Eliciting Insight within Complex Polyhedral Configurations of Concepts

AI-assisted clarification of cognitive challenge of fundamentally elusive metaphors

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Introduction

The future may note the remarkable conventional tendency to articulate matters of importance as checklists -- possibly with some clustering of items in the checklist. This is evident in declarations of various kinds, most notably the Universal Declaration of Human Rights and the various regional charters of human rights. The pattern can be recognized in the presentation of the Earth Charter and the articulation of a proposed Global Ethic. The UN's set of Sustainable Development Goals, as with it predecessors, also takes list form. Such lists are typically unmemorable. Curiously this evokes little concern in a period in which coherence is desperately sought by many.

The exploration described here follows directly from earlier attempts to configure complex sets in a coherent and memorable manner through mapping them onto polyhedra of appropriate complexity -- polyhedra whose geometric symmetries offer mnemonic keys as well as implying systemic coherence. The previous exercise, which is the primary trigger for this further investigation, focused on the possibility of memorable organization of the classic 48-fold set of Zen koans of the Mumonkan on a truncated cuboctahedron (Configuring the Paradoxical Insights of 48-fold and 100-fold Sets of Koans, 2024). That noted in passing the potential value of mapping the traditional set of 147 Delphic Maxims.

Earlier explorations focused on the 64-fold pattern of distinctions shared by the set of genetic codons, the branches of mathematics distinguished by the Mathematics Subject Classification, the Kama Sutra, and the Chinese classic of the "Book of Changes" (Yi Jing / I Ching) as variously highlighted (Triadic correspondences between Topology, Kama Sutra and I Ching, 2011). These were variously mapped onto the relatively unique 64-edged drilled truncated cube (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015; Configuring the 64 subjects of mathematics as a 64-edged drilled truncated cube, 2021). Its relation to the hypercube (or truncated tesseract) -- of concern in studies of oppositional logic -- was also noted (Implicate order through hypercube and drilled truncated cube? 2022).

It is one thing to shift from a minimally ordered list format to any such three dimensional mapping configuration. It is another matter to derive new forms of significance from their more integrative order. The concern in what follows is how any such significance might be associated memorably with the geometry of more complex polyhedra. That such a diversity of topics should share a 64-fold pattern recalls the arguments from the perspective of cognitive psychology (George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

The focus here is on the patterns of connectivity "within" any "superficial" mapping of an array of conceptual distinctions onto a polyhedron -- however that mapping may be organized "without" from an "external" perspective as argued in a presentation the World Academy of Art and Science illustrated by a variety of human rights charters (Topology of Valuing: dynamics of collective engagement with polyhedral value configurations, 2008 and Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors (2008). By contrast here, the geometry of selected polyhedra was used to explore the "underlying" connectivity between "superficial" distinctions. It is assumed that these would be of potential systemic significance -- as well as being a key to the memorability and credibility of the pattern as a whole.

As noted, a point of departure was the array of 48 koans on a truncated cuboctahedron. An effort was made to apply the method to the array of 16 Sustainable Development Goals mapped onto the 16 vertices of the 1-frequency truncated tetrahedral geodesic sphere to clarify their potential internal systemic connectivity. Its applicability to the 30 articles of the Universal Declaration of Human Rights was explored to highlight underlying patterns of connectivity between them. In the light of the relative success of those exercises, the possibility of applying it to the above-mentioned 64-fold arrays was considered.

Of particular interest was the possibility of detecting any "inner" substructures (or patterns of coherence) within each such array -- as implied by the structure of the truncated tesseract, and as explored in some measure with respect to the exploratory mappings in 3D of the UN's SDGs onto the 64-vertex truncated tesseract (Visualizing the Coherent Configuration of Incommensurable Cognitive Modalities, 2024). That structure offers the tantalizing image of a form of "inner chamber" of connectivity suggestive of a higher order of coherence and integration. The question here is whether any such "inner chambers" could be identified and visualized in the case of the SDGs or the Universal Declaration of Human Rights -- or indeed with the set of 48 koans.

The detection and visualization of such connectivity, and any formation of an "inner chamber", depended to a large extent on the facilities offered by AI in previous exercises -- in the form of ChatGPT 4o and Claude 3.5 -- given the limited competence of the author (and the relatively inaccessibility of professional mathematical and graphical expertise). In consequence, and somewhat ironically, the interaction with AI did not result in immediate solutions to the question posed and the visualization sought. It became evident that AIs were constrained to a curious degree in responding to a challenge for which they "optimistically" claimed ability. The very lengthy iterative process enabled by artificial intelligence -- with a degree of relative success -- could then be usefully recognized as a metaphor for collective ability to identify higher orders of connectivity -- whether within strategies, human rights, or elusive insights.

The nature of the challenge can then be clarified in terms of the difficulty for AI to transform a series of purportedly connected "points" -- comparable to any conventional checklist of concepts -- into a superficially connected polyhedral configuration, and then to distinguish the patterns of relationship between those points. This can be recognized as the challenge of detecting the connectivity between any array of human rights or of the SDGs. Little effort is made to do so from a systemic perspective -- to the extent that the possibility is even considered to be of any relevance. In detecting and visualizing such connectivity in 3D, the difficulties encountered by AIs could be understood as a metaphor for the human cognitive difficulties in recognizing such patterns.

Despite what is claimed for AI capacity, including imagery, it became very clear that AIs have as yet very limited ability to "think in 3D" -- as might well have been expected. All the more problematic was effective erosion of AI memory in that learnings in an earlier exchange were not retained and applied in a later part of the exchange (for resource management and marketing reasons). Errors previously encountered were simply repeated -- and repeatedly corrected -- undermining any convergence on a viable solution previously achieved successfully. The challenge proved to be all the greater with respect to the detection and visualization of any "inner" patterns of coherence framed by such connectivity -- any "inner chamber".

Metaphorically such a "chamber" might be recognized as a cognitive sanctum sanctorum to which only the cognoscenti would currently claim access. The challenge could be comprehended as the quest for a kind of Rosetta Stone enabling a form of "translation" between distinctive cognitive modalities (as previously suggested) -- variously understood to be incommensurable or mutually irrelevant. Such a configuration has been previously explored in the light of that metaphor (Memorable Packing of Global Strategies in a Polyhedral Rosetta Stone, 2023).

In cognitive terms the Rosetta Stone might be more appropriately recognized as the archetypal philosopher's stone -- or even

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