Remembering the Disparate via a Polyhedral Carousel

TRANSCEND MEMBERS, 22 Sep 2025

Anthony Judge | Laetus in Praesens - TRANSCEND Media Service

Memorable configuration of transformations between core polyhedra of strategic relevance.

Introduction

22 Sep 2025 – Polyhedra indicative of coherence: A configuration of the symbolic Platonic polyhedra and Archimedean polyhedra was elaborated previously and presented as a 3D model — together with their Catalan duals and the Kepler-Poinsot stellations (Cognitive Fullerene as a Rosetta Stone for Patterns of Systemic Constraint, 2025). This was seen as an exercise in the reconciliation between disparate frameworks — ironically in the the light of global familiarity with the stitching pattern of the football, namely the truncated icosahedron.

If only in the light of their symbolism, particular emphasis can be given to the cognitive significance of regular polyhedra as indicative of “ways of thinking” coherently (Interrelating Multiple Ways of Looking at a Crisis, 2021). Whether as the 5-fold Platonic configuration (4+1), or as the 12-fold configuration of the Archimedean polyhedra configured around a 13th, the patterns associated with the configuration as a whole are suggestive of orders of coherence and integrity which remain to be fully recognized. Especially remarkable is the range of 12-fold articulations, notably of mathematical, religious, and strategic nature (Checklist of 12-fold Principles, Plans, Symbols and Concepts, 2011). The implications can be variously explored (Clarifying the Unexplored Dynamics of 12-fold Round tables, 2019; Enabling a 12-fold Pattern of Systemic Dialogue for Governance, 2011; Eliciting a 12-fold Pattern of Generic Operational Insights, 2011).

The earlier configuration of polyhedra emphasized an issue previously indicated (Map of Systemic Interdependencies None Dares Name: 12-fold challenge of global life and death, 2011). Curiously little is widely known of the relationships between the 35 polyhedra so configured — beyond the essentially obscure insights of geometricians, topologists and specalists in polyhedral combinatorics. The question is whether such a configuration can be rendered inherently memorable — given its strategic and other implications, and especially their cognitive dimensions. The latter concern was necessarily inspired by the magnum opus on “synergetics” of Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975-1979) — which unfortunately makes little reference to its cognitive implications, as argued separately (Geometry of Thinking for Sustainable Global Governance, 2009).

Memorable symmetry preserving operations: The relation between such polyhedra has been formally described by the Conway Polyhedron Notation — of which only the simplest visual indications are offered for the core set of polyhedra. The “pattern that connects” is indeed known, but it has not evoked much attention to its memorable visualization (Hidetoshi Nonaka, Visualization of Conway Polyhedron Notation, World Academy of Science, Engineering and Technology, 50, 2009). To the extent that this may well be indicative of the relationships between disparate cognitive modalities, this is clearly a challenge it is appropriate to address in a society ever more characterized by fragmentation. That argument is developed and illustrated separately in the light of correspondences between their polyhedral representation (Comparable Modalities of Aesthetics, Logic and Dialogue, 2021).

The focus in what follows is specifically on the visualization of that pattern of connectives between the 35 polyhedra in the core configuration as a mnemonic aid. How do iconic members of any 12-fold collective dialogue with each other — whether apostles, deities, tribes, etc — and why is no effort made to explore that dynamic, as exemplfied by the secrecy of 12-member juries?

The connectives are otherwise known to specialists as sysmmetry preserving operations (Pieter Goetschalckx, etc, Local orientation-preserving symmetry preserving operations on polyhedra, Discrete Mathematics, 344, 2021, 1; Pieter Goetschalckx, et al, Generation of Local Symmetry-Preserving Operations on Polyhedra, arXiv, 1908, 1162292). These operations are discussed and illustrated separately in terms of the metaphorical comprehension of complexity enabled by graph theory and polyhedra (Connecting the Multiple Voices of the Pattern that Connects, 2024). As stressed, the particular bias in this exploration is comprehensibility and memorability — mnemonic issues of remarkably little interest to those claiming most familiarity with such operations.

Strategic and discourse relevance: Of particular significance is the manner in which selected polyhedra are central to the representation of the set of connectives of logical operations and are thereby fundamentsal to the manner in which computers process queries — as with the operation of AIs (Oppositional Logic as Comprehensible Key to Sustainable Democracy, 2018). It might then be asked whether related polyhedra might usefully extend the understanding of such operations and the reframing of insight into those logical processes — in a period in which rational discourse is fundamentally challenged..

A fundamental question of relevance to governance is whether and how the connectives are related to the potential feedback loops between disparate perspectives as embodied in agency, ministerial and departmental mandates. There is some irony to the fact that the regularity of the core polyhedra offers a reminder of the role of regulatory bodies and their sense of order. Which connectives might then be vital, which are neglected or ignored, and which are little known? Addressing such questions does however require that the context in which they are configured should offer mnemonic aids to comprehension of the complexity of the whole. That question offers a challenge to reconfiguration of the UN’s 8-fold Millennium Development Goals (within the memorability constraints of George Miller’s famed Magical Number Seven, Plus or Minus Two, 1956) into the current 16+1 Sustainable Developement Goals, now faced with “inexplicably” limited uptake.

Use of AI: Although this experimental exploration has been variously enabled by AI, most of the responses of AI have been framed as grayed areas in the text. Given the length of the document to which the exchanges gave rise, the form of presentation has itself been treated as an experiment — in anticipation of the future implication of AI into research documents. Many responses may be irrelevant to the outcome (rather than of interest to the process), and can therefore be readily ignored — especially given questionable use by AI of “algorithmic flattery”.

Only the “questions” to AI are rendered immediately visible — with the response by AI hidden unless specifically requested by the reader (a facility not operational in PDF variants of the page, in contrast with the original ). Readers are of course free to amend the questions asked, or to frame other related questions — whether with the same AI, with others, or with those that become available in the future. In endeavouring to elicit insight from the world’s resources via AI, the dependence on “leading questions ” calls for critical comment in contrast with more traditional methods for doing so. The original responses by AI may include citations of multiple sources which have not been systematically in the responses presented.

It should be emphasized that the following exercise with AI is experimental and tentative in anticipation of a more considered approach. As a proof of concept, the questions and results call for iterative refinement — if only with respect to the designs of the animations produced. This continuing experimental use of responses of several AIs to prompts, and their reproduction as optionally visible to readers, occurs in a period in which there is ever increasing criticism from various sectors and perspectives regarding the appropriateness, risks and validity of any such engagement with AI. A response to such criticms is appropriate (as articulated below).

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