Visualization of Polyhedral Inner Chambers with Psychosocial Implications

TRANSCEND MEMBERS, 7 Apr 2025

Anthony Judge | Laetus in Praesens - TRANSCEND Media Service

AI-assisted Exploration of Neglected Underlying Patterns of Order in 3D

Introduction

7 Apr 2025 – There is a basic set of symmetrical polyhedra of which many were known in Ancient Greece. Their degrees of symmetry have long made them a focus for architecture and the symbolism of sacred geometry — as well as an inspiration for patterns of order and coherence in many domains. Those considered here are the 5 regular Platonic polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and the more complex set of 13 semi-regular Archimedean polyhedra.

As geometrical structures in 3D, most have diagonals between diametrically opposed points. The focus here is not on those axial diagonals, nor on the links between vertices on the surface of the structure (known as face diagonals). The internal links between all the many vertices of a polyhedron are known as space diagonals. Of those, the focus here is on the non-axial diagonals and the manner in which they may intersect, thereby defining and framing one or more internal polyhedra — typically concentric (around the centre through which pass the axial diagonals). Of potentially greatest interest is the internal polyhedron — an “inner chamber” — framed by the longest internal non-axial diagonals.

The question is what such a “hidden” inner polyhedron might imply symbolically and cognitively. Being “inside” and underlying, the question is how that perspective relates to what is “outside” and superficial (Interface challenge of inside-outside, insight-outsight, information-outformation, 2017; Alternation of worldview between “inside-outside” and “outside-inside”, 2013; Cognitive osmosis through topological eversion and interlocking tori — framing outside-inside otherwise, 2017). The nesting of polyhedra in this way is reminiscent of the iconic visualization in 3D presented by the astronomer Johannes Kepler (Mysterium Cosmographicum, 1596) — virtually the first attempt since Copernicus to indicate the physical truth of heliocentrism. Any 2D representation, as by a mandala or yantra, could then be recognized as a “flattened” perspective of entangled 3D configurations.

Although apparently abstruse, the challenge of perspective such inner structures represent could provide a key to vexatious issues of “oversight” and “coordination” in governance. To the extent that multi-dimensional polyhedral structures have been recognized by neuroscience as configurations of neurons within the brain, visualization of such changing patterns offers a distinctive approach to degrees of comprehension. More provocatively this suggests a case for exploring “nuclear memetics” in contrast to nuclear weaponry and defensive preoccupation with nuclear warfare.

It appears that relatively little is referenced concerning the space diagonals, although it is acknowledged that they have application in a variety of domains — with implications for others. There do not appear to be catalogues of the inner polyhedra which they may frame and subtend, notably with respect to the fundamental set of regular polyhedra. Whilst formula may indeed exist defining the number of space diagonals, the further challenge is how any information about a fundamental polyhedron may be analyzed by computer in order to provide comprehensible visualizations of any inner structures. How many inner polyhedra are framed by the iconic regular polyhedra — and how are they configured? This exercise follows from a less systematic approach by which a series of 3D visualizations were generated (Eliciting Insight within Complex Polyhedral Configurations of Concepts, 2025). Those results are reproduced in what follows.

The approach taken here is to use AI facilities to analyze information on each regular polyhedron, to identify and define the space diagonals, to determine the coordinates of their intersections, and to visualize the results in 3D accessible via the web. The challenge was variously submitted to four AIs (ChatGPT 4o, DeepSeek, Claude 3.5, Perplexity).

The exercise was undertaken with limited geometrical insight (or the associated mathematical skills) — and with only a degree of knowledge of the requisite programming in 3D for web representation (in this case X3D). To compensate for these limitations, the question was how AI skills could be applied to the challenge, notably the generation of a Python script which could do the analysis and generate the requisite X3D file enabling visualization (using H3DViewer or FreeWRL).

What follows is therefore a report on what was achieved and its limitations — in what is effectively a work in progress — despite considerable assistance from various AIs. The results could encourage others with greater skills and insight to improve the process. Portions of the report will only be of interest to those seeking to replicate the procedure and improve upon it — with or without the rapidly developing skills of AI. Of more general interest are the visualizations presented and the psychosocial implications of the “inner chambers” detected with respect to neglected patterns of order.

The responses from AI in this exploration have been framed as grayed areas. 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. 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).

Reservations and commentary on the process of interaction with AI to that end have been discussed separately (Methodological comment on experimental use of AI, 2024). Editing responses has focused only on formatting, leaving the distractions of any excessive “algorithmic flattery” for the reader to navigate (as in many social situations where analogous “artificial” conventions are common). 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 process calls for critical comment in contrast with more traditional l methods for doing so.

In making very extensive use of AIs to develop a generic approach to the visualization of the internal structure of regular polyhedra, the limited success offers insights into the current constraints in the use of AI as a “cognitive prosthetic” in a domain where its advantages might be assumed to be evident. Whilst the visualizations presented are an indication of a degree of success, the failure to enhance the device with operational features to facilitate its use is interesting in its own right — as with the failure to provide analytical geometric data in which confidence is warranted. Many such features have been built into the device, but their operation is currently questionable. This is an invitation to those with greater skills and insights — or more capable of debugging the device (with or without AI assistance).

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