commentary.
As noted by Kauffman (2001):
Peirce wrote a remarkable essay [The New Elements of Mathematics] on the Boolean mathematics of a Sign that combines the properties of addition and negation.... Peirce went on to write an essay on the formal properties of his Sign of illation and how it could be used in symbolic logic... The Sign of illation enables a number of notational conveniences, not the least of which is that the implication "A implies B" usually written as "A>B" is expressed as "[A]B" using the Sign of illation.... In the end, it is important that the portmanteau Sign can be decomposed back into its component parts, for this allows the translation between Peirce's thought and the Boolean algebra. It is these issues of translation, from one formalism to another and from meaning in natural language to meaning in the formalism, that he holds with great sensitivity.
Kauffman continues his review with a discussion of Peirce's notion of the more widelycited "Sign of itself":
Peirce speaks as a topologist. He tells us that if we overlay or in any (continuous) way place the map of a territory upon that territory then there must be a point on the map that coincides with the corresponding point on the territory.... He is using topology as an amplifier for our thought about selfreference.... It has been said that "the map is not the territory" and this is indeed correct. But the most interesting terrain is that territory where we have no choice but to use the territory in the course of the construction of the map. And this is exactly what is done in mathematics, linguistics and science. In order to study language one must use language. In order to study mathematics one must use mathematics, and indeed we use mathematics to elucidate mathematics. Map and territory grow and evolve together in the course of time. In this view it is obvious that any attempt to fully explain anything will cause the map and territory to expand into a new domain in which further explanation will be needed.
Further insight is offered by Paul Ryan (A Sign of Itself, 2001) who interprets such a sign as a relational circuit in elaborating the Earthscore Notational System for Orchestrating Perceptual Consensus about the Natural World (2011).
The possibility of interest here is the relation of any selfreferential notation to the depictions and animations above, and notably those articulated by other cultures and having stood the test of time. More generally, how might a selfreferential notation be designed (and interpreted) and what variety of designs might it usefully include? There is then the issue of how one such formalism translates into another  or can be mapped into it such as to highlight the differences. The case of the elaborate I Ching enoding, and that of the associated Taixuanjing, merit particular consideration, especially given the manner in which they have been valued in governance and decisionmaking  perhaps in contrast to the appreciation of Boolean notation in that context.
Truth tables? To the extent that the conventional relevance of truth has now been called into question in the political arena at least, and the advertising process more generally, there is a case for considering the nature of a "posttruth table" in the light of the apparent limitations of the truth table informed by Boolean connectivities. Exploiting an I Ching style notation, this has been speculatively explored separately discussed (Towards articulation of a "posttruth table"? 2016).
Fake news from beyond the realm of logic? 
I believe that legends and myths are largely made of 'truth', and indeed present aspects of it that can only be perceived in this mode. 
(The Letters of J. R. R. Tolkien, 1981, page 131) 
The above argument begs the question as to why disparate understandings of unity and integration should be so distinctively (and desperately) embodied in unrelated static structures  whether the symbolic stars associated with authorities in conflict or the frameworks proposed by logic. This contrasts with the ability of mathematics to explore a wide variety of frames in generic terms  but with total indifference as to their psychosocial relevance to a world in crisis. Curiously it is only in the deprecated domains of symbolism, especially its esoteric forms, that a degree of relationship is acknowledged. As noted above the issue is reinforced by the emphasis on static structures lending themselves most readily to depiction in 2D  as with many "global plans". This is a strange bias in a period in which information tchnology offers the capacity to represent and interact with structures in 3D, if not 4D, as currently promoted.
Given reference to the paradigm shift implied by Eppur si muove, it is appropriate to confront a seminal image of Galileo's time by Johannes Kelpler (left below) with a dynamic model in 3D (below right). The latter offers a unique embedding of the Platonic polyhedra within a rhombic triacontahedron the Platonic polyhedra depicted by Kepler (dodecahedron (blue), icosahedron (red), cube (grey), octahedron (yellow), with tetrahedron (cyan) and tetrahedron (magenta)). Commentary on the model on the right, and the justification for use of the rhombic triaontahedron, are presented separately (Nesting polyhedra to enable comparison of patterns of discourse, 2015). Especially noteworthy are the distinctive implications of mutual rotation and a "pumping" action resulting from relative changes of scale of the nested polyhedra.
Nested polyhedral model of solar system of Johannes Kepler in Mysterium Cosmographicum (1596) 
Rhombic Triacontahedron (green) as a nesting framework (virtual reality variants static: vrml or x3d; mutual rotation: vrml or x3d; "pumping": vrml or x3d; videos: "pumping" mp4; "rotation" mp4) 
A similar approach to exploring such dynamic relationship may be undertaken with respect to the 13 Archimedean polyhedra, most notably by animating them in a "pumping" motion of contraction to the common centre and expansion from it, as discussed separately (Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra, 2015)
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) 

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 
If logicians are fixated on the mapping facility of the rhombic dodechedron, as a dual of the cuboctahedron, what is being systematically neglected if the latter offers an unusual mapping facility for the set of Archimedean polyhedra? Such dynamically ordered interrelations between 3D projections recalls the relevance of other metaphors, especially "conceptual gearbooxes" with a requirement for systems of "transmission" (The Future of Comprehension: conceptual birdcages and functional basketweaving, 1980).
Cognitive bias? Like the rhombic dodecahedron of such interest to depiction of logical connectivity, the rhombic triacontrahedron is a Catalan solid  a dual of the icosidodecahedron. The challenge for any universal logical framework is perhaps illustrated by the references to disparate Platonic solids (and their polygonal equivalents) as a key to such modelling  without considering how they might be related:
Having bypassed the triangle privileged by semiotics (for example), it is striking to note the absence to any reference to the the pentagon, Perhaps "logically" there is little need for interest in the pentagon  despite many references to the strategic logic of The Pentagon. This presumably calls for greater insight into its oppositional logic, as indicated separately (Reframing NATO and The Pentagon? 2017), given the importance accorded to it in various symbols and other approaches to logic, most notably the Chinese Wu Xing (as noted above).
The discipline of logics eemingly restricts itself to evennumbered frameworks  thereby reinforcing the questionable limitations of binary thinking. Curiously it is the conventionally deprecated "sacred geometry" which encompasses the reality of both even and oddnumbered modalities. It is not clear how such restrictive logic can aspire to being "universal".
The logic of governance is only too obviously defined primarily by the line  you're either with us or against us (Edward de Bono, I Am Right; You are Wrong, 1991). There is little consideration in practice of the "square of opposition" and even less of the cube. No consideration of the abovementioned tesseract of 16 connectives is therefore to be expected.
Quest for diversity in ordering the ways of knowing? Again it would appear that the fundamental question is what ways of knowing are carrried or enabled by other configurations, most notably in 3D? Framed otherwise, what are the cognitive flaws in the approach to such matters which ensure avoidance of other forms and the ways of knowing they may represent, however "distorted" and "faulty" and "incomplete":
Given that set of hundreds of forms, is it a type of group think or "conformation bias" [sic] that has focused the attention of specialists in logic on forms which have 14 vertices, 36 edges, and 24 faces, as indicated by the following animations? If so, why? Is this an indication of the logical trap within which aspiraions to global governance are envisaged?
Animations of "Logical stars" or "Stars of logic"? (excluding the rhombic dodecahedron indicated above) Forms with characteristics of primary current relevance to Boolean logic connectives namely their 14 nontrivial forms, and excluding 2 associated with tautology and contradiction (5 forms with 14 vertices, 36 edges, and 24 faces, excluding the Facetted stella octangula and the 1Frequncy cubic geodesic sphere) 

Tetraicosahedron / Tetrahexahedron ?  Great triakisoctahedron  Triakisoctahedron  Stella octangula  
Tetrakishexahedron  Augmented cube  
Prepared using Stella Polyhedron Navigator 
Combining the tetrahedron with its dual then forms the stellated octahedron  the only stellation of the octahedron. It is also called the stella octangula , discussed separately in terms of the Merkabah symbol (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017). Note that its vertices may be numbered to total 8, or 14 (if the nonprojective types are included).
Relevance of polyhedra with 16 vertices for cognitive organization ? Given the focus on the rhombic dodecahedron with 14 vertices, it is interesting to consider polyhedra which permit a mapping of the 16 Boolean connectives onto their vertices  to frame discussion of the relevance of those with 32 or 64 features, namely the logical progression from 2^{3}, via 2^{4}, through 2^{5} to 2^{6}, namely 64.
Animations of selected polyhedra with 16 vertices  
"Simplest torus" 12f 28 e 16v 
Octachoron tesseract (4D) 24f 32e 16v 
Compound of 2 cubes 12f 24e 16e 
Compound of 4 tetrahedra 16f 24e 16v 
Trunc. tetra. geodesic sphere 28f 42e 16v 
Prepared using Stella Polyhedron Navigator 
Relevance of polyhedra with 32 and 64 features for cognitive organization? In considering the seemingly desirable extension of the patterns of knowing beyond the 16 (or 14) which are the preoccupation of logic, a question is why "connectives" should be associated with the vertices of a polyhedron rather than with its edges  evoking the sense of "connection" more readily. It might also be asked how "faces" are to be interpreted in any such framework  especially when they may frame a field of dialogue constrained by connectives.
In the light of such considerations, the identification of coherent configurations could usefully take account of several alternative attributions. There are relatively few "interesting" configurations offering a pattern of 32 vertices  if interesting includes a sense of memorability. Perhaps appropriately, the examples below have figured in the argument above. The rhombic triacontahedron (dual of the icosidodecahedron) was found to be unisue in defining a framework for the nesting of the Platonic polyhedra. In providing 32 vertices, the drilled truncated cube also offers 64 edges  potentially extending its value to issues relating to the 2^{6} so common in computer systems.
Selected polyhedra with 64 vertices  
Rhombic triacontahedron 
Drilled truncated cube 32f 64e 32v 

Prepared using Stella Polyhedron Navigator 
To the extent that a preoccupation with a 16fold pattern can be usefully challenged in the light of the wider challenges of governance, of particular interest (as noted above) is its recognitions as a nested subset of the toroidal drilled truncated cube. This has 64 edges (9 types), 32 faces (5 types), and 32 vertices (4 types), The separate presentation, with animations, can be understood as a Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements (2015). As a hollow toroidal cube, its shape recalls both that of the cubical representation of the I Ching Bagua (which feature in the animations) and the tesseract of Zellweger's Logical Alphabet (shown above). It is especially suggestive of the possibility that the muchstudied "cube of opposition" merits a larger framework. The following animations are reproduced from a previpous exercise (Decomposition and recomposition of a toroidal polyhedron  towards vortex stabilization? 2015).
Animation of symmetrical edge movements of drilled truncated cube  
Framed inner cube movements (#1/2) Access X3D variant 
Framed outer cube movements (#3/4) Access X3D variant 
Angled movements Access X3D variant 
Alternative views of selected cycles of movement of parallels along edges of the drilled truncated cube Video version (.mp4); interactive virtual reality (.x3d; .wrl) 
The following images are those of several of the relatively few forms which have 64 vertices, if the conditions of change could best be associated with them  as encoded by the I Ching hexagrams. They derives from the list of uniform polychora in Stella4D. Together with that above, there are striking commonalities. The fact that it is of toroidal form suggests that the more fundamental understanding of "holes" might merit careful consideration, as remarkably discussed by Roberto Casati and Achille C. Varzi (Holes and Other Superficialities, 1994)  with respect to the borderlines of metaphysics, everyday geometry, and the theory of perception (as they summarize in the entry on holes in the Stanford Encyclopedia of Philosophy).
Selected polyhedra with 64 vertices based on truncation of a tesseract (Reproduced from Visual representations of globality of requisite variety for global governance, 2015) 

3D projection of the 4D polychoron Truncated tesseract ("Tat")  3D projection of the 4D polychoron Rectified tesseract ("Rit")  
of 48 faces (7 types), 112 edges (15 types), 64 vertices (8 types) (animation)  of 64 faces (4 types), 96 edges (4 types), 64 vertices (4 types) (animation)  
Prepared using Stella Polyhedron Navigator 
The model on the right is especially interesting, if not uniquely memorable, in having both 64 faces and 64 vertices. The characteristics of both are also of significance in relation to the 4dimensionality of the logical tesseract (mentioned above). Their visualization, through projection into 3D, is only achieved by the operations of truncation and rectification  whose cognitive implications merit attention. Given the abovementioned convention adopted by Tony Phillips with respect to depiction of 4dimensionality through angled lines in a 3D model of a tesseract, there is a case for exploring how contrasting orientations in the above 3D models could be used to distinguish 6 dimensions (especially in the case of the rectified tesseract)..
Selected facets of individual polyhedra (presented above)  
Rectified tesseract  Drilled truncated cube  Truncated tesseract 
Prepared using Stella Polyhedron Navigator 
Cognitive "resonance hybrids"? Assumptions regarding attribution of logical connectives and cognitive implications to sets of vertices, rather than sets of edges or faces, can be usefully challenged. Of particular concern is the capacity to comprehend the coherence of complex sets  and to render them widely communicable. To this consideration might be added issues relating to the aesthetics of enabling symmetry as constituting a strange attractor of a particular kind  by which any such model is rendered "interesting" rather than "confusing".
It is in this sense that it could be argued that the attribution to the 3fold option of vertices, edges or faces could be explored in terms of alternation. This offers the implication that mappings could be based on a set of complementary models  recalling the many distinctive projections through which the globe is mapped. Given the biomimetic inspiration of doublestranded DNA for the argument above, there is a case for recalling that the structure of the "benzene molecule"  most fundamental to organic life  is understood in terms of the dynamics of a resonance hybrid (of which there are many depictions, starlike and otherwise, notably inspired by August Kekulé).
Ouroboros benzene molecule as envisaged by Kekulé (from Wikipedia) 
Cognitive resonance between modes of attribution to polyhedra (suggestive schematic) 
Negative curvature? The case for topological complexification in the quest for more fundamental order can be made otherwise in terms of the significance accorded by astrophysicists to recognition of negative curvature and its implications for understanding the shape of the universe, as discussed separately (Eliciting a Universe of Meaning  within a global information society of fragmenting knowledge and relationships, 2013). Recent research by Stephen Hawking and colleagues (Accelerated Expansion from Negative Lambda, 2012) has shown that the universe may have the same surreal geometry as some of art's most mindboggling images (Lisa Grossman, Hawking's 'Escherverse' could be theory of everything, New Scientist, 9 June 2012). This offers a way of reconciling the geometric demands of string theory, a stillhypothetical "theory of everything", with the universe as observed  through a negativelycurved Escherlike geometry (essentially a hyperbolic space).
The insight relies on a mathematical twist previously considered impossible, namely the use of a negative cosmological constant rather than a positive one. The new approach provides a description of "all the possible universes that could have been  including ones in which the solar system never formed, or in which life might have evolved quite differently". Making conventional use of a positive cosmological constant, it had proven impossible to describe universes that were "anything more than clunky approximations to reality". A plethora of universes have now been generated from wave functions with negative cosmological constants.
Arguably, whether discovered by artificial intelligence or otherwise, analogous topological breakthroughs may have significance for connectivity in the ways of knowing, as argued separately in relation to deprecated symbol systems (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016; Quest for a "universal constant" of globalization? Questionable insights for the future from physics, 2010). Might viable global governance require some analogue to negative curvature to render global order coherent?
Rotation of "existential" perspective? As noted above, considerable controversy is engendered in society by the inversion of some symbols (as with any cross or pentagon), or between left and rightfacing (as in the case of the swastika). This can be variously explored (Unquestioned Bias in Governance from Direction of Reading? Political implications of reading from lefttoright, righttoleft, or topdown, 2016; Reversing the Anthem of Europe to Signal Distress: transcending crises of governance via reverse music and reverse speech? 2016).
It is significant that the attributions by logicians to polyhedra are indifferent as abstractions to what is implied by such inversion  as understood in popular culture and in the depiction of symbols on flags (for example). This suggests that dimensions of existential significance are indeed "missing" from conventional academic understanding  in accord with the abovementioned argument of Terrence Deacon (2001, 2010).
The point may be explored otherwise in terms of the neglect in conventional models of any consideration of the role of some form of "corruption"  notably in relation to the trilogy of academia / government / business in the Triple Helix model, for example. And yet the extent of such corruption (and its denial) is a matter of daily record in the media  most notably at the highest level. In addition, just as the inversion of some symbols is associated with "evil", it is indeed the existence of "evil" which is frequently proclaimed and deprecated by the leaders of the world, notable for attributing that quality to their peers (Existence of evil as authoritatively claimed to be an overriding strategic concern, 2016; Which world leaders have (not) been labelled "evil"? 2015). The models above partially address this issue through the helical dynamics associated with a "cognitive wormhole" (perhaps usefully recognized as a "cognitive sinkhole") and reference to the process of enantiodromia highlighted by William Irwin Thompson (From Nation to Emanation: Planetary Culture and World Governance, 1982),
Encompassing the elusive: The question is then how the reality of such missing dimensions might be cognitively "encompassed", especially if they are "compactified" and "curled up" in a manner beyond the overriding preference for 3D frameworks (as for the rhombic dodecahedron), whatever necessity is perceived by logicians for the 4dimensionality represented by a tesseract. This concern presumably includes the need to engage effectively with the reality of "fake news" and "spin", however much their truth may be called into question from a logical perspective.
It is therefore intriguing to note the efforts to depict the 4dimensionality of a tesseract by animation, whilst at the same time begging the question of the potential relevance of hypercubes of higher dimensionality (penteract, hexeract, etc). The question is most clearly framed by the 6level encoding of the I Ching hexagrams in contrast with the 4level encoding required for the 16 Boolean connectives.
Arguably the 4level encoding is completely adequate for encoding an essentially "mechanical" world view  3D entities in movement  as framed by the tesseract. Higher orders of selfreference are effectively "designed out"  notably in the light of the paradoxes they pose for physics in the observerobserved relationship. Given widespread institutional reference to strategic pillars (as noted above) as mechanical models, there is clearly a missing link in practice to experience of the implied value nexus  although effectively aspiring to starlike configuration as "cognitive keystones" (Challenge of "soullessness"  beyond the "pillarization of Europe", 2004; Paul Pardi, Has Science Become 'Soulless'? Philosophy News, 15 August 2013),
To the extent that such higherorder relationships call for 32 and 64 conditions  beyond the 16 Boolean connectives  it is interesting that these can be fruitfully mapped onto 3D polyhedral projections of the 4D polychoron  the tesseract (in motion)  in addition to the potential exception of the drilled truncated cube. However it is the cognitive significance of the process of projection which merits attention  namely how 4dimensionality (or higher) can be reduced to 3dimensionality through truncation, rectification and other such processes.
It is in this sense that recognition of the necessary rotation of the corresponding starlike symbols enables a degree of 4dimensionality to be encompassed and compehended in a 3dimensional context. Hence the reference to eppur si muove.
Cognitive implication in the "missing dimensions": It is intriguing to note that the 16 Boolean connectives can be encoded by 4 levels of an I Ching hexagram  a trigram plus one. The four values which can be taken by the last two levels  the 5th and 6th  then pose the question as to what they might signify, especially as modifiers of the preceding four (to encode a total of 64 "conditions of change"). The manner in which these are excluded from conventional thinking offers its own clue in that the last two levels four could be understood in terms of the tetralemma highlighted with respect to a necessary shift beyond formal logic by Kinhide Mushakoji (Scientific Revolution and Interparadigmatic Dialogue, United Nations University, 1979). The tetralemma is notably a feature of Indian logic as: affirmation, negation, both affirmation and negation, neither affirmation nor negation.
This pattern of comprehension might be usefully understood otherwise and more generally, perhaps such as to include western interest in enactivism and the psychosocial construction of reality (Peter L. Berger and Thomas Luckmann, The Social Construction of Reality, 1966), as separately discussed (Enminding the Extended Body: enactive engagement in conceptual shapeshifting and deep ecology, 2003). The pattern would then take the form of enaction (and the implied responsibility assumed), being enacted by (and the implied denial of responsibility), both conditions, and neither of them. This has the merit of engaging with currently problematic issues of radical choice, radical attribution of blame (identity politics), and plausible deniability  all of which are beyond the scope of formal logic.
Another approach is through understanding use of any binary coding system to represent "filling up", namely with respect to any initial assumption as to whether broken and complete lines in the (hexagram, for example) appropriately represent 0 and 1 or, conversely, 1 and 0 (or the order in which a trigram might be read, top down or down up). The fundamental nature of this seemingly trivial assumption has been explored by Xavier Sallantin (L'épistémologie de l'arithmétique, Communication aux Seminaires internationaux d'épistemologie de l'Abbaye de Senanque, 1976). Sallantin exaplains this assumption in terms of how agreement is established when, depending on culture, "yes" may be denoted by a "nod" or a "shake" of the head (for example, by a referee tossing a coin for captains before a football match). The ambiguity is also recognized in the distinction between perceiving a glass as being "halffull" rather than "halfempty".
The question can be approached otherwise through the fine distinctions encoded and comprehensible through music  potentially far beyond any limit at 32 or 64. In the various analyses by musicologist Ernest McLain (The Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato, 1976), his focus is on the distinctions made arising from patterns of multiples: 2^{p} x 3^{q} x 5^{r} and beyond. The cases of 64 (as 2^{6}), 72 (as 2^{3} x 3^{2}), or 108 (as 2^{2} x 3^{2}) are clearly only at the beginning of any such pattern, subject to comprehension constraints (Tonal patterns of Rg Veda poetry, 1984; Patterns of Nfoldness: comparison of integrated multiset concept schemes as forms of presentation, 1980; Comprehension of Numbers Challenging Global Civilization, 2014).
Any related reference to myth serves to emphasize that it is the patterns of myth  and their associated symbols  that tend to be especially memorable over extended periods of time, as catalysts for creative imagination for which appeals are made at this time (Engendering 2052 through Reimagining the Present, 2012; Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007).
Comprehension, insight and "catching the light"? In emphasizing the absence of selfreference in conventional logic it is interesting to note concern with the role of the observer in the orthographic projections  by which Boolean connectives might be mapped, as argued above especially in the case of connectives of a higher dimensional order (Ian O. Angell, The Observer and the Orthographic Projection, 1990). This calls for clarification in relation to facet diagrams as indicated below.
Disambiguation of faceting 
As noted in the Stella4D Manual, the faceting diagram of a polyhedron is reciprocal to the stellation diagram for its dual. There is a different faceting diagram for each different type of vertex of a polyhedron  to be variously viewed (as with stellations). In their presentation they have a starlike resemblance to the orthographic projections  both potentially having a striking aesthetic quality (as indicated above). Both have a resemblance to the gemstone faceting diagrams basic to the mechanical process of gemstone faceting: cutting surfaces on a piece of crystal in order to create beautiful gems. As diagrams, all three only imply an observer and their potential roles as a starlike symbol for that observer.
The process of developing a gem through cutting merits exploration as the elaboration of a symbol, given the major role that gems are widely held to have in embodying and expressing value  whether on the crowns of royalty or on highly valued jewelry of various kinds. This is most notable in the case of the diamond  a symbol of wisdom  as may be speculatively explored (Current metaphoric applications of gemstones, 2002; Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). Gems can be readily recognized as the ultimate aesthetic expression of starlike symbols presented otherwise  being especially attractive in that respect.
The fundamental point to be made in support of the argument above is the necessity of sequential eye movement to recognize the elegant symmetry of an individual gemstone from different perspectives, or that of orthographic projections and faceting diagrams  of which a number may be relevant to any one polyhedron. Appreciation of symmetry may well require a rotary pattern of eye movement through which meaning is elicited and the pattern is held to be meaningful as a whole  and to be valued as symbolic of a higher order. Gemstones are appreciated through moving them in order to "catch the light". Symbols can be understood as requiring movement to enable insight into the cognitive light of which they are indicative. Their use on national flags recalls the role of prayer flags fluttering in the wind.
Movement necessarily implies the dimension of time and a timetranscendent perspective, as discussed separately (Aesthetics of human understanding through embodiment, 2011). Mark Johnson, in particular has developed this understanding following his earlier work (The Meaning of the Body: aesthetics of human understanding, 2007; The Body in the Mind: the bodily basis of meaning, imagination, and reason, 1987). The argument is taken further with respect to embodiment of the mind in movement by Maxine SheetsJohnstone (The Primacy of Movement, 2011). In contrast with the proccupations of logicians, the process is especially well illustrated by sports involving acrobatics and aerobatics.
If movement is primarily associated with 4dimensionality, what is the role of selfreferential insight in relation to the 5th and 6th dimension? Although the question may be framed as irrelevant to the conventions of "western" logic, it is appropriate to note the ever increasing insightful contributions of those of "eastern" cultures to the paradoxes of fundamental physics, as argued by Susantha Goonatilake, Toward a Global Science: mining civilizational knowledge, 1999). The traditional insights of those cultures may yet subsume the conventions of logic within a subtler framework of greater relevance to global governance (Enhancing the Quality of Knowing through Integration of EastWest metaphors, 2000). However the cognitive frameworks of hypothetical extraterrestrials could well be an even greater challenge.
Stafford Beer. Beyond Dispute: the invention of team syntegrity. Wiley, 1994
John G. Bennett. The Dramatic Universe: search for a unified vision of reality, Coombe Springs Press, 1956
Peter L. Berger and Thomas Luckmann. The Social Construction of Reality. Anchor, 1966
JeanYves Béziau. Around and Beyond the Square of Opposition. Springer, 2012
JeanYves Béziau and Gianfranco Basti. The Square of Opposition: a cornerstone of thought. Birkhäuser, 2017
A. G. E. Blake. The Intelligent Enneagram. Shambhala, 1996
R. Blanché:
G. J. Chaitin. Meta Math! The Quest for Omega. Vintage, 2006 [text]
Saloua Chatti and Fabien Schang. The Cube, the Square and the Problem of Existential Import. History and Philosophy of Logic, 34, 2013, 2, pp. 101132 [abstract].
G. Clark. New light on Peirce's iconic notation for the sixteen binary connectives. Paper presented at the Sesquicentennial Congress at Harvard University. 1989
Terrence W. Deacon:
Lorenz Demey and Hans Smessaert:
Lynnclaire Dennis, Jytte Brender McNair, Louis H. Kauffman (Eds.). The Mereon Matrix: unity, perspective and paradox. Newnes, 2013
Guoping Du, Hongguang Wang and Jie Shen. Oppositional Logic. In: He X., Horty J., Pacuit E. (Eds), Logic, Rationality, and Interaction. LORI 2009. Lecture Notes in Computer Science, 5834, 2009. [abstract]
Amy C. Edmondson:
Elias Carayannis and David F. J. Campbell. Triple Helix, Quadruple Helix and Quintuple Helix and How Do Knowledge, Innovation and the Environment Relate To Each Other? International Journal of Social Ecology and Sustainable Development, 1, 2012, 1, pp. 4169 [abstract]
R. Buckminster Fuller. Synergetics: Explorations in the Geometry of Thinking. Macmillan, 1975
Susantha Goonatilake. Toward a Global Science: mining civilizational knowledge. Indiana University Press, 1999
Louis H. Kauffman. The Mathematics of Charles Sanders Peirce. Cybernetics and Human Knowing, 8, 2001, 12, pp. 79110 [text]
George Lakoff and Rafael Núñez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2001
Dominique Luzeaux, Jean Sallantin and Christopher Dartnell. Logical Extensions of Aristotle's Square. Logica Universalis, 2, 2008, 1, pp. 167187 [abstract]
Ernest G. McClain:
A. Moretti:
Kinhide Mushakoji:
R. Pellissier. Setting nopposition. Logica Universalis, 2, 2, 2008 (prefinal draft) [text]
Edward S. Popko. Divided Spheres: geodesics and the orderly subdivision of the sphere. CRC Press, 2012
Pierre Sauriol:
Bernd Schmeikal. The Emergence of Orientation and the Geometry of Logic. Quality and Quantity, 32, 1998, pp. 119154 [abstract].
Hans Smessaert:
Hans Smessaert and Lorenz Demey. Visualising the Boolean Algebra B4 in 3D. In: Diagrammatic Representation and Inference, Springer, 2016; pp. 289292 [abstract].
René Thom. Structural Stability and Morphogenesis: an outline of a general theory of models. W. A. Benjamin, 1972 [text]
William Irwin Thompson:
William H. Warren, Daniel B. Rothman, Benjamin H. Schnapp and Jonathan D. Ericson. Wormholes in Virtual Space: from cognitive maps to cognitive graphs. Cognition, 166, 2017, pp. 152163 [text]
Arthur M Young. The Geometry of Meaning. Delacorte Press, 1976
Shea Zellweger:
For further updates on this site, subscribe here 