Dynamics of systemic connectivity as a challenge to invariance

Cognitive Embodiment of Patterns of Governance of Higher Order (Part #11)

Towards attribution of cognitive significance to complex patterns: It is one thing to elaborate complex symmetrical structures as an indication of potentially significant coherence. It is quite another matter to articulate a large array of distinctions and associate them meaningfully with features of that structure as a mnemonic aid. In this respect the Chinese binary coding system which inspired Gottfried Leibnitz merits attention (Mary von Aue, How the 'I Ching' Inspired His Binary System, Inverse, 2 July 2018). It was presented to him 1701 in the form of the Shao Yong circle of 64 hexagrams -- of which an adaptation is presented below right, and discussed separately in the light of its recognized capacity to order the 64 genetic codons (Changing Patterns using Transformation Pathways, 2015).

Of interest in relation to the drilled truncated cube is the manner in which the 64 hexagrams are traditionally clustered in 8 "houses" on the basis of one of the component trigrams (Organization of I Ching hexagrams in terms of traditional "houses", 1995). The clustering by "house" is evident from the two "houses" in each quarter of the circle (below right). The basic 8 trigrams of the BaGua pattern have been associated with a cubic configuration, as illustrated above (Z. D. Sung, The Symbols of Yi King or the Symbols of the Chinese Logic of Changes, 1934, p. 12). In the image below left the manner in which the lines of those distinctive trigrams are associated with the outer corner features of the drilled truncated cube are shown.

The elements of the second set of 8 distinctive trigrams can be associated below with inner features of each corner. These 8 inner trigrams are then presented dynamically in the central animation below in relation to the 8 static corner elements indicative of the distinctive "house" clusters. Fundamental to the association of significance with the hexagrams is whether they are to be read from top down or from bottom up -- an issue of inner-versus-outer highlighted by the dynamics of the hypercube presented above (Unquestioned Bias in Governance from Direction of Reading? 2016). The animation presents the dynamics of each of the 8 "houses" in the same sequence, as indicated in the circular representation. The dynamics of the structure as a whole thus cycles through all 64 hexagrams.

Association of traditional circular hexagram encoding with drilled truncated cube in 3D
Correspondence of trigram coding to 8 corners of drilled truncated cube	Animation of upper and lower trigrams represented on drilled truncated cube	Circle of hexagrams surrounded by a circle of codons

	Interactive 3D variant

Although somewhat simplistic, the use of 3D web technology (X3DOM) in the experimental development of the animation above does demonstrate that it is possible to control separately all the individual elements of the polyhedron as a mnemonic aid. With greater skill, such an animation could be further developed to hold a greater range of significance -- adjusting timing, colours, sound, viewpoint perspectives, and any labelling -- according to interactive and aesthetic preferences. The pattern of attribution of significance could be variously adjusted experimentally. To the extent that the structure can be considered to be a simple model of a "global brain", of interest is the manner in which the polyhedron is segmented into "lobes" -- especially evident when rotated in 3D.

Significance is traditionally attributed to the 64 hexagrams through metaphor, allowing for many adaptations to sustainable dialogue, vision, conferencing, policy, network, community and lifestyle (Transformation Metaphors derived experimentally from the Chinese Book of Changes (I Ching), 1997).

Coherence of sustainable development? An earlier exercise with respect to the SDGs can be seen as anticipating the following argument (Interrelationships between 64 Complementary Approaches to Sustainable Development , 2002). The challenge of systemic coordination and coherence of the 16 SDG goals is indicated by the 17th: Strengthen the means of implementation and revitalize the global partnership for sustainable development . As explained, given its "keystone" role, this goal has been omitted from the 16 mappings above, as with the omission of the tetrahedron and the truncated tetrahedron in the polyhedral configurations.

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

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

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

Fundamental to Tesla's innovative discoveries was the rotation of a magnetic field, potentially suggestive of psychosocial analogues (Potential implications of alternation and rotation in psychosocial fields, 2014). This is especially evident in the operation of a commutator in motors and dynamos. in addition to concern with with positive and negative, the insights into operation of electrical systems merit consideration for the operation of psychosocial systems (Electrical Systems as a Guiding Metaphor for Stages of Group Dialogue, 2001). It is intriguing that mathematics uses commutator to give an indication of the extent to which a binary operation fails to be commutative. The challenge of global governance could well be seen metaphorically as one of "commutator design" -- by whatever metaphors this is enabled.

Also of mnemonic value in the case of the tetrahedral animation (left above) is the alternation between a degree of "top-down" and "bottom-up" -- as might be a democratic ideal. The tetrahedron and its dual may be understood as either counter-rotating, or co-rotating then potentially forming a stella octangula (or stellated octahedron). Symbolic significance attributed to this form is discussed separately with other animations, as a 3D version of the Star of David (Framing Global Transformation through the Polyhedral Merkabah, 2017; Polyhedral Merkaba Symbol in Virtual Reality: experimental interactive 3D display).

As with a commutator, the manner in which the vertices of the rotating polyhedra successively contact the surrounding structure is suggestive of how processes of alternation in governance could be understood as variously triggered. The more complex animations below indicate how the four corner parts of the surrounding structure are successively coloured distinctively following such contact, thereby highlighting particular "information pathways" through the structure of categories it might map (as noted above).

Screenshots of animations of alternative views of tetrahedra rotating within drilled truncated cube (rotation changes colours of external parts in a cycle)
Co-rotation of central tetrahedra in the form of a stella octangula (stellated octahedron)		Counter-rotation of central tetrahedra

See access to interactive 3D animations below; x3d variant

Technically the animations above, as extracts from a video, are only indicative and are variously unsatisfactory. In particular the phasing of the colour changes is not well-coordinated with the rotation of the central polyhedron -- given that the video and 3D versions emphasize a smooth rotation rather than a step shift in position. Slower speeds may well be far more meaningfully appealing. However, with greater skill, the structure enables further modification to the changing colours, possibly extended from four to eight colours with further segmentation of the structure accordingly. This would be appropriately consistent with preferences for 8-fold organization (Framing Cognitive Space for Higher Order Coherence, 2019; Cubic organization ofÂ I ChingÂ trigrams -- an eightfold way, 2019).

Corresponding sound effects could be added, given the case for facilitating complex pattern comprehension through sonification. Such experiments are consistent with arguments made separately (Envisaging a Comprehensible Global Brain -- as a Playful Organ , 2019). Another intriguing possibility is to use 3D web technology to enable "rays of light" to pass between facets of a structure with rotating elements, as an indication of information pathways and connotations -- thereby potentially enhancing memorability in aesthetic terms. Visualization of the interplay of sets of voices in discourse (24-fold Pattern Implied by Dynamics of the Lauburu in 3D, 2016).

Usefully reframing the stasis implied by the "philosophers stone" is the forgotten chess-like game Rithmomachia (The Battle of Numbers), which combined the pleasures of gaming with mathematical study and moral education (Ann E. Moyer, The Philosophers' Game: rithmomachia in Medieval and Renaissance Europe, 2001). Complexifying the framework of a simple abacus, the drilled truncated cube evokes the possibility of a framework in 3D for such a game (as with three-dimensional chess) -- with the aesthetic and cognitive connotations evoked by Hermann Hesse (The Glass Bead Game, 1943).

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

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

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

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

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

Cognitive dynamics through cycles of flow: Symmetry within the drilled truncated cube suggests its use in framing distinctive toroidal rings indicative of contrasting modes and orientations of flow. These can be understood as a a contrasting expression of dimensions otherwise taking the form of axes. Thus that on the left can be understood as relating the extremes of horizontal axes and diagonal axes. That in the centre is indicative of the manner in which the central cube is framed by 6 distinctive flows.

The animation on the right is an experiment on the assumption that the Mereon toroidal knot, with which fundament significance has been associated, might indicate a mode of glow through the cubic form in some manner. The knot features in earlier commentary (Cyclic Representation of Coherence as Knots and Otherwise , 2022; Louis Kauffman, Pattern, Sign and Space: Mereon Thoughts. 2003; Louis H Kauffman, et al, The Mereon Matrix: everything connected through (k)nothing, 2018).

The toroidal presentation evokes consideration of what could be variously understood as "flowing" in a cognitive cycle (Circulation of the Light: essential metaphor of global sustainability? 2010). The "cognitive locus" can then be represented by a sphere (as on the left) or as a spherical polyhedron (as with use of the truncated icosahedron in the other animations below). This follows from the experiment above in which all the Archimedean polyhedra flow through a torus -- thereby suggesting the variety of perspectives or cognitive phases in play around an archetypal roundtable. Use is made below of the truncated icosahedron because of its familiarity as a soccer ball with all that its dynamics may imply in a game.

Toroidal flow is potentially of the greatest importance with respect to future global energy requirements -- framed by experimental development of toroidal nuclear fusion reactors, most notably that of ITER. To the extent that these invite technomimicry, the animation on the left is potentially suggestive of the operation of a psycho-social analogue -- an Imaginal Transformation of Energy Resourcing (ITER-8) (Enactivating a Cognitive Fusion Reactor, 2006).

Animations indicative of toroidal circuits of interest in relation to the drilled truncated cube (using the truncated icosahedron as an indicator of one dynamic)
3 mutually orthogonal toroidal rings	6 toroidal rings framing the central cube	Interweaving of Mereon toroidal knot

Interactive 3D variant	Interactive 3D variant	Interactive 3D variant

Cognitive design challenges beyond 64? The transformations from patterns of 4 to 64 are potentially taken further in numerical terms by the animations above. In the case of the 4-sided tetrahedron (with a tetrahedron as its own dual), the 8 sides (or vertices) together extend the 64 edges of the drilled truncated cube to 72 . This anticipates the traditional emphasis on the dynamics of that pattern, as discussed below in various Western traditions. Given the reference above to Jacob's Ladder, its degrees were to the number of 72 (according to the Zohar). The traditional central importance of the 8-edged square pyramid -- architecturally -- offers a similar extension to 72, although questionable in terms of symmetrical elegance.

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

A design assumption to be explored is how the inner portion of the drilled truncated cube acts as a form of "housing" for a cognitive nexus -- as a metaphorical commutator. Metaphorical alternation could be considered a potential requisite of any function as a cognitive Rosetta Stone -- recognized by physicists to a degree in terms of wave-particle duality and the Uncertainity Principle. The insight is usefully emphasized by the Sanskrit adage Neti Neti (Not this; Not that), suggestive of a fundamental cognitive dynamic and driver. Somewhat ironically this is appropriately challenged by the the insight of Alfred Koryzbski regarding the problematic consequences of assertions of what "is" (Science and Sanity: an introduction to Non-Aristotelian Systems and General Semantics, 1994).

From a design perspective there is a challenging loss of pattern continuity by switching from the focus on 64 edges to dependence on 8 sides of the stella octangula or the tetrahedron and its dual. In terms of cognitive "commutator design", this is only partially remedied by embedding the stella octangula in another cube (again with 8 vertices) nested at an angle within the inner cube (screenshots below, with access to 3D animations). It is appropriate to emphasize that any such fundamental cognitive nexus is necessarily associated with paradox, as explored through the illusory ambiguity of the Necker cube discussed by Steven M. Rosen (Splitting the Atom: the paradox of proprioceptive dialogue, May 2022; The Strange Nature of Quantum Perception: to see a photon, one must be a photon, Journal of Mind and Behavior, 42, 2021, 3-4).

Indicative animations of "commutator designs" for global governance (preliminary screenshots)


Interactive 3D variant	Interactive 3D variant

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Dynamics of systemic connectivity as a challenge to invariance