Encasement within Contrasting Polyhedral Baskets

TRANSCEND MEMBERS, 20 Jul 2026

Anthony Judge | Laetus in Praesens - TRANSCEND Media Service

Psychosocial Transcendence through Dynamic Exchange and Sonification with AI Assistance

Introduction

In the quest for coherence in a society faced with polycrisis, consideration can be given to the coherent patterns identified through geometry and variously appreciated as symbols. Most obviously these are the well-known forms of the 5 Platonic solids (tetrahedron, octahedron, cube, dodecahedron, and icosahedron). Others, identified as “semi-regular”, include the 13 Archimedean solids and their 13 Catalan duals. Together these could be understood as templates for potential psychosocial coherence of increasing complexity, especially since they can be defined in relation to a circumsphere. The regular and semi-regular solids share the property of a circumsphere containing all vertices; their Catalan duals, which lack one, can be associated with the same sphere by radial projection of their vertices. That set excludes the infinite prism and antiprism families.

As a set, that collection invites the question as to how they might be collectively understood as forming a pattern together. Is there a pattern to the individual patterns of coherence together? More to the point, if such a pattern can be recognized, how is it to be rendered more widely comprehensible within a fragmenting society challenged by coherence and arguments for “unity”? Whilst aspects of the challenge are routine orbit-counting exercises in mathematics of standard group theory, the knowledge currently sits below the threshold of interest and motivation. That knowledge exists compressed in the reflection-group formalism, without any reason to decompress it into any particular shape. Mathematics indeed offers ways of describing such a pattern, most notably through “languages” which are incomprehensible to most and a challenge to visualization. The contribution of AI in the following exercise was one of decompression, verification, and visualization

An early approach to the matter had been that of Johannes Kepler (Mysterium Cosmographicum, 1596). Another has been the configuration offered by Keith Critchlow (Order in Space A Design Source Book, 1969). Both suggest a degree of relevance to psychosocial organization, as with the work of Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975-1979). Although variously appreciated, unfortunately such implications have not been extensively explored despite the challenges of polycrisis and interdisciplinarity (Geometry of Thinking for Sustainable Global Governance, 2009). With the rapid development of visualizing technology and its accessibility via the web, the possibility of configuring together the set of polyhedral templates of coherence now invites further experiment. More recently, with the aid of AI, this gave rise experimentally to a “Carousel model” of the set (Remembering the Disparate via a Polyhedral Carousel: memorable configuration of transformations between core polyhedra of strategic relevance, 2025). An earlier experiment, inspired by Kepler, explored how Platonic polyhedra could be nested dynamically within a rhombic triacontahedron (Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra, 2015).

The challenge can be explored otherwise by taking as a point of departure the large number of vertex points — as the distinguishing feature of those regular and semi-regular polyhedra — and mapping them together (rather than separately) onto the sphere with which they are all associated — exemplifying coherence and unity. The question is then whether the result offers a more complex (but still memorable pattern), indicative of a degree coherence — which has not been widely recognized or explored. In particular can this pattern be visualized to enable widespread recognition? This is the kind of problem which AI is able to address with some ease, but is otherwise somewhat beyond the techniques of mathematics or the conventional motivation of mathematicians.

In the exercise which follows the challenge was put to Fable-5, the latest version of Anthropic’s series of large language models — which has featured in considerable controversy with the US government. The Pentagon designated Anthropic a “supply chain risk” in early 2026 after the company refused to drop its prohibitions on mass surveillance and autonomous weapons.Days after launch the Commerce Department’s export-control order led Anthropic to suspend Fable 5 whose public availability, leaving subscribers with only lower-capacity models (notably Claude Opus 4.8) until access was restored in July 2026 (Technology Scoop: Powerful Anthropic model, Fable 5, on track to return soon, Axios, 27 June 2026). The exercise has therefore been conducted within a temporary window of opportunity of unpredictable duration.

The exercise with Fable-5 clarified the fact that the approach enabled representation and visualization in 3D of two incommensurable models. It is appropriate to note that everything was computed and programmatically verified, including the correction of errors the AI itself made en route (including misaligned textbook coordinates; and an initial misstatement about mirror circles).

Of particular interest — from a psychosocial perspective — is that the two models are immediately reminiscent of distinctive basket weaving patterns. The “basket” metaphor as a container featured in an extensive earlier exchange with Fable-5 (of which this is effectively a sequel). That highlighted the various connotations of the metaphor, notably including “basket-case”, the fundamental significance of basketry in indigenous cultures, and the role of the “baskets of issues” through which the Helsinki Accords emerged during the Cold War (Engaging with a Planetary Basket-Case during a Singularity? Potential reframing of perception with the assistance of AI, 2026). An emphasis on “basket weaving”, recalls the considerable emphasis of Mahatma Gandhi on the relevance of spinning to swaraj and governance — and by extension to weaving, as discussed separately (Warp and Weft of Future Governance: ninefold interweaving of incommensurable threads of discourse, 2010).

The two contrasting polyhedral patterns are then usefully indicative of the widely experienced incommensurability of binary perspectives — of “us” and “them”, and their reciprocation. This is a prominent feature of polycrisis — from the problematic relations between opposing political ideologies, to the “two cultures” of sciences and the arts, and the failures of interreligious dialogue which to continue to trigger violence,. The two patterns then articulate and frame the challenge of such incommensurability — offering a language in which it can be explored and transcended, if only through visualization, dance or sonification. The case for sonification, explored experimentally in what follows, is based on the ability of the “ear” to hear relationships which are typically a challenge for the “eye” — especially in the light of the widespread appeal of music, and its omnipresence, in a society challenged by incoherence (A Singable Earth Charter, EU Constitution or Global Ethic? 2006)

An intriguing question in psychosocial terms is whether there is any understanding of how many distinctive perspectives are extant in society. Relatively simple answers to such a question feature in the distinction between the limited numbers of “ways of looking” (Interrelating Multiple Ways of Looking at a Crisis, 2021), the array of some 10,000 religions worldwide (Pam Wasserman, World Population by Religion: a global tapestry of faith, 2024), or their secular “analogue” in the extensive variety of disciplines (Intellectual Disciplines and Sciences, 1976). These are potentially consistent with biological and cybernetic arguments for requisite variety — although notions of “memetic variety” are seldom evoked, except through rhetorical references to “diversity”. Characteristic of polycrisis is the framing of any “other” as necessarily problematic — if not “evil” — which the incommensurability of the contrasting models (as highlighted here) suggests might be considered otherwise. By contrast, there is considerable emphasis on singular models and the hegemony which they justify in the face of alternatives.

An unexpected development of the exercise associated the radical distinction between the binary perspectives modelled in terms of the two prime numbers characterizing their geometry, namely 13 and 31. This resulted in further investigation of a set of four pairs of prime numbers — paired paradoxically as reversible emirps — with which sets of seemingly incommensurable strategic concepts were associated to some degree. This fruitful approach to incommensurability and coherence is developed and illustrated in a sequel to the polyhedral modelling focus here (Mirrored Primes as Bridges between Incommensurable Strategic Frameworks, 2026).

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