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Constraints on recognition of subtler patterns of order indicated by the periodic table


Comparable Modalities of Aesthetics, Logic and Dialogue (Part #13)


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Periodicity of cognitive engagement? As noted above, another clue is offered by the pattern of order exhibited by the Periodic Table of Chemical Elements in which 14 is the maximum number of electrons that can fit in the f-sublevel of an atom (How many electrons can occupy the f orbitals at each energy level? 2018). The relevance to this argument derives in part from consideration of the possibility of a degree of periodicity in human engagement with different levels of knowledge -- or modalities of knowing (Periodic Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order, 2009;  Periodic Pattern of Human Life: the Periodic Table as a metaphor of lifelong learning,  2009).

The periodic table can be variously presented, whether relatively simply or to reveal otherwise hidden complexities using complex hypergraphs (Michelle Starr, Mathematicians Have Proposed a New Structure to The Periodic Table, Science Alert, 18 June 2019). This suggests simpler ways of developing the argument with the possibility of more complex refinements in the future. It is also noteworthy that there have been many attempts to render the periodic table comprehensible as a whole, given the subtle interrelationships it is possible to highlight (Alternative periodic tables, Wikipedia). It could be usefully assumed that the subtlety in the case of human cognition is potentially equivalent, if not greater,

With respect to any 14-fold pattern, this features in green in the tables below left as the f-sublevel (or subshell). It can be argued that human comprehension is most readily associated with the 2-fold pattern in red in the tables -- most obviously in the reliance on binary thinking under most circumstances, whether personal or global. This has been most succinctly argued by Edward de Bono (I Am Right, You Are Wrong: from this to the New Renaissance, 1968; Water Logic: the alternative to I Am Right You are Wrong, 1993). There is a considerable challenge in shifting to the 6-fold pattern in yellow in those same tables, which corresponds to Edward de Bono's other arguments (Six Thinking Hats: an essential approach to business management, 1985; Six Frames for Thinking about Information, 2003).

Presenting the 2-fold and 6-fold patterns at the centre of the concentric schematic on the right is then suggestive of a succession of cognitive challenges -- to which the 10-fold (in blue) and the 14-fold (in green) may then be added. Again it can be asked why the 14-fold is favoured in so many contexts, and how this relates to preference for the 10-fold. At a simpler level, the cognitive constraint at each stage is usefully argued by Ron Atkin with respect to a recognition of the coherence of a triangle of disparate elements using the mathematics of q-analysis (Multidimensional Man; can man live in 3-dimensional space?, 1981), as separately summarized (Comprehension: Social organization determined by incommunicability of insights).

The periodic table can be considered indicative of the relative occurrence of elements in nature, those indicated in green being appropriately named "rare earths". As a metaphor this suggests a way of thinking about 2-fold, 6-fold and 10-fold cognitive modalities -- relative to a rarer 14-fold modality. As implied by the periodic table, the 2-fold and 6-fold are readily recognizable as vital to the viability of organic life (at least at some level). Less evident is the importance of the 10-fold, although the principles articulated by that pattern have become fundamental to socio-economic conduct and organization -- as exemplified by the 10 Commandments. The question raised by the checklist of 14-fold examples above is whether the 14-fold pattern constitutes a more fundamental recognition of coherence -- potentially vital to global governance in a manner as yet to be clarified.

Potential correspondences between periodic tables
Periodic table of chemical elements
with indication of shell and subshell capacity
Cognitive modalities suggested by periodic table shells
with indication of transitions by polyhedral configurations
Periodic table of chemical elements Cognitive modalities suggested by periodic table shells with transitions enabled by polyhedral configurations
Periodic table shell and subshell capacity
Reproduced from Wikipedia Provisional in anticipation of animation enhancement

The concentric schematic invites interpretation through another metaphor, namely as a transmission system -- as with the gearbox of a vehicle. This would then frame the cognitive challenge of how to shift from gear to gear according to circumstances -- specifically how to "get" from a 6-fold modality to a 14-fold modality, for example, or from 14-fold to 6-fold. The argument is that the 14-fold offers greater coherence over time whereas the others are more responsive to the immediate dynamics of a situation. This clarifies the advantage of the binary thinking associated with the 2-fold, despite the associated oversimplification and loss of sensitivity.

Enabling symmetry? Of particular interest in the animation is the indication of polyhedra with characteristics matching particular subshells. The suggestion is that shifting cognitively and coherently between subshells is enabled by correspondences between symmetry elements in the associated polyhedra. The potential role of symmetry in this respect has been extensively referenced by B. Pavlovic and N. Trinajstic (On Symmetry and Asymmetry In Literature, Computers and Mathematics with Applications, 12 1986, 1â--2). The fundamental role of correspondences in the sciences and the arts has been clarified separately especially in relation to "monstrous" forms of symmetry (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007; Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

Of primary significance to such a succession of shifts are the transformations associated with what was named by Buckminster Fuller as the vector equilibrium -- allowing for so-called "jitterbug" transformations from the cuboctahedron, via the icosahedron to the octahedron. Such shifts have been indicated by white arrows in the animation (Vector Equilibrium and its Transformation Pathways, 1980; H. F. Verheyen, Computers and Mathematics with Applications, 17, 1989, 1-3).

Associated with this pattern is the dual of the cuboctahedron, namely the rhombic cuboctahedron (as discussed above) -- necessarily sharing 14 as a symmetry characteristic. Duality also features in the indicated relation between icosahedron and dodecahedron. The dodecahedron is positioned on the 10-element subshell, since its 10 primary axes of symmetry are those linking its 20 vertices.

Examples of jittterbug transformations
Single cuboctahedron to an octahedron and back again
Antiprism: Jitterbug Animation Antiprism: Jitterbug Animation
Reproduced from Antiprism: Jitterbug Animations, as developed by Adrian Rossiter
See also Buckminster Fuller's Jitterbug (YouTube)

Coherence of a viable system: A particular concern in any quest for coherence is that few, if any, of the checklists of 14-fold patterns address the issue of the systemic relations among the 14 items identified. This tends also to be characteristic of the 10-fold and 6-fold patterns. This is even in the case of the remarkable set of 14 process elements of a viable system model as indicated discussed below.   Curiously, although that study cites Stafford Beer, as the originator of such modelIing, it fails to note the role played in his understanding of a 30-fold pattern, as articulated in Beyond Dispute: the invention of team syntegrity (1994) -- one of his final studies.

In that study Beer uses the 30-edged icosahedron as a way of representing and holding the dynamics of a viable system -- effectively as a tensegrity. Arguably what is required is a 30-fold pattern of relationships as a context or container for the set of distinctive elements identified in the 14-fold pattern. It is in that respect that the relationship with the preoccupation of Buckminster Fuller merits attention, especially his focus on the cuboctahedron -- recognized by him as a vector equilibirum. In particular he focused on the transformational dynamic between the 14-faced cuboctahedron and the 30-edged icosahedron -- and the various intermediary structures .

Transformative dialogue through symmetry preservation: Framed in this way, a further clue is offered by the much-cited Conway polyhedron notation, as discussed separately (Encoding Coherent Topic Transformation in Global Dialogue: memorability of cognitive implication in symmetry-preserving operations on polyhedra, 2021). Presented schematically (as below), this indicates the distinctive symmetry preserving operations through which polyhedra may be transformed between one another (Hidetoshi Nonaka, Visualization of Conway Polyhedron Notation, World Academy of Science, Engineering and Technology, 50, 2009).

Conway relational chart
Showing 12 forms created by 3 operations on the cube
Conway relational chart for polyhedra
Tomruen at English Wikipedia, Public domain, via Wikimedia Commons

Missing from such a depiction is any sense of the "symmetry preserving cognitive operations" so fundamental to memorability and comprehension -- and the recognition of correspondences as discussed further below. Such depictions tend to omit any indication of the numeric characteristics of the polyhedra through which the symmetry is preserved and transformation is enabled -- potentially fundamental to Bach's 14 canons, for example. Some sense of this is evident in an alternative pattern highlighting this, reproduced from an earlier exercise (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).

Tentative map of relationships between spherically symmetrical polyhedra
(regular and semi-regular)

(numbers indicate: F=faces, E=edges, V=vertices;
total of these in parenthesis, with indication of prime factors)
[Product in square brackets with indication of prime number factors]
Map of relationships between spherically symmetrical polyhedra

The wider appreciation of such transformations through music is suggested by mapping the 14 canons onto the faces of a cuboctahedron, and noting the manner in which it unfolds. Of relevance to the above argument, Fuller's preoccupation with the management of global resources (notably in the light of the image of Spaceship Earth) was informed by geometric insights derived from the cuboctahedron (Operating Manual for Spaceship Earth, 1968). However, despite the intention implied by the title of his 2-volume magnum opus (Synergetics: Explorations in the Geometry of Thinking, 1975), this has not translated into cognitive facilitation of "living with the enemy" in the New Renaissance (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).


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