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14 October 2019 | Draft

Framing Cognitive Space for Higher Order Coherence

Toroidal interweaving from I Ching to supercomputers and back?

- / -

Torus interconnect -- as used in supercomputers
Cubic organization of I Ching trigrams -- an eightfold way
Pattern language and polyhedral mapping
Higher dimensionality, polyhedral packing and transformation
Brain organization, cognition, comprehension -- and music
Oppositional logic and its requisite polyhedral geometry
Reflexivity in multi-loop thinking and higher order learning
Toroidal constraint -- nuclear fusion as metaphor of cognitive fusion
Supercomputers, hypercomputing and superquestions?
Framing the space for conscious creativity?
Dancing cognitively inside the box -- and beyond


There is no lack of references in the history of mathematics and computing to the role of the encoding of the I Ching (Yi Jing) in providing inspiration to Gottfied Leibniz in 1701 with regard to the development of binary logic, as variously noted (Mary von Aue, Gottfried Wilhelm Leibniz: how the 'I Ching' inspired his binary system, Inverse, 1 July 2018; Will Buckingham, Forget Prophecy: the I Ching is an uncertainty machine, Aeon, 11 October 2013). The latter argues that using the I Ching is a weirdly useful way to open one's mind to life's unexpected twists.

It is now extraordinary to recognize that the current development of supercomputers is notably based upon a mode of organization -- torus interconnect -- which is itself evident to a degree in the diagrammatic patterning of the I Ching. Such development necessitates a level of coherence now framed as six-dimensional (Tomohiro Inoue, The 6D Mesh/Torus Interconnect of K Computer, Fujitsu, 2016; Yuichiro Ajima, The Tofu Interconnect D for Supercomputer Fugaku, Fujitsu, June 2019; K supercomputer). "Tofu" stands for "torus fusion".

Any understanding of supercomputing or the I Ching is necessarily a challenge -- defined as they are as separate domains of expertise, typically characterized by mutual deprecation. Such conceptual frameworks merit a degree of comparison with the results of recent neuroscience research which indicate the remarkable possibility of cognitive processes of up to 11-dimensional form in the light of emergent neuronal connectivity in the human brain.

Also curious is the parallel dependence on geometrical frameworks in the development of the formal logic of opposition, notably extending into those of higher dimensionality which are not readily visualized -- such as the hypercube (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018; Alessio Moretti, The Geometry of Logical Opposition, 2009). Seemingly quite unrelated is the preoccupation of cybernetics with feedback loops, fruitfully explored as interrelating thousands of problems and strategies of global concern (Feedback Loop Analysis in the Encyclopedia Project, 2000).

A degree of comprehension of the potential "cognitive confluence" between these disparate preoccupation can be recognized in the widespread familiarity with the Rubik Cube -- and, to a far lesser degree, with its more complex forms (Interplay of Sustainable Development Goals through Rubik Cube Variations: engaging otherwise with what people find meaningful, 2017).

This argument follows from previous exploration of a range of animations in which a toroidal perspective is a common feature (Imagining Toroidal Life as a Sustainable Alternative: from globalization to toroidization or back to flatland, 2019). The question here is how better to comprehend what appears to be a pattern of confluence of cognitive concerns probably only detectable through complementary perspectives, as argued by Magoroh Maruyama (Polyocular Vision or Subunderstanding? Organization Studies, 25, 2004).

The challenge is to avoid excessive emphasis on textual articulations characteristic of these patterns by focusing on visual representations which are more suggestive of potential correspondences, as justified separately (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007; Imagining Order as Hypercomputing: operating an information engine through meta-analogy, 2014). In so doing, the intention is to avoid premature (and presumptuous) closure in order to evoke imaginative reflection towards greater coherence -- perhaps usefully to be understood as a form of "hypercomputation", as speculatively envisaged by Alan Turing with respect to an oracular function. With the emphasis on imagination, the quest is for indications that might trigger it (In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007).

The argument traces a pattern connecting disparate cubic frameworks variously evident or implied. This includes the suggestion that global civilization is currently trapped within such a cubic metaphor -- if not incarcerated therein -- recalling the insight of Geoffrey Vickers: A trap is a function of the nature of the trapped (Freedom in a rocking boat: changing values in an unstable society, 1972). With the future degrees of urbanization now envisaged, a cubic context will become preferred to ever higher degrees in contrast with the subtle complexities of nature.

Torus interconnect -- as used in supercomputers

The pattern inspiring the following exploration has been the torus interconnect fundamental to one approach to memory organization in what are generically now known as high performance computers. The array of possibilities for such computation is usefully presented in the following accessible texts:

With respect to the high degree of competition in this rapidly evolving arena, the torus interconnect has been central to the development of the Fujitsu initiative, of which the K supercomputer has been ranked as the world's fastest, although recently superseded. The K version (Tofu1) has itself been superseded by the TofuD version, as variously indicated in the following.

Torus fusion: "Tofu" stands for "torus fusion" -- a feature central to the followings argument. Such development necessitates a level of operational coherence now framed as six-dimensional. As described in 2012, Tofu is an interconnect for massive parallel computers, connecting more than 80 000 nodes with scalability beyond 100,000 nodes. The network topology is a highly scalable six-dimensional mesh/torus.with node communicating in four directions simultaneously.

The image on the left below is a schematic from Wikipedia of the principle underlying torus interconnect a 3D torus network as often used by high performance computing systems.

Torus interconnect schematic
(in cubic array)
3D reconstruction of schematic on left
(animation with indicative cube)
Torus interconnect schematic in Tofu supercomputer Animation in 3D of torus interconnect schematic in Tofu supercomputer
Reproduced from Wikipedia  

The images above suggest the manner in which the array of nodes can be extended from 2x2x2 (8 nodes) to 16 nodes, with those below showing completion of the torus interconnection process with the addition of further nodes to 24. Given the original cubic array, it is useful to note how colour coding is used in the 3D variants to distinguish x-axis (blue), y-axis (red) and z-axis (green). The design artifice used in the virtual reality application orients the plane of each torus above towards the centre. Given the requirement to stretch each torus to encompass the increasing number of nodes, in those below a further artifice was used, namely slightly flattening the cross-sections rather then using toroidal ellipses. A variety of improvements and alternatives could of course be envisaged. The node colouring is arbitrary to clarify and increasingly complex structure. Note that each loop of toruses of similar colour frames a "pathway" (below right) which is irrelevant to the supercomputer metaphor but is discussed further below.

Indication of progressive increase and interlinking of nodes
Interlinking of 16 nodes
(only partially unlinked)
Addition of a further set of nodes
(only partially unlinked)
Interlinking of a 3x3x3 set of 24 nodes
(each linking 3 orthogonal loops)
Addition of nodes to 3D torus interconnect schematic in Fujitsu supercomputer Addition of nodes to 3D torus interconnect schematic in Fujitsu supercomputer Animation of additional nodes in torus interconnect schematic in Fujitsu supercomputer

Scalability and dimensionality: The pattern of "torus fusion" is necessarily developed much further in a supercomputer. The image on the left of the Fujitsu conceptual model of Tofu shows the 6 coordinate axes through which 6-dimensionality is achieved. Each node has a 12-fold 3D pattern embedded within it. The X-Y-Z axes vary in size according to the system configuration; the A-B-C axes are fixed at 2x3x2 as shown. Each pair of adjacent nodes in then connected by a pattern of 12 links, as shown below right

Whereas the conceptual model of Fujitsu, as indicated below, necessarily emphasizes a rectilinear matrix and its potential for scalability, the reconstruction of that model in 3D emphasizes a degree of spherical organization in relation to the centre of the cube. This implies that scalability is better understood, for the purposes of this argument with respect to comprehension, as spherically scalable. Aspects of the argument for the merits of such organization are presented separately (Spherical Accounting: using geometry to embody developmental integrity, 2004).

The higher dimensionality is considered below in terms of nesting.

Conceptual model of Tofu supercomputer memory organization
Overview Detail
Details from depiction of supercomputer conceptual model of Fujitsu
Details from copyrighted presentations by Fujitsu (above)

Cubic organization of I Ching trigrams

There is a considerable literature, traditional and recent, on the manner in which the 64 hexagrams of the I Ching can be fruitfully organized. A traditional focus has included the manner in which the 8 constituent trigrams can be organized through the BaGua mirror.

Trigram organization: The BaGua mirror is traditionally presented in two circular forms. Of interest here is how that configuration of 8 trigrams can be related to the cubic format basic to the torus fusion approach presented above.

Comparison of octant organization of BaGua trigram system
Octants with signs
(according to Wolfram)
8-fold BaGua of trigrams
(according to Sung)
Trigram octant configuration
(following Wolfram)
Octants with signs Cubical representation  of BaGua pattern of I Ching Octant configuration with signs and trigram encoding
Reproduced from Wolfram MathWorld Reproduced from Z. D. Sung, The Symbols of Yi King or the Symbols of the Chinese Logic of Changes (1934, p. 12)  

Other experimental animations may be used to suggest other ways of comprehending the cognitive dynamics. As shown below, the moving cubes could be understood as the "cognitive tanks" discussed in the preceding argument (Tank Warfare Challenges for Global Governance: extending the "think tank" metaphor to include other cognitive modalities, 2019). That on the left is reproduced from an earlier discussion (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016; Neglected recognition of logical patterns -- especially of opposition, 2017).

Comparison of octant organization of BaGua trigram system
Animation of virtual variant
(wireframe rendering)
Screen shot of virtual reality variant in 3D
(solid rendering)
Number of trigram line
transformations between nodes
Octant organization of BaGua trigram system on cube Octant organization of BaGua trigram system on cube Octant organization of BaGua trigram system on cube
Virtual reality variants: vrml/wrl; x3d.  

Correspondences and analogies: A cubic configuration of the BaGua trigrams draws attention to the nature of the correspondences between them, as is evident from the complementarity between the patterns of lines. Of particular relevance to this argument is the distinctiveness of these patterns, despite a degree of similarity. This is further emphasized by the number of step changes across the cubic configuration (above right), from 3 (across the diagonal through the centre) to 1 (along the edge of the cube). Within the cultural framework of the I Ching, these relationships are articulated through poetic metaphors.

Within the conventions of the various disciplines, correspondences and analogies between domains tend to be viewed with suspicion in the quest for articulations which are natural to the particular domain -- and make no reference to other domains. Of interest in this respect are the various theories of correspondences (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). The latter clarification was elaborated in the light of the role of correspondence in a fundamental mathematical discovery with regard to the so-called monster group (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007). A valuable discussion of the related nature of analogies has been made by Douglas Hofstadter and Emmanuel Sander (Surfaces and Essences: analogy as the fuel and fire of thinking, 2013), as a further development of Hofstadter's earlier work (Fluid Concepts and Creative Analogies: computer models of the fundamental mechanisms of thought, 1995) and an extension of his seminal work on music and self-reference (Gödel, Escher, Bach: an Eternal Golden Braid, 1979). The importance of metaphor is specifically highlighted with respect to the creativity of Albert Einstein.

Given the tragically divisive dynamics of global society at this time, despite the desperate quest for coherence and unity, there is a case for reframing that quest as explored here. Rather than an obsessive focus on similarity, there is a case for exploring difference otherwise -- as is otherwise held to be vital to requisite variety in cybernetic terms. Rather than some kind of "similarity engine" through which to engender unity in conventional terms, the argument is for a form of "difference engine" through which an unconventional form of unity might be engendered.

In this sense each of the trigrams can be considered as indicative of an archetypal form of difference. Organized as a cube, as shown above, the toroidal links between the nodes are then indicative of correspondence relationships -- distinct from the forms of relationship typically sought within any conventional framework. The cube is then an archetypal configuration of correspondences.

Various graphical metaphors can be explored to suggest relationships within that framework. Those on the left below use the toroidal links as tunnels though which spheres travel -- suggestive of a form of "interdisciplinary" thinking. A central sphere is added at the origin -- indicative of the confluence of perspectives with which any cognitive integration might be associated. Those animations on the right suggest other design possibilities.

Alternative representations of cubic 8-fold pattern
Spheres in nodes Animations of edge movement
Spheres in nodes of cube Spheres in nodes of cube Animation of edge movement in 8-fold cubic pattern Animation of edge movement in 8-fold cubic pattern
Access X3D variant Access X3D variant Access X3D variant

Eightfold way? The cube of nodes in the animations on the left above can be usefully understood otherwise in terms of various 8-fold patterns, most notably those described as an "8-fold way". There is of course no lack of references to an "eightfold way" with which a pattern of octants might be appropriately associated (Reframing an eightfold way by entangling imagination and reality? 2019). These are seemingly inspired in whatever way, mnemonic or otherwise, by the Noble Eightfold Path of Buddhist practices leading to liberation from samsara, namely the painful cycle of "rebirth" (Eightfold Path, aka: Eightfold Way; 5 Definition(s), Wisdom Library). They include include frameworks as fruitfully diverse as:

Of particular relevance to the emphasis on comprehension of psychosocial organization in this argument are other "eightfold ways" through which distinctiveness is credibly recognized:

An obvious concern is the extent to which such articulations are proven, rather than determined by tradition or convenience. The question applies to any articulation of N-foldness to which credibility is attached for some purpose (as discussed further below with respect to "chunking" and sets of strategic goals). This issue is highlighted by subsequent challenges to the replicability of research on which such sets are based, as in the case of multiple intelligences (Lynn Waterhouse, Multiple Intelligences, the Mozart Effect, and Emotional Intelligence: a critical review, Educational Psychologist, 41, 2006, 4; Trenton Knauer, Psychology's Replication Crisis, Areo, 1 October 2019).

Given any sense of "cognitive valence", what chunks are held to "work" in practice by some, irrespective of the existence of proof or its quality. Whether or not proof is held to be required, how solid is the evidence for frameworks such as the following, otherwise deemed to be credible and to "feel right":

Deadly questions and catalysts for creativity:With the nodes of the cube indicative of mutual irrelevance, according to the conventions of the thinking relating to each node, each can be specifically recognized as incompatible or incommensurable with the others -- each an "anathema" to the others. In terms of creativity however, any combination may well prove to be a trigger to creativity through any perceptible analogies or correspondences. The mix may then be recognized as fruitful with the nodes recognized in this way understood as complementary.

One of the difficulties of any given conventional thinking framework -- as curated by the peer review system -- is that any external reference (to another "nodal" domain) tends to be recognized as a "kiss of death" effectively defining the irrelevance of an argument. Succinctly any such reference is a "turn off". It can be understood as killing creativity within the framework of a particular convention. Such a "kiss of death" is perceived as a threat to the coherence of the domain, outside the scope of appropriate comprehension -- perhaps to be framed as characteristic of a pseudoscience. The pattern of claims and counter-claims is intimately related to the dilemmas of "fake news" (Varieties of Fake News and Misrepresentation, 2019).

Unfortunately, but perhaps appropriate for the times, as argued by Jim Baggott:

There is no agreed criterion to distinguish science from pseudoscience, or just plain ordinary bullshit, opening the door to all manner of metaphysics masquerading as science. (But is it science?, Aeon, 7 October 2019)

The distinctive modalities, perhaps to be confirmed by "non-citation analysis", could be understood as "think tanks" -- Cognitive boundaries of cognitive process containers, as speculatively argued (Tank Warfare Challenges for Global Governance: extending the "think tank" metaphor to include other cognitive modalities, 2019).

In the quest for innovation, one approach is the cultivation of "nasty questions", typically disruptive to practitioners of a given discipline (Checklist of 'Nasty Methodological Questions' -- regarding development analyses and initiatives, 1981; In quest of the most deadly question, 2013). In the current period of global crisis -- seemingly beyond the capacity of conventional thinking -- the uncomfortable, unasked questions merit particular attention (Coping Capacity of Governance as Dangerously Questionable: recognizing assumptions and unasked questions when facing crisis, 2019).

Design of a "difference engine" for innovation? In this light, there is a case for confronting a set of differences as a means of engendering creativity transcending the preoccupations of a particular domain.

Mapping of distinctive preoccupations onto 8-fold supercomputer/BaGua schematic
The pattern of the "conceptual model" of the supercomputer, as depicted above, may therefore used to distinguish (and map) mutually contrasting nodes as discussed here. The image on the right shows the first two sections of this argument presented as two nodes in that schema. The six other domains, discussed below, are also indicated in that schema. Understood as complementary, the suggestion is that these are potentially related by a pattern of correspondences -- although any such pattern is necessarily tentative at this stage. Whether these are sufficiently distinct to constitute requisite variety remains to be explored. Mapping of distinctive preoccupations onto 8-fold supercomputer/BaGua schematic

There are numerous references to unity, global, integration, interdisciplinarity, and the like, as separately profiled (Integrative Knowledge Project). Despite their attraction over decades, the times are most notably characterized by toxic fragmentation of a high order. What then is the "pattern that connects" such distinctive quests for a coherent sense of identity? As articulated by Gregory Bateson: The pattern which connects is a meta-pattern. It is a pattern of patterns. It is that meta-pattern which defines the vast generalization that, indeed, it is patterns which connect (Mind and Nature: a necessary unity, 1979). To which he added in a much-cited phrase: Break the pattern which connects the items of learning and you necessarily destroy all quality.

The question is how to elicit, comprehend or engage with such a pattern, as can be variously discussed (Engaging with Elusive Connectivity and Coherence, 2018; Walking Elven Pathways: enactivating the pattern that connects, 2006; Hyperspace Clues to the Psychology of the Pattern that Connects, 2003). The argument here is that the pattern is of necessity elusive necessitating a complementarity set of cognitive modalities, as otherwise argued by Magoroh Maruyama (Polyocular Vision or Subunderstanding? Organization Studies, 25, 2004, pp 467-480).

The disparate modalities recognized -- with methodologies alien to each other -- may then be explored as suggestive metaphors, implying the need for a form of "discourse through metaphor" (Metaphorizing Dialogue to Enact a Flow Culture: transcending divisiveness by systematic embodiment of metaphor in discourse, 2019). To "make a difference", the argument here is that the metaphors in a difference engine are primarily visual. It is visual metaphors which offer particular advantages in terms of comprehension of the pattern that connect -- and in navigating that pattern.

A degree of recursion is evident in the cubic pattern as explored here in that the elements of that pattern are also of cubic form. Any paradoxical inconsistency with the requisite variety merits exploration in that light.

Understood otherwise, the quest here is for the nature of the design of a "comprehension computer" in which no particular cognitive modality is either primary or dependable. This is somewhat consistent with the Sanskrit adage: Neti Neti (Not this, Not that).

Pattern language and polyhedral mapping

Pattern language: The major insight in this respect has been the work of Christopher Alexander in identifying patterns that frame the "quality without a name" with which a "place to be" is associated ( (Pattern of transformations as a dynamic quality without a name, 2012). For him, a "place to be" in a building or a town is only viable to the extent that it is governed by the "timeless way" (The Timeless Way of Building, 1979). Alexander (and his team) identified 254 interlinked patterns as providing a language by which it could be framed. The approach could be extended to cognitive environments, as argued separately (5-fold Pattern Language, 1984).

Recent surveys of the possibility have been made by Helene Finidori (Patterns that Connect: exploring the potential of patterns and pattern languages in systemic interventions towards realizing sustainable futures, ISSS, 2016; Configuring Patterns and Pattern Languages for Systemic Inquiry and Design, Proceedings of the 25th Conference on Pattern Languages of Programs (PLoP), 2018).

Mapping on polyhedra: A valuable overview of the use of polyhedra for mapping patterns is provided by Ulrich Brehm and Egon Schulte (Polyhedral Maps, In: J.E. Goodman, et al. eds, Handbook of Discrete and Computational Geometry, CRC Press, 2017). This includes mapping onto non-orientable surfaces and convex polyhedra.

The possibility of mapping content (other than the surface of the Earth) onto polyhedra is the subject of a variety of separate discussions with images and animations:

The argument was notably illustrated in the latter through the tentative mapping onto polyhedra of the articles of various charters (Universal Declaration of Human Rights, European Convention on Human Rights, and Arab Charter on Human Rights).

Mapping strategic proposals: Another approach has been taken through the experimental mapping of the elements of strategic proposals onto polyhedra:

A particular consideration is the completely mysterious process through which global strategic initiatives are framed in terms of patterns of N-foldness -- in which the reason for whatever is chosen as N remains unexplained in systemic terms. As an extension of the traditional "laundry list" approach to the articles of international conventions, at best it is claimed to be a matter of political convenience. The size of N is seemingly unquestionable and not subject to further comment -- whether or not it is memorable.

Examples include:

Systemic inexplicability? This suggests the existence of an unexplored (unconscious) process through which psychosocial organization is most conveniently and comfortably organized in terms of the cube -- even to the point of being "locked into" one such pattern or another (possibly described as "feeling right"). The various patterns of strategic articulation can be considered in this light -- if only for mnemonic purposes. Understood in this way, the question is framed as to how systemically complete is any explicit pattern of N-foldness -- whether or not additional elements are effectively implicit.

Unusual (and relatively obscure) leads are offered with respect to N-fold strategic articulations of particular importance. For example:

The question could then be asked how any such locking is reinforced (or exemplified) by typical transformations of the cube into related polyhedra -- then to be recognized as "cognitive locking patterns". Is the simpler and more fundamental structure, the tetrahedron, to be considered "systemically meta-stable" with a greater number of dimensions implied rather than explicit.

The question is discussed separately (Global Coherence by Interrelating Disparate Strategic Patterns Dynamically: topological interweaving of 4-fold, 8-fold, 12-fold, 16-fold and 20-fold in 3D, 2019). The concern has been highlighted in more general terms (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980; Examples of Integrated, Multi-set Concept Schemes, 1984).

Higher dimensionality, polyhedral packing and transformation

Multidimensionality: This argument has further implications in the light of the assumptions too readily made regarding personal human experience of "globality" and "wholth" (Wholth as Sustaining Dynamic of Health and Wealth: cognitive dynamics sustaining the meta-pattern that connects, 2013).

To the extent that one assumes oneself to be "rounded", is this as a 2-sphere -- superficially, as with a bubble? Or is there an understanding of depth for which a 3-sphere would be more appropriate? More challenging -- if not inspiring -- is human identity better understood in terms of an N-sphere, with N being commensurate with the insights of physics? Such topological considerations could clarify the manner in which spherical experience of coherence could be transformed into toroidal experience -- with corresponding topological distinctions between 2-torus and N-torus.

Are human aspirations to "freedom" far more usefully recognized as being associated with such multidimensionality, as implied by arguments from various perspectives:

For some, speculation extends to the nature of "multidimensional humanity" and to "transdimensional humanity" (Alice Bryant and Linda Seebach, Multidimensional Potential of Human Beings).

Complexification through hexagram organization: There exist four traditional approaches to the organization of the 64 hexagrams of the I Ching as an 8x8 matrix, as presented and discussed separately (Classical Chinese Arrangements of 64 Hexagrams in Squares, 2008). The four patterns can be experimentally superimposed as indicative of alternation between frameworks, as presented separately (Fractal comprehension of coherence requiring an 8-fold uncertainty principle? 2019).

There are examples of a circular organization, as shown below left and discussed separately (Diagram of 384 Relationships between I Ching Hexagrams, 1983; Bagua and the sequence of 64 hexagrams, Shanghai Daily, 20 December 2015). One of these is named as the circle of Shao Yong (1011-1077) or the I Ching hexagram circle. It was an influential feature of the communication to Leibniz in 1701. A qualification to a prevailing conclusion by science (predictable as argued above) is offered by James A. Ryan:

Leibniz thought he had discovered evidence of a forgotten mathematical science in the Chinese past, and, in spite of his sinological knowledge, he never found evidence to the contrary. Thus, it has been left to contemporary scholars to explain the apparent correspondence. While the general trend in scholarship has rightly presumed that no forgotten mathematical science existed in ancient China, the conclusion that the Yijing / Binary System Episode was a mere coincidence has perhaps not satisfied scholars, in view of the intricacy of the binary geometrical progression and the temporal and spatial isolation of the Chinese Diagram from the European. (Leibniz' Binary System and Shao Yong's "Yijing", Philosophy East and West, 46, 1996, 1)

The pattern of 64 is relatively unique within the variety of polyhedra. However one interesting candidate for mapping purposes is the toroidal drilled truncated cube with 64 edges -- with which any set of 64 elements could be associated, as discussed separately (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015). The issue is whether the manner in which they can be positioned on that framework constitutes a configuration which is meaningful in relation to particular cases, such as the hexagrams (or genetic codons). Furthermore, is it possible that known constraints in the patterning in such particular cases can together offer guidance in the attribution of the distinct elements -- of relevance to each case? The correspondence between hexagrams and genetic codons offers a provocative possibility that the hexagrams could be understood in terms of memetic codons, as notably argued by M. Pitkänen (Could one find a geometric realization for genetic and memetic codes? Semantic Scholar, 2013; Three new physics realizations of the genetic code and the role of dark matter in bio-systems, Semantic Scholar, 2018).

Preliminary experiments with the drilled truncated cube have been undertaken previously with respect to the hexagrams alone -- but only to get a sense of the possibility, as a "proof of concept" (Enabling Wisdom Dynamically within Intertwined Tori: Requisite resonance in global knowledge architecture, 2012).

Traditional circular configuration of hexagrams
(augmented with transformation pathways between them)
Drilled truncated cube of 64 edges with random attribution of hexagram names
Selected faces transparent All faces transparent
Logo of Laetus in Praesens Drilled truncated cube of 64 edges with hexagram names Drilled truncated cube of 64 edges with hexagram names
Animated variant at Dynamic Exploration of Value Configurations (2008) Animations prepared using Stella Polyhedron Navigator

(Topology of a Renaissance "Stargate" of Higher Dimensionality: complementary ways of imagining engagement with otherness, 2018) ***

Nesting polyhedra and nested cubes: As noted above, the Fujitsu torus fusion memory organization involves a form of nesting, as shown in the image below left. Such nesting also features in the diamond cubic structure as shown in the animation below centre. The structure of the hypercube or tesseract (as discussed below) can be represented by nesting of cubes to a higher degree as shown in the animation below right -- where 7 cubes are nested within an eighth

Supercomputer memory organization
(detail from Tofu)
Diamond cubic crystal structure
Nested cubes -- N-fold hypercube?
Detail of higher order sFujitsu supercomputer memory organization Animation of diamond cubic crystal structure Animation of 8-fold nested cubes as an N-fold hypercube
Detail from Fujitsu image above By MarinaVladivostok -- Own work Link  

Such a pattern invites reflection on yet another pattern for the periodic table of chemical elements of which there are some thousand indicated in the Internet Database of Periodic Tables. Of relevance however is the absence of nesting in such configurations with few exceptions (Tomás A. Carroll, Spherical and Russian Doll Formulations, 2008; Anthony Grainge, Elemental Periodicity formulation with concentric spheres intersecting orthogonal planes, 2019). Of particular relevance are recent arguments for a hypercube formulation (Ramon Carbó-Dorca and Tanmoy Chakraborty, Divagations about the periodic table: Boolean hypercube and quantum similarity connections, Journal of Computational Chemistry, 40, 2019, 30). This considers the possibility of a seven-dimensional Boolean hypercube.

The image below left suggests how the groups and periods might be related within a nested hypercube framework. It is however important to recall that such nesting is a representation of higher dimensionality. The image below centre shows another technique for the depiction of the four-dimensionality of a cube by projection into 3D. Another metaphor is offered by use of h polyhedra nested within one another, as discussed separately (Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra, 2015). Two cases are considered there Relative movement of nested Platonic polyhedra: pumping and rotation and Packing and unpacking of 12 semi-regular Archimedean polyhedra. The animation on the right is an illustration of the first.

24 cells
Nested Periodic Table of Chemical Elements
(Groups I to VIII in Periods 1 to 8)
4D: Uniform polychoron from 3D vertex
24 cells, 96 faces, 96 edges, 24 vertices
Nesting 5 Platonic polyhedra
"Pumping" motion (video mp4)
within Rhombic Triacontahedron (green)
Nested Periodic Table of Chemical Elements 4D: Uniform polychoron from 3D vertex Platonic polyhedra nested within Rhombic triacontahedron
Developed with X3D Edit and Stella Polyhedron Navigator

There are many references to nesting of cubes, including:

Of particular interest, given the emphasis here on comprehension of complexity, are references to the nested cube with respect to memory, as discussed in the Art of Memory Forum in relation to the technique of Gregor von Feinaigle (The New Art of Memory, 1812). There is also the sense in which links between the nested cubes can be recognized as directions of perception of inner or outer coherence. This is suggestive of the understanding associated with degrees of access, as cultivated with respect to classified or secret knowledge.

Inversion of cube: As discussed separately, so much of psychosocial organization is framed by the static architecture of the cube in 3D -- or through its compression into a square in 2D (Eliciting the dynamics of the cube: reframing discourse dynamics, 2018). This is the favoured modality for most explanatory tables. Through its 12-edges, the cube potentially offers clues to a relationship within any 12-fold pattern, but has not been extensively explored in that respect, although it is a feature of studies of oppositional logic, and a relationship to the 8-fold pattern valued in Chinese thinking (as discussed below). Missing are the paradoxical insights justifying reference to the Necker cube and the "4D" Klein bottle of relevance to this argument ((Steven M. Rosen, Topologies of the Flesh, 2006, Dreams, Death, Rebirth: a multimedia topological odyssey into alchemy's hidden dimensions, 2014).

The question is therefore whether the form that Paul Schatz extracted from the cube -- through the dynamics of its possible eversion -- offers indications of a way of transforming conventional preoccupation with its static form. The following images offer some indication of this.

Explorative 3D animations of the image on the left above are presented separately (Succinct mapping of multidimensional psychosocial dynamics? 2016). ****

In terms of the argument with respect to features hidden from the observer, this is especially evident in the case of the central image above. In that phase, the 24 sides are visible through the animation. But in the case of the static blue-green perspective or the static red-yellow perspective, only 12 sides are visible. Being hidden, the other 12 can only be inferred unless the structure was rendered transparent. In the reality of sociopolitical discourse opposing sides are never "transparent" to one another -- whatever the claims that are made. Cognitively each could be interpreted as a form of shadow for the other in the Jungian sense. The wireframe image on the right is indicative of the commercial product widely marketed as Hexyflex.

Schatz cube prior to inversion   Rotation of views of a phase
in inversion of cube
Animation of selected phases
in inversion of cube
Schatz cube inversion Sergey Bederov of Cortona3D has produced an interactive vrml version of the complete cycle of the original, with formulae kindly provided by Charles Gunn.
Thanks to both.
See video of the complete cycle
Rotation of views of a phase  in inversion of cube Cube inversion animation

Brain organization, cognition, comprehension -- and music

Tuning systems and music: Whilst the complexity indicated above is variously beyond ordinary comprehension, as formally presented or illustrated, it is surprising to note the capacity of many to engage with higher orders of complexity through music.

Given the emphasis above on the apparent inexplicability of different patterns of N-foldness in the distinction of sets of concepts and strategies, there is a strong case for exploring the extent to which the sense of coherence ("feeling right") may derive in part from unrecognized preference for particular musical tuning systems. This possibility merits exploration in the light of Cognitive constraints in comprehending strategic coherence (2019) and chunking required for memorability, dating from the famed paper of George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review. 63, 1956, 2).

The validity of this widely accepted argument is now challenged (Nelson Cowan, The Magical Number 4 in Short-term Memory: a reconsideration of mental storage capacity. Behavioral and Brain Sciences. 24, 2001, 1; Nelson Cowan, et al., The Legend of the Magical Number Seven, 2007; Timothy Brady, et al., A Review of Visual Memory Capacity: beyond individual items and toward structured representations, Journal of Vision, 11, 2011, 4).

A tuning system is the system used to define which tones, or pitches, to use when playing music, namely the choice of number and spacing of frequency values used. The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones -- arguably, as with the creation of sets of concepts or strategies. As the number of tones is increased, conflicts arise in how each tone combines with every other -- again, as in the case of strategies. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses. The distinctive sets of global strategies could then be explored in this light.

The unexplained preference for 12-fold sets was noted above (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011), Any quest for clarity could then be explored in terms of the 12-note chromatic scale, each note being unique, and the various compromises in tuning it. It is not known why there are 12, as stressed by Daniel White (Potential Mathematical Models for the Western Musical Scale A Historical and Empirical Comparison, 2007; summary). This presents a number of explanatory theories.

A musical scale is any set of musical notes ordered by fundamental frequency or pitch. Scales may be described according to the number of different pitch classes they contain. The notes of a scale form harmonic intervals with each of the other notes of the chord in combination.

musical scales notes per octave harmonic intervals usage polyhedra strategies
Chromatic, or dodecatonic 12        
Octatonic 8 28 used in jazz and modern classical music    
Heptatonic 7 21 the most common modern Western scale    
Hexatonic 6 15 common in Western folk music    
Pentatonic 5 10 common in folk music, especially in Asian music    
Tetratonic 4   generally limited to prehistoric ("primitive") music    
Tritonic 3   generally limited to prehistoric ("primitive") music    
Ditonic 2   generally limited to prehistoric ("primitive") music    
Monotonic 1   limited use in liturgy, and for effect in modern art music    

Given the confusion with regard to such chunking, the provocative question could be raised -- in the light of the self-referential perspective -- as to whether attention could then be usefully accorded to "14 plus-or-minus-2" rather than "7 plus-or-minus-2". This would neatly encompass the range of strategies from 12 to 16 at least.

Of particular interest to the argument with regard to musical tuning systems is the hexany invented by Erv Wilson. This can be thought of as analogous to the octahedron (geometric dual of the cube). The notes are arranged so that each point represents a pitch and every edge and interval with each face represents a triad. It thus has eight just intonation triads where each triad has two notes in common with three of the other chords. Each triad occurs just once with its inversion represented by the opposing 3 tones. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series. The points represent musical notes, and the three notes that make each of the triangular faces represent musical triads. Wilson also pointed out and explored the idea of melodic hexanies. (Robert Walker, Hexany) Hexany
  Modified by Robert Walker from Tilman Piesk's Hypercubestar on Wikipedia

Polyhedra and music: This association dates back to the reflections of the pythagoreans and the Harmony of the Spheres. A recent exercise includes that of W. Douglas Maurer (A Musical Suite Based on the Platonic Solids, Bridges: Mathematical Connections in Art, Music, and Science, 2002). Other references include (Polyhedra: frozen music for the eyes; Caspar Schwabe, The Zonohedra Music Chart; Iulia Millesima, The Pythagorean Acoustic: Geometry and Music of Sirens, Labyrinth Designers; Peter Pesic. Music and the Making of Modern Science, MIT Press, 2014; Bruce Stephenson, The Music of the Heavens: Kepler's harmonic astronomy, Princeton University Press, 2014).

Consideration has also been given to organization of musical scales in terms of a hypercube by R. W. Peck (A Hypercube-Graph Model for n-Tone Rows and Relations. In: J. Yust, et al, (eds), Mathematics and Computation in Music, MCM 2013. Lecture Notes in Computer Science, 7937, 2013).

Music and I Ching: One extensive discussion of this relationship is provided by Richard O. Burdick (I Ching as a Structural Foundation for Music). The author offers a relationship of musical scales to the Shao Yong circle of hexagrams (Richard O. Burdick, I Ching Music -- Shau Yung's Circle).

With the 8-eements of the BaGua associated with the musical octave, the images above can be further modified to suggest a mapping of the hexagram houses associated with each trigram, as shown below (Organization of I Ching hexagrams in terms of traditional "houses", 1995).

Indication of BaGua "sub-cubes" positioned outward along diagonals
Screen shot of virtual reality image
(as above with trigram coding)
Trigrams as "houses" from image on left
(with associated hexagrams)
Sub-cube movement outward on diagonals
Octant organization of BaGua trigram system Hexagrams associated with octant organization of BaGua trigram system of houses Schematic animation of house hexagrams of BaGua trigram system

Strategic credibility: Such arguments usefully frame the challenge of comprehending the nature of "belief" in any N-fold strategic framework or set of "goal" -- and its credibility when communicated widely. Since equivalent sets have long been a feature of the theology of different religions, there is a case for exploring such belief otherwise (Mathematical Theology -- Future Science of Confidence in Belief, 2011; Gregory Benford, Applied Mathematical Theology, Nature, 440, 2006; James Bradley, Theology and Mathematics, Theology and Science, 9, 2011, 1)

The chunking issue can be usefully related to the central preoccupation of this argument as framed by the cube:

  • 3 body diagonals / 3 axes of symmetry
  • 6 faces
  • 8 vertices
  • 12 edges
  • 12 face diagonals
  • 14 faces+vertices
  • 15 face+body diagonals
  • 20 vertices+edges
  • 24 torus half-loops (feed-back/feed-forward)

In arguing the point through a musical metaphor, there is an irony to any criticism of the study by Jacques Attali ( Demain, qui gouvernera le monde? 2011) in that Attali himself (in a previous study) specifically indicated that cultures articulated their social organization through the musical structure favoured in the immediate past (Noise: the political economy of music, 1977). Thus he specifically relates the currently favoured pattern of organization to that of classical Western music. As discussed separately, with respect to "tomorrow", should then at least take account of the pattern of music currently favoured by the voters of the future (Tomorrow, Who Will Govern the World? 2011).

Neuroscience: The arguments above merit comparison with the results of recent neuroscience research indicating the remarkable possibility of cognitive processes of up to 11-dimensional form in the light of emergent neuronal connectivity in the human brain. As summarized:

Using mathematics in a novel way in neuroscience, the Blue Brain Project shows that the brain operates on many dimensions, not just the three dimensions that we are accustomed to. For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions - ground-breaking work that is beginning to reveal the brain's deepest architectural secrets..... these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object. ...

The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner. It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates. (Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks, Frontiers Communications in Neuroscience 12 June 2017)

Oppositional logic and its requisite polyhedral geometry

4D Tesseract: The challenge to comprehension is all the greater in that any dynamic suggests that it is at least in 4D that a 3D framework merits consideration. With respect to the Logic Alphabet Tesseract (below left), as noted by Louis Kauffman:

Shea 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 and Human Knowing, 8, 2001)

Zellweger's depiction is usefully complemented by that of the 4D tesseract as in the other images below, as discussed separately (Oppositional logic? 2018; Tony Phillips, Topology of Venn Diagrams, AMS, June 2005).

With regard to the argument above, one provocative approach is to consider how the polyhedral forms employed in the extensively developed literature on logical geometry might be presented in relation to a 3D representation of the Tao symbol. An important key to the significance for governance of hyperdimensionality is that work on logical geometry. This notably featured in the presentations at the IV International Congress on: The Square of Opposition -- Vatican City, 2014, compiled by Jean-Yves Béziau and Gianfranco Basti (The Square of Opposition: a cornerstone of thought, 2017):. This constitutes a collection of new investigations and discoveries on the theory of opposition (square, hexagon, octagon, polyhedra of opposition), ranging from historical considerations to new mathematical developments of the theory of opposition including applications to theology, theory of argumentation and metalogic.

Of particular relevance, beyond the Aristotelian square of opposition is its relationship to the rhombic dodecahedron featuring in Hasse diagrams -- both involving discussion of the hypercube, as featured in the work of Lorenz Demey and Hans Smessaert (The Relationship between Aristotelian and Hasse Diagrams, 2014; Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation, Symmetry, 9, 2017, 204; Geometric and Cognitive Differences between Logical Diagrams for the Boolean Algebra B4) and by Hans Smessaert (On the 3D Visualisation of Logical Relations, Logica Universalis, 3, 2009, 2).

Arguably if there is one characteristic of psychosocial reality which is a fundamental challenge to governance it is that of "opposition" and the framework within which it can be appropriately integrated. The argument for doing so is that literature is particularly focused on the geometrical representation of opposition as articulated in truth tables through the set of Boolean connectives. A key polyhedron used to map the 16 (-2) Boolean logical connectives in that approach is the rhombic dodecahedron of 14 vertices and 12 faces.

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

Tesseract animation
simulating requisite 4-dimensionality?
Topologically faithful 4-statement Venn diagram
is the graph of edges of a 4-dimensional cube
as described by Tony Phillips
Organization of contingent bitstrings
on a rhombic dodecahedron
The Logic Alphabet Tesseract by Shea Zellweger Tesseract animation Topologically faithful 4-statement Venn diagram Rhombic dodecahedron with contingent bitstrings
Diagram by Warren Tschantz
(reproduced from the Institute of Figuring) .
by Jason Hise [CC0], via Wikimedia Commons A vertex is labeled by its coordinates (0 or 1) in the A, B, C and D directions; the 4-cube is drawn as projected into 3-space; edges going off in the 4th dimension are shown in green. Adapted from Lorenz Demey and Hans Smessaert (2017)

The images above recall that of the diamond cubic crystal structure with its repeating pattern of 8 atoms (presented above). This is all the more intriguing because of the value attached to diamonds in society and the particular value associated with it in Buddhism. Major symbolic importance is associated with the diamond, notably in Buddhist traditions, as a metaphor of a particular emergent order of the mind and the understanding of that order as a 'vehicle', or 'body', for the spirit (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). The terms 'diamond mind' and 'diamond body' are widely used in Buddhism and are notably a focus for Diamond Way Buddhism. This metaphor seems however to focus on the individual and not on the ordering of society.

Other experimental animations may be used to suggest other ways of comprehending the cognitive dynamics. As shown below, the moving cubes could be understood as the "cognitive tanks" discussed in the preceding argument (Tank Warfare Challenges for Global Governance: extending the "think tank" metaphor to include other cognitive modalities, 2019). That on the left is reproduced from an earlier discussion (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016; Neglected recognition of logical patterns -- especially of opposition, 2017).

The point can be developed further with respect to oppositional logic (Guoping Du, Hongguang Wang and Jie Shen, Oppositional Logic, Logic, Rationality, and Interaction, Springer, 2009, pp 319-319) and discussed separately in terms of a 4-dimensional polyhedral configuration of directions (Neglected recognition of logical patterns -- especially of opposition, 2017).

Of particular relevance to the argument here, relating to a fundamental pattern of distinctions, are the commentaries of Louis Kauffman on the much-cited study by George Spencer-Brown (Laws of Form, 1969) and the remarkable connection between Laws of Form, polyhedral geometry, mirror symmetry and the work of Zellweger (indicated above). As noted in an insightful review by Kauffman (Laws of Form: an exploration in mathematics and foundations):

... this is an approach to mathematics (and to epistemology) that begins and ends with the notion of a distinction. Nothing could be simpler. A distinction is seen to cleave a domain. A distinction makes a distinction

Although Kauffman specifically indicates that the further implications of such fundamental insights remain to be explored, it is far from clear whether appropriate effort has been devoted to their relevance to the "poisonous" distinctions which are such a prominent feature of psychosocial dynamics at this time. Specifically there is the question of how such a subtle understanding of distinction and difference might help to reframe relations between partisan political extremes, religions in conflict ("interfaith discourse"), and disciplines cultivating fragmentation ("interdisciplinarity"). Given the many related studies of Kauffman on knot theory, the relevance of such insights have yet to be applied to the sense in which society has "tied itself into a complex knot" -- even an archetypal Gordian knot, as discussed separately (Engaging globally with knots and riddles -- Gordian and otherwise, 2018).

Given the higher dimensional implications of the cubic metaphor which inspired this exploration, Kauffman's diagrammatic application of the notaion of Laws of Form are potentially of the greatest relevance to the possibility of transcendent modes of discourse.

Application of the notation of the Laws of Form to logical connectives
16 Binary Boolean Connectives Planar graph of the rhombic dodecahedron Logical Garnet (Zellweger)
Reproduced from Louis H Kauffman (Laws of Form: an exploration in mathematics and foundations)

There are very few examples of application to any degree of such insights to the challenges of inter-domain discourse. Exceptions appear to include:

Reflexivity in multi-loop thinking and higher order learning

There seem to be a number of somewhat unrelated approaches to what might be characterized as reflexivity or self-reference (Hilary Lawson, Reflexivity: the post-modern predicament, 1986). Binary challenge ***

Multi-loop thinking and learning: Distinctions are variously made between:

Bottom lines in accounting: In business and accounting, net income is a measure of the profitability of a venture. Other "bottom lines" are however proposed, as discussed separately (Spherical Accounting: using geometry to embody developmental integrity, 2004):

Helical models of innovation: A remarkable range of research and other initiatives have emerged under the banner of the Triple Helix model of innovation (Marina Ranga and Henry Etzkowitz, Triple Helix Systems: an analytical framework for innovation policy and practice in the knowledge society, Industry and Higher Education, 27, 2013; Loet Leydesdorff, The Triple Helix: an evolutionary model of innovations. Research Policy, 29, 2000):

Causal and feedback loops: The cybernetics of control systems has long highlighted the distinctive roles of positive feedback loops and negative feedback loops in control systems. This preoccupation has been extended to the "cybernetics of cybernetics", namely the recursive application of cybernetics to itself, or second-order cybernetics..This is distinguished from first-order cybernetics (namely of "observed systems") as being the cybernetics of "observing systems". It has implications for creativity (Cybernetics of cybernetics: complex adaptive systems? 2007; Relevance to change, learning and creativity, 2014; Magoroh Maruyama, Causal Loops, Interaction, and Creativity, International Review of Sociology, 13, 2003, 3).

Higher orders of feedback can also be envisaged, although the distinction between the "orders" of cybernetics is currently a matter of controversy, most recently addressed by Maurice Yolles and Gerhard Fink (Generic Agency Theory, Cybernetic Orders and New Paradigms, Kybernetes, 44, 2015, 2). A valuable interpretation of related distinctions is provided in the discussion of Cadell Last (Towards a Big Historical Understanding of the Symbolic-Imaginary, 2017):

Viable system modelling: The viable system model (VSM) is a model of the organizational structure of any autonomous system capable of producing itself. A viable system is any system organized in such a way as to meet the demands of surviving in the changing environment through adaptation. A useful overview is provided by QualTechSys (15 January 2018) in the light of thinking with regard to the human brain by Stafford Beer (Brain of the Firm: the managerial cybernetics of organization, 1972). This was inspiration to develop the Viable System Model by leading him to identify the following inter-related systems governing the human body. A VSM is composed of five interacting subsystems which may be mapped onto aspects of organizational structure. In broad terms Systems 1–3 are concerned with the 'here and now' of the organization's operations, System 4 is concerned with the 'there and then' – strategical responses to the effects of external, environmental and future demands on the organization. System 5 is concerned with balancing the 'here and now' and the 'there and then' to give policy directives which maintain the organization as a viable entity:

It can be understood as embodying many of the considerations above with respect to cybernetics (Andrew Pickering, The Science of the Unknowable: Stafford Beer's cybernetic informatics, Kybernetes,33, 2004, 3/4). Thus for Frank van Caspel (VSM as a Tool for Organizational Change? A Critical Examination, Nijmegen School of Management, 2011): *** 7 functions cubocta?

It is an appealing idea to use the Viable System Model as a tool to guide organizational change. In doing so, however, the risk of exceeding the VSM's 'jurisdiction' is quite real. This article consists of an analytical examination of the degree to which the VSM can meaningfully contribute to organizational change. A functional definition of organizational change is introduced, in the form of the 3D-model of organizational change. It defines organizational change as consisting of three dimensions: functional, social and infrastructural. Next, the VSM is described. It is (also) a functional model, specifying five necessary and sufficient functions for organizational viability. Having acquired both a definition of organizational change and knowledge of the VSM, its suitability to contribute to change in different phases is examined.It is concluded that because the VSM is purely functional, it can only be used for diagnosis of existing or proposed organizational infrastructures. It cannot contribute to the design of concrete organizational infrastructures. This is a direct criticism of those cases in which the VSM was used during post-diagnostic change phases, some of which will be discussed. Post-diagnostic usage cannot be guided using only the VSM, but must rely on knowledge external from it. Researchers should be aware of the functional nature of the VSM, and its associated limitations. This will help to prevent the misattribution of the success or failure of change efforts to the VSM, where in fact other sources have implicitly steered the process

Strategic decision-making: Aspects of the considerations above are evident in the reality of decision-making. One valuable articulation is that of Arthur Young (The Geometry of Meaning, 1976). The sense of a learning cycle is fundamental to that articulation in discussing the sufficiency of a fourfold pattern, Young relates this to the necessity of feedback (in the light of piloting a helicopter):

    1. To know the position of a body in space, we need one instantaneous observation...
    2. To know its velocity, which is computer from the difference in position of the body and the difference in time between the two observations, we need two such observations
    3. To know its acceleration, we need three observations
    4. To know that a body... is under control, and to distinguish it from one in which the controls are stuck, we need at least four observations...
    5. To know the destination, provided the operator does not change his mind or try to fool us, we need five observations
    6. To know the operator has changed his mind or is trying to fool us, we need six observations

      Note that the fifth observation is to establish a position... and the sixth a change of position. Thus categories five and six repeat the cycle, the fifth falling into the position category and the sixth into the velocity category... the sufficiency of four categories is demonstrated. (p. 18)

Young's 12-phase learning / action cycles. can be variously adapted (Typology of 12 complementary strategies essential to sustainable development, Typology of 12 complementary dialogue modes essential to sustainable dialogue). The particular merit of the approach is the explicit distinction between the 12 elements of the pattern offering insights into their cyclic relationship in practice.

Of related interest is the articulation of the OODA loop developed by John Boyd as a military strategist.

Catastrophic WH-questions: It would not be surprisng, in a civilization dominated by cubic environments, to discover that its conceptual and strategic challenges and dilemmas lend themselves to fruitful cubic configuration. This could follow from the potential interplay of two less obviously interrelated 7-fold sets, together with a third set of "dilemmas" (questionably "sevenfold"):

Any such configuration reframes the question of 4-fold through 7-fold patterning and how this might be recognized and held in a cubically informed context -- the cognitive space for daily reflection, otherwise recognized as "thinking inside the box", as variously advocated (Bruce Bueno de Mesquita, et al, Thinking Inside the Box: a closer look at democracy and human riights, International Studies Quarterly, 49, 2005; John Gerring, The Mechanismic Worldview: thinking inside the box, British Journal of Political Science, 38, 2008, 1). Such a cognitive box offers six "windows" on external catastrophes but leaves comprehenion of the seventh catastrophe mysterious -- as with the traditional "seventh seal".

Speculative clues are offered by the following:

Commensurate with the "irrational" nature of questions, catastrophes and dilemmas, especially in a strategic context, is the relation to the 7-fold offered by the unusual Szilassi polyhedron -- usefully symbolic of the cognitive challenge of the times. This has 7 faces (of 4 types), 14 vertices (of 7 types), and 21 edges (of 12 types) -- 42 features together suitably reminiscent of the widely cited Answer to the Ultimate Question of Life, the Universe, and Everything (Douglas Adams, The Hitchhiker's Guide to the Galaxy).

The association of the Szilassi polyhedron with cube inversion is discussed separately (Time for Provocative Mnemonic Aids to Systemic Connectivity? 2018), notably in relation to the cuboctahedron. Its value with respect to configuration of questions can be similarly explored (Mapping of WH-questions with question-pairs onto the Szilassi polyhedron, 2014).

Multi-loop representation in 2D, 3D, and more? There is no lack of imagery representing multi-loop thinking, learning and decision-making in 2D -- in a manner typical of readily reproducible systems diagrams and mind-maps. As might be expected, many are subject to copyright, with all the irony this implies with respective to collective learning. Examples include the following :

Observe–Orient–Decide–Act cycle
of John Boyd
Circular configuration of 12 "measure formulae" of physics correlated with the pattern of the zodiac
(combining representations by Arthur Young  from The Geometry of Meaning, p. 102 and 119)
Observe–Orient–Decide–Act cycle  of John Boyd Zodiac tripliciities (Geometry of Meaning) Zodiac quadruplicities (Geometry of Meaning)
Reproduced from Wikipedia Reproduced from Rosetta stone of meaningful cycles? (2018) .

There are very few indications that the complexity of such relationships lend themselves to representation in 3D -- or may require it -- nor to the possibility that representations of higher dimensionality may be required.

The helical approaches described above can be usefully explored as embedded in polyhedra (Biomimetic embedding of N-tuple helices in spherical polyhedra, 2017; Contrasting the implications of "triple helix" -- cognitive and otherwise, 2017). Implications of a third dimension (or more) are evident in Arthur Young's use of the Rosetta Stone metaphor (Insights into Dynamics of any Psychosocial Rosetta Stone: standing wave understood dynamically rather than statically, 2018) For Arthur Young, the correspondence between the measure formulae of physics and learning cycles can be significantly presented mnemonically in terms of the signs of the zodiac. For him, this cyclic pattern then constitutes a form of Rosetta stone (Geometry of meaning: an alchemical Rosetta Stone? 2013). ***

There is a case for recognizing that the cybernetic "orders" above could be associated with distinct topological surfaces whether for representation, mapping or symbolic purposes:

(Cognitive Osmosis in a Knowledge-based Civilization: interface challenge of inside-outside, insight-outsight, information-outformation, 2017)

Toroidal constraint -- nuclear fusion as metaphor of cognitive fusion

The widespread focus on cubic organization tends to obscure how, if at all, that static pattern relates to what flows within the pattern that connects -- and effectively defines that connectivity. The distinctive encoding of the 8 trigrams of the BaGua (discussed above) is indicative of transformations between whatever they can be considered to represent metaphorically. However the sense of what flows or circulates in experiential terms within that cubic "container" is relatively obscured, however it may be discussed (Circulation of the Light: essential metaphor of global sustainability? 2010). The question is one of how the transformation perspective between a static cubic pattern and a circular dynamic is to be comprehended.

Two indications are offered through technomimicry, given the fundamental importance of circulation to particular technologies (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011). One is provided by the design constraints of the International Thermonuclear Experimental Reactor (ITER). This depends on a toroidal construction in which the circulation of plasma is constrained by a pattern of 6 magnets and 18 toroidal field coils to prevent the plasma coming into contact with the walls of the torus (In the lair of the ring magnets, ITER, February 2016). Those design constraints can be explored with respect to a hypothetical cognitive analogue (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006).

The cubic framework above can be used to indicate how its toroidal elements might be used to frame circular pathways within the cube (below left). These could be interpreted in terms of information, attention or enthusiasm -- given the challenge of containing and focusing such flows, and avoiding dissipation. Flow could be notably explored in the light of the psychology of flow ( Mihály Csíkszentmihályi, Beyond Boredom and Anxiety: experiencing flow in work and play, 1975).

A second indication is offered by the fundamental insights of Nikola Tesla into the rotation of a magnetic field, vital to the functioning of electrical generators and motors (Reimagining Tesla's Creativity through Technomimicry: psychosocial empowerment by imagining charged conditions otherwise, 2014). That includes a discussion of the Potential implications of alternation and rotation in psychosocial fields, Imagining a method for adapting Tesla's insights to a psychosocial context and Detecting a meta-pattern of connectivity amongst Tesla's insights. Especially relevant are Tesla's insights into fruitfully interrelating oppositely charged conditions ("positive" versus "negative"), as discussed there Encycling positive and negative for future sustainability. The current capacity for doing so in psychosocial systems is negligible, as is now only too evident in the toxic dynamics of democracies (Being Positive Avoiding Negativity: management challenge of positive vs negative, 2005; Barbara Ehrenreich, Bright-Sided: how the relentless promotion of positive thinking has undermined America, 2009).

Such considerations recall the mythical references to the role of the Ouroboros. As a challenge to the imagination in this context, this is further discussed separately (Explanation vs. Inplanation: multiversal embodiment through the Ouroboros, 2012).

Comparable circuit organization within a toroidal framework
Mutually orthogonal pathways (white)
framed by the torus loops
Suggestive animation of circulation of relationships
based on BaGua trigrams
Animation of helical coils embedded within helical coils indicative of an engendered "magnetic field" Indicative association of Ouroboros with 64 interrelated conditions of change encoded by the I Ching, as an indication of the circular configuration of the variety of pathways of choice and decision
Mutually orthogonal pathways  framed by the torus loops Animation of an Arrangement of Patents / Inventions Animation of Ouroboros schematically  embedded in  helical coils Schemati indication of  Ouroboros  and  a circle of I Ching hexagrams

As a design metaphor, the exploration can be taken further using the 64-edged drilled truncated cube (indicated above), combining it with a complete rotation of the traditional Shao Yong circle of 64 hexagrams -- each circle being composed of 8 "houses" of 8 hexagrams. The inner trigram of each hexagram is common within each such house. The metaphor follows from the question of "what flows" cognitively within the framework of a hypothetical "cognitive fusion reactor". Variants of this particular design metaphor are clearly possible. The house hexagrams could be distinctively coloured, for example. The circles, with others, could be oriented in terms of other features of of the polyhedron.

Drilled truncated cube with rotation of mutually orthogonal circles of 64 hexagrams
Single circle of hexagrams Double circle of hexagrams Triple circle of hexagrams
Drilled truncated cube with rotation of single circle of 64 hexagrams Drilled truncated cube with rotation of 2 mutually orthogonal circles of 64 hexagrams Drilled truncated cube with rotation of 3 mutually orthogonal circles of 64 hexagrams
X3D model; prepared using X3D-Edit and Stella Polyhedron Navigator

An incidental consequence of the software used to represent the circle in 3D (or a result of incompetence in its use) is that the orientation of the individual hexagrams (upper vs lower) is reversed when the circle is viewed from its other side (lower becomes upper). This has the merit of being a reminder of unresolved assumptions associated with directionality (Unquestioned Bias in Governance from Direction of Reading? Political implications of reading from left-to-right, right-to-left, or top-dow, 2016).

The following experimental animations use a design metaphor with the circle of hexagrams augmented by indication of many of the transformation pathways between them. Clearly variants with lower rates of rotation offer more reflective insights whereas at higher rates this is reminiscent of Tesla's rotation of a magnetic field (through which electricity is generated in dynamos).

Drilled truncated cube with animations of circles of hexagrams (augmented by transformation pathways)
Fast rotation of single circle Circles associated with faces and verticals Circles on faces of polyhedron
Drilled truncated cube with animation of single circle of hexagrams augmented by transformation pathways Drilled truncated cube with animation of multiples circle of hexagrams augmented by transformation pathways Drilled truncated cube with 6 circles of hexagrams associated with faces
  video; X3D  

Supercomputers, hypercomputing and superquestions?

Human brain as a "supercomputer"? It has long been remarked that the brain is an extremely sophisticated device by the standards of computing. The most accurate simulation of the human brain ever carried out took one of the world's largest supercomputers 40 minutes to calculate a single second's worth of activity (Supercomputer models one second of human brain activity, The Telegraph, 19 September 2019).

It is a matter of continuing debate as to whether the brain is in fact a computer:

By contrast the remarkable advances in the technology of supercomputing are better recognized in terms of speed, parallel processing (multitasking) and error avoidance. The criteria shift with respect to applications and intelligence. It is only recently that artificial intelligence has been developed to the point of being able to compete successfully with humans in classic games such as chess, go and poker. So-called neural learning promises many other competitive advantages.

Ronald W. Dworkin asks: What should worry us most about artificial intelligence: losing our jobs to cheaper labor or losing our lives to killer robots? He argues the real threat may lie in yet another danger: losing our minds (Artificial Intelligence: What's to Fear? The American Interest, 8 October 2018). Related concerns have been expressed (Michael Klenk, Are We Being Manipulated By Artificially Intelligent Software Agents? 3 Quarks Daily, 9 September 2019; Microsoft's Bill Gates insists AI is a threat, BBC News, 29 January 2015; Elon Musk: artificial intelligence is our biggest existential threat, The Guardian, 30 October 2015).

Trivial applications of supercomputers? It is however curious to note the relatively trivial applications for which supercomputers are so enthusiastically developed. Few are of immediate significance to human well-being; many will result in unemployment -- although dubious claims to the contrary are made (Earl C. Joseph, et al, Real-World Examples of Supercomputers Used For Economic and Societal Benefits: a prelude to what the exascale era can provide, International Data Corporation, 2014). High levels of techno-optimism are evident in events such as Modeling the World's Systems (MWS 2019) with their focus on the science, technology and applications of modeling and managing complicated, interacting systems, at all scales (from molecular to global processes) and in multiple domains. Unfortunately these tend to avoid the challenge of "wicked problems" and the remedial capacity to address them.

Priority has clearly been given to adaptation of supercomputers to the challenges of cyberwarfare. Much more controversial are the applications to surveillance -- soon to become universal. Adaptation to profiling for marketing purposes and the systematic manipulation of collective opinion are similarly controversial.

It could be said that the development of exascale supercomputers epitomizes the need to go faster and faster in order to sustain a process dependent on evermore busyness. This is consistent with the dependence on economic growth at all costs -- challenged most recently in a declaration to the UN General Assembly as being a "fairy tale" (David Brancaccio and Daniel Shin, Sustainable growth and whether it's realistic or a "fairy tale", Marketplace, 24 September 2019).

It is vigorously argued that supercomputers are vital to climate and weather modelling (Sabine Hossenfelder, Is Climate Change Inconvenient or Existential? Only Supercomputers Can Do the Math, The New York Times, 12 June 2019). Whilst this is indeed the case, the emerging climate crisis frames a counter-argument in that their capacity to engender models of decision-making to enable appropriate strategic responses is seemingly virtually zero. This is further highlighted by the highly divisive politics of societies upheld as the epitome of democracy -- dynamics described as "toxic". No effort is made to apply supercomputers to develop more fruitful social models of greater credibility.

The World Economic Forum frames the potential proactively:

Supercomputers have the potential to turn us into superhumans. Our potential and power increase in lockstep with the tools we have to serve us. The World Economic Forum's Global Future Council on the Future of Computing is aiming to shape the direction of that power. Our goal is to define a positive, inclusive, and human-centric future of supercomputing. (Georgia Frances King, Supercomputing could solve the world's problems, and create many more, 20 February 2019)

In metaphorical terms it could be said that supercomputing has proven to be of little significance to modelling the "weather" and "climate" of societies and smaller groups -- and indicating viable means for transcending binary limitations and other challenges of social "turbulence" and "overheating" (Climate of Change Misrepresented as Climate Change: insights from metaphorical confusion, 2008; Weather Metaphors as Whether Metaphors Transcending solar illusion via a Galilean-style cognitive revolution? 2005). Of relevance to this argument are the design constraints and ambitions of three contrasting exascale supercomputer projects seeking funding from the European Union -- with limited focus on psychosocial challenges:

It could also be vigorously argued that supercomputers are vital to astrophysics and astronautics -- to the extent that these are framed as relevant to the challenges of life on Earth, however questionably (Challenges More Difficult for Science than Going to Mars -- or exploring the origins of the Universe or of Life on Earth, 2004). In metaphorical terms again, missing is their relevance to "noonautics" (Towards an Astrophysics of the Knowledge Universe -- from astronautics to noonautics? 2006). Exceptions include:

There has been notable cultivation of the metaphor of a "global brain", a neuroscience-inspired and futurological vision of the planetary information and communications technology network that interconnects all humans and their technological artifacts. Curiously, despite this, there is no attention to what might constitute the "corpus callosum" to reconcile the preoccupations of the disconnected "hemispheres" of global society (Corpus Callosum of the Global Brain? Locating the integrative function within the world wide web, 2014).

Abacus, suanpan and soroban: It is appropriate to this argument to note the importance of the abacus as a calculating tool that has been in use in the ancient Near East, Europe, China, and Russia -- and still remains in common use in some countries. Mention is now made of use in ancient Aztec culture of a form of abacus called a nepohualtzintzin. The suanpan is a form of abacus developed in China; the soroban is an abacus developed in Japan, derived from the Chinese suanpan. China and Japan continue to use their variants in education -- despite the availability of electronic calculators.

Of relevance to the argument developed here, an imaginary abacus can also be used, as mentioned by Wikipedia:

By learning how to calculate with abacus, one can improve one's mental calculation which becomes faster and more accurate in doing large number calculations. Abacus-based mental calculation (AMC) was derived from the abacus which means doing calculation, including addition, subtraction, multiplication, and division, in mind with an imagined abacus. It is a high-level cognitive skill that run through calculations with an effective algorithm. People doing long-term AMC training show higher numerical memory capacity and have more effectively connected neural pathways. They are able to retrieve memory to deal with complex processes to calculate. The processing of AMC involves both the visuospatial and visuomotor processing which generate the visual abacus and perform the movement of the imaginary bead. Since the only thing needed to be remembered is the finial position of beads, it takes less memory and less computation time.

Glass Bead Game: Of potentially greater inspiration is the fictional study by the Nobel Laureate Hermann Hesse (The Glass Bead Game, 1943) -- which continues to evoke emulations and commentary. As described by Rainer E. Zimmermann (The Modeling of Nature as a Glass Bead Game), Hesse introduces an essentially cosmic game within a symbolical Universe which recently has been compared with a "neuronal network of the cosmic mind". In his game all those games which are known to us today appear to be summarized. Hence, by closer inspection, according to Zimmermann, the glass bead game shows up as a meta-game, a proto-game, and as a playful paradigm of playing, at the same time. As its rules it encompasses all what characterizes the reflexive activity of humans within the worldly environ-ment. It thus aims towards all those fields of the sciences and arts which are a-vailable as the inventory of human orientation.

Hesse describes the rules as:

These rules, the sign language and grammar of the game, represent a kind of highly developed secret language, in which several sciences and arts, in particular mathematics and music (or the science of music) participate which are able to express the contents and results of practically all of the sciences and relate them to each other.

This systematic form of approach to a universal game, conceptualized in a global manner, implies an important methodological consequence. Hesse continues therefore:

The glass bead game is thus a game with all the contents and the va-lues of our culture, it plays with them, as in the heyday of the arts a painter may have played with the colours on his palette. What humankind has produced in its creative epochs in terms of knowledge, noble thoughts, and works of art, what the successive epochs of learnt reflexion have conceptualized and claimed as intellectual property, all of this extraordinary material of intellectual values is being played by the glass bead player like an organ is played by the organist, and this organ is of a hardly graspable perfection, its manuals and pedals are scanning the whole spiritual cosmos, its registers are almost uncountable, theoretically, the complete intellectual contents of the world could be reproduced by playing.

The degree of association with philosophy recalls the early enthusiasm for Rithmomachy -- a board game previously rivalling chess -- otherwise known as the Philosophers' Game.

Enhancing creative capacity and imagination: Whereas the abacus can indeed be understood as enhancing cognitive capacity, the question is what applications of computers -- and of supercomputers -- can be recognized as doing so. It could perhaps be argued that it is the application of ever greater computer power to social media dynamics and online gaming which is of greater relevance to the exploration and enhancement of psychosocial processes.

The specific focus of the Triple Helix model of innovation seeks to foster economic and social development through enhancing the interactions between academia, industry and governments. It could be asked to what degree such innovation is enabled to a higher degree through the use of supercomputers. It can be argued that virtual reality, enabled by supercomputers, is of specific relevance to enhancement of creativity, although there is little attention to this possibility (Psychosocial Learnings from the Spiral Form of Hurricanes, 2017; Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity, 2017).

Despite the remarkable possibilities of supercomputers, it is clear that they will enngender a new array of problems with which human creativity will be faced (Peter Dockrill, Computers Are Making Huge Mistakes Because They Can't Understand Chaos, Scientists Warn, Science Alert, 27 September 2019).

Stuart Russell How to Stop Superhuman A.I. Before It Stops Us, The New York Times, 8 October 2019) The answer is to design artificial intelligence that's beneficial, not just smart. ***

Metacomputing and hypercomputing? As implied by the term, metacomputing encompasses forms of research on the development and applications of computing, including socio-cognitive engineering (despite its questionable nature). From a systemic and philosophical perspective, it frames a focus on the limits of the transformation of human knowledge and individual thinking to the form of computer programs.

Of far greater relevance to this argument is what has been associated with hypercomputing, notably in the light of the speculations of Alan Turing (as noted above). This has been discussed separately (Imagining Order as Hypercomputing: operating an information engine through meta-analogy, 2014) under the following headings:

Paradoxical locus of nonlocal oracular hypercomputing
Engendering oracular hypercomputing through investing significance
Necessary impossibility of explaining oracular hypercomputing
Imagining as key to oracular hypercomputing
Hypercomputing as imaginative enactment
Nescience as a mode of hypercomputing?
Imagining order and pattern "re-cognition"
Exercise in imagining hypercomputing via hexagram patterning
Hypercomputer operation clarified through metaphors of engine design
Recognizing a confluence of imaginative possibilities

The question of interest is the manner in which higher orders of "computation" can facilitate, enhance and sustain the kinds of imagination of relevance to the extreme challenges with which society is purportedly faced. Arguably the focus of exascale computation to date is on relatively simple (if not trivial) problems which demand myriads of simple calculations -- "quantitative" problems rather than the "qualitative" problems more intimately related to individual and collective learning. Supercomputers are ranked according to such quantitative measures. The TOP500 project ranks and details the 500 most powerful non-distributed computer systems in the world.

There is little indication that supercomputers are especially skilled in detection of pattern correspondences of significance to imaginative innovation and creativity. Or is success in game-playing to be understood as a valid counter-indication, despite lack of other evidence?

Superquestions -- beyond the trivial? Provocatively it can therefore be asked how "superquestions" might be identified and configured in a time of crisis -- questions meriting a higher order of investment, as argued separately (Coping Capacity of Governance as Dangerously Questionable: recognizing assumptions and unasked questions when facing crisis, 2019; Superquestions for Supercomputers: avoiding terra flops from misguided dependence on teraflops? 2010; ). The latter included the following sections:

Precedents and complementary initiatives
Cognitive implications?
Higher order questions: quality vs. quantity?
Potentially relevant superquestions
Clustering superquestions in terms of WH-questions
Refining and reconfiguring the set of superquestions
Possibilities of artificial intelligence

Whilst search engines are increasingly remarkable in responding to specific queries, curiously there is no requirement for them to be able to detect the unasked questions and assumptions implied by such queries -- the "unsaid" (Global Strategic Implications of the "Unsaid", 2003; Question Avoidance, Evasion, Aversion and Phobia, 2006).

Is it hypercomputation, using supercomputers or otherwise, which could enable "deadly questions" to be identified and highlighted (as noted above)? Is it precisely such questions which engender and enable fruitful change and innovation -- rather than reinforcing conventional patterns of thinking? Noteworthy in this regard is the subsequent sidelining of population growth as a problem driver featured in the early use of computers for global modelling -- resulting in the report on The Limits to Growth (1972). Is it expected that artificial intelligence can formulate questions in the light of contradictions and dilemmas? Potentially understood as wicked problems, a notable example is the complex of resource constraints, corruption, population growth, arms race, inequality, and criminalization of remedial options (migration, contraception, abortion, or suicide). Is there a need for hypercomputation to formulate adequately "wicked" strategies (Encycling Problematic Wickedness for Potential Humanity, 2014; Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016)?

Framing the space for conscious creativity?

Polyhedral metaphor: The argument above explored the fundamental role of the cube (and more complex polyhedra) in framing a creative cognitive space -- inspired by the design constraints of supercomputer development. The question is how to enable a more conscious focus on the creative phases prior to integrative comprehension. Simple experiments with virtual reality applications are suggestive of this possibility. As illustrated below, the focus is on the cuboctahedron as a complexification of the cube. That polyhedron was a particular focus of the extensive work of Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975/1979), notably associated with his development of geodesic domes.

Arguably society has yet to develop approaches to computation to engender psychosocial analogues to geodesic domes, as discussed separately (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009; Interweaving Thematic Threads and Learning Pathways: noonautics, magic carpets and wizdomes, 2010). As illustrated by the following animations, the cognitive struggle is to enable disparate simpler configurations to cohere -- into the form of a cuboctahedron in this case (left-hand image).

The various animations offer cases in which the challenge is the configuration of the edges separately, the 6 square patterns, the 8 triangular patterns, or the 4 hexagonal circuits. The challenge is simplified by assuming that each colour rotates around one of three axes -- or is stationary in the case of a fourth. The challenge to coordination at the present time is obvious where the colours are indicative of political parties, the edges are indicative of their preferred projects, and the spheres are indicative of distinctive goals. The animations are helpful as indicating the strategic confusion (or "mess") of the times, whether for the individual or the collective.

Proto-coherence represented by animation of modes of convergence on a cuboctahedral configuration
Cuboctahedron Animation of edge movement Animation of squares of edges Animation of triangles of edges Animation of circuits of edges
Cuboctahedron Animation of edge movement of a cuboctahedron Animation of squares of edges  of a cuboctahedron Animation of triangles of edges  of a cuboctahedron Animation of circuits of edges  of a cuboctahedron

Engaging with complexity: The limited number of elements in the cuboctahedron -- already greater than the simplicity of the cube -- offers a reminder that the complexity of the psychosocial system calls for a capacity to engage with a far greater number of elements and their dynamics. The complexification of the 8-fold cube of the original BaGua pattern of trigrams into the 64-fold pattern of the I Ching hexagrams is a reminder of the dynamics between the elements in each case.

As noted above, a polyhedron appropriate to mapping the 64 elements is the drilled truncated cube -- a complexification of the cube and the cuboctahedron. The following animations are indicative of the cognitive challenge in such a case.

Alternative views of selected cycles of movement of parallels along edges of the drilled truncated cube
Video version (.mp4); virtual reality (.x3d; .wrl)
cycles of movement of parallels along edges of the drilled truncated cube cycles of movement of parallels along edges of the drilled truncated cube
Access X3D variant  


Framed inner cube movements (#1/2)
Framed outer cube movements (#3/4)
diagonl movements
cycles of movement of parallels along edges of the drilled truncated cube cycles of movement of parallels along edges of the drilled truncated cube cycles of movement of parallels along edges of the drilled truncated cube
Access X3D variant Access X3D variant Access X3D variant

Implying hypercomputation through mandalas and circlets of beads? Arguably the quantitative focus of the abacus has been traditionally complemented by the qualitative focus of the mandala and circlets of beads. The question is how the development of information technology might engender equivalents enabling greater psychosocial integration (Designing Cultural Rosaries and Meaning Malas to Sustain Associations within the Pattern that Connects, 2000; Concordian Mandala as a Symbolic Nexus, 2016; Visualization in 3D of Dynamics of Toroidal Helical Coils: in quest of optimum designs for a Concordian Mandala, 2016).

Arguably mandalas and bead circlets are "two-dimensional" devices with "hyper-dimensional" implications -- with mandalas effectively constituting a cognitive analogue to the forms of dimensional compactification so fundamental to physics. Relevant arguments are evident in the work of Douglas Hofstadter (Gödel, Escher, Bach: an Eternal Golden Braid. 1979; I Am a Strange Loop, 2007).

It could also be argued that the peculiar attraction of precious stones and their faceting is indicative of intuitive comprehension of a higher order (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). Diamond mind and diamond vehicle are common themes of Buddhism, for example (Rob Nairn. Diamond Mind: psychology of meditation, 1998).

The question calling for continuing reflection is what "circulates" in the process of "individual hypercomputation". Understood metaphorically as the "circulation of the light", the formation of the diamond body has been addressed by Carl Jung in his commentary on the Taoist classic The Secret of the Golden Flower (V. Walter Odajnyk, Gathering the Light: a Jungian view of meditation, 2011). Unfortunately the metaphor overemphasizes the sense of a closed circular circuit, thereby failing to encompass the varied dynamics so characteristic of individual and collective reflection -- as discussed in the light of weather metaphors, as characteristic of coherent chaos. The metaphor can be usefully considered more generally (Circulation of the Light: essential metaphor of global sustainability? 2010):

Circulation of the light, its inhibition and its surrogates
Cognitive and strategic implications
Hidden dynamics of the "circulation of the light"
Experiential implication in the "circulation of the light"
Sustaining the circulation dynamic
Circulation around what?
Circulation engendering circulation?
Symbolic indications
Completing the circuit

"Brain box"? There is considerable irony to the fact that the term "think tank" originated from reference to "brain box" -- suggesting an intuitive understanding of a cognitive cube of some kind (Thinking "inside-the-box" as reinforced by think tanks, 2019). What are indeed the processes within a "think tank" -- whether in the case of an individual or a collective, with either to be understood as a form of cloud chamber for concepts, or an incubator (Varieties of cognitive "tank" beyond the conventional "think tank", 2019).

The concern above is with how that space might be configured and framed -- bearing in mind the design constraints for "cognitive fusion" (as suggested above), with a paradoxical requirement to avoid inappropriate closure. The space so framed could be understood as one which facilitates "conjugation" as more generally and etymologically understood. In this sense it could be understood as engendering cognitive characteristics which could be explored as the emergent modality of what might be described as homo conjugens (Authentic Grokking: emergence of Homo conjugens, 2003). This emergent species could be compared with insights into the nature of homo udulans, as the theme of a penultimate chapter of the very detailed study by Daniel Dervin (Creativity and Culture: a psychoanalytic study of the creative process in the arts, sciences, and culture, 1990) and discussed separately (Emergence of Homo undulans -- through a "grokking" dynamic? 2013).

Given the emphasis on recognizable difference in preferences for order and organization, in contrast to "subunderstandings" of unity and integration, of particular interest is any creative focus through a difference engine on superordinate patterns, however these might be understood through complementary metaphors (Using Disagreements for Superordinate Frame Configuration, 1992). How supercomputing, meta-computing or hyper-computing might enable this is clearly a matter for speculation and for the future. Is this essentially a question of intuition, as explored separately (Patterning Intuition with the Fifth Discipline, 2019)?

Recent research on the cognitive significance of the body in movement merits association with the nature of the dynamics within any extension of the "brain box" space (Maxine Sheets-Johnstone. The Primacy of Movement, 2011; George Lakoff and Mark Johnson, Philosophy in the Flesh: the embodied mind and its challenges to western thought, 1999; Mark Johnson, The Body in the Mind: the bodily basis of meaning, imagination, and reason, 1987). There is a sense in which the dynamics within that cognitive space might be most fruitfully compared to a dance -- if not a "sword dance" across categories.

Of notable relevance to such dynamics is the manner in which any alphabet, especially one deemed sacred, may variously establish relationships between categories. Most significant in this respect is a remarkable experiment in visualization dynamics in 3D strongly constrained by copyright -- and therefore unmentionable -- reminiscent of the obsessive preoccupation of the traditional Golem, so fruitfully embodied in the tale of The Lord of the Rings.

With respect to reference to "supercomputation", there is clearly a case for emphasizing the capacity of the individual human being (to say nothing of the collective) to engage in cognitive modalities of equivalent or greater significance. This suggests recognition of individual (or collective) identity as a "supercomputer" in its own right -- DIY supercomputing, distant only in terms of focusing capacity and discipline? The widespread engagement with music is itself suggestive of the accessibility of a modality which is elusively related to the challenges of navigating the pattern that connects.

Dancing cognitively inside the box -- and beyond

Within the cubic framework: Given the cubic framework of this exploration, the point was made above that the six sides of the cube offered six windows through which the wider world may be seen. Using the Platonic polyhedra, the relationship between that 6-foldness, as with the 6-foldness of the I Ching hexagrams, can be explored with respect to the 8-foldness of the cube vertices highlighted above. Each of the polyhedra can be undestood as a "cognitive box" by which one is "boxed in".

Platonic polyhedra as "cognitive boxes"
  tetrahedron octahedron cube dodecahedron icosahedron
edges 6 12 12 30 30
sides/faces 4 8 6 12 20
vertices 4 6 8 20 12
axes   6 edged-centered 6 edged-centered 6 face-centered 6 vertex
great circles          

The cube is the easiest to comprehend in this respect. As mapping surfaces any of the 6 windows could be transparent or opaque -- or possibly distinctively tinted. In the hexagram encoding, transparency could (for example) be denoted by a broken line and opacity by an unbroken line. The four vertices of any window could then be understood as distinctive perspectives. Opacity could then signify a lack of connectivity between those 4 perspectives -- whether as beliefs, faiths, disciplines, models, ideologies, or the like. Transparency of the window could then imply fruitful 4-fold connectivity between them -- enabling a view through the window.

Clearly any combination of transparent and opaque windows is then possible for the 6 windows -- possibly to be understood as "glass ceilings" or "glass doorways" of some kind. The glass cube then offers conditions ranging from complete non-transparency to total transparency. Which windows are imagined to be up or down, front or back, or right or left, is necessarily another matter -- depending on one's orientation. The transparency can be imagined as changing over a period of time in response to various conditions -- although the condtions could be usefully understood as cognitive rather than externl.

The octahedron (as dual of the cube) can be used otherwise -- with the vertices holding the 6-fold distinction between openness and closedness, and the sides a 3-fold pattern of connectivity between perspectives. In the case of the tetrahedron, it is the edges which then hold the conditions of openness and closedness. In the case of the dodecahedron, it is 6 face-centered axes which could hold those distinctions -- also held by the 6-vertex axes of the icosahedron.

The following are exercises in imagining the cognitive "dance". Rather than a tetrahedron, the equivalent number of edges of the Star of David can be used for a 2-dimensional animation. Of interest is the convention regarding allocation of trigram lines to triangle positions and whether alternative allocations are anyway of significance in their own right. The experimental animation on the right has the signs of the zodiac "dancing" within the constraints of a 12-sided "cage" in 2D, suggestive of the connectivity of such signs if represented in 3D within a dodecahedron (or in relation to the edges of a cube). The exercise derives from arguments of Arthur Young (Geometry of Meaning, 1976) regarding an experiential system of interwoven creative processes -- embodied in alchemical processes encoded by the forms of zodiacal signs, as discussed separately (Geometry of meaning: an alchemical Rosetta Stone? 2013; Representation of Creative Processes through Dynamics in Three Dimensions, 2014).

Representation of I Ching hexagram line codes Use of 2D 12-fold pattern

Mapped onto edges of Star of David
Reproduced from Mapping of I Ching hexagram coding onto Star of David (2008)

Use of signs of zodiac to
suggest patterns of creative conectivity through "dancing" within a dodecahedron
64 I Ching hexagrams
configured as double triangles
(as in animation on the right)
Animation of cycle of 64 hexagrams
suggestive of dynamics
of triadic bonding
Mapping of I Ching hexagrams onto Star of David Zodiac of alchemical processes with Geometry of Meaning
The following animations are exercises in representing a hexagram by the sides of a cube, one line per side. The design metaphor chosen was to use transparency and opacity to indicate unbroken or broken lines. This offers two alternatives represented below. An unbroken line could indicate opacity with a broken line indicating transparency (as on the right, as suggested by the openning). This would not however emphasize the "openness" associated with the first hexagram in red. In the reverse case an unbroken line could indicate transparency with a broken line indicating opacity (as on the left).
Experimental animations of one group of 8 hexagrams (a "house") using sides of a cube
(alternative conventions with lower trigram unchanging -- 3 sides at bottom left of cube)
Convention: hexagram unbroken lines "transparent"
(therefore lower left sides always transparent)
Convention: hexagram broken lines "transparent"
(therefore lower left sides always opaque)
Animations of one group of 8 hexagrams using sides of a cube Animations of one group of 8 hexagrams using sides of a cube

The experiment is indicative but relatively unsuccessful. Unlike the Star of David animations in 2D of all 64 hexagrams (above), it is not practical to cycle through more than a small group. Hence the use of the 8 in a "house" with one invariant trigram. More problematic is the default shading in the 3D rendering which effectively undermines the distinction between opaque and transparent (unless using a 3D viewer), since the opaque is rendered dark grey according to the orientation of the cube to the light.

It has been noted that in the much-cited poem Wallace Stevens (Thirteen Ways of Looking at a Blackbird, 1917), the description of a Turdus in a snowy autumn landscape alludes to the Cubist painting tradition of observing subjects simultaneously from numerous viewpoints to present a novel perspective. Of some relevance is the sense of a thirteenth perspective beyond the 12-fold pattern of the sube or the dodecahedron, as discussed separately (Ways of looking at ways of looking -- in a period of invasive surveillance, 2014; Post-modern challenge to simplistic binary framing of the other, 2014)

Beyond the cubic framework:

Maps indicating relationships between Platonic and Archimedean polyhedra
Polyhedra flow chart Distinctive relationships pathways between spherically symmetrical polyhedra
Polyhedra flow chart Route maps of psychosocial life suggested bysymmetrical  polyhedra
Reproduced from Reddit (2018) From Pathway "route maps" of potential psychosocial transformation? (2015)


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