Existential remembrance via a tetrahedral cognitive keystone?

Being Spoken to Meaningfully by Constructs (Part #20)

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The 8-faced truncated tetrahedron plays a central role in the organization of the 12 other Archimedean polyhedra, as presented by Keith Critchlow (Order in Space: a design source book, 1969) and further discussed by Robert C. Meurant, with a twofold extension of the order to include the regular and semi-regular tilings of the planar surface (A New Order in Space: Platonic and Archimedian Polyhedra and Tilings, International Journal of Space Structures. 6, 1991, 1). The 13 (or 15) polyhedra are distinguished from others in that all their vertices lie on the surface of a surrounding sphere (as with the 5 Platonic polyhedra) -- thereby suggesting a degree of global relevance.

With respect to the ambiguity of "16 plus or minus 2", a valuable insight is the fact that the Archimedean polyhedra can be understood as numbering 15, if the two chiral "twins" of the snub cube and snub dodecahedron are considered distinctive as an instance of chiral polytopes (mirror images or enantiomorphs). Together with the Platonic polyhedra they then number 18 (or 20). Critchlow illustrates the keystone role in the image below left, as discussed separately with an interactive 3D version reproduced below (Packing and unpacking of 12 semi-regular Archimedean polyhedra, 2015).

The six immediately surrounding the truncated tetrahedron are formed by truncations of the octahedron. The outer six are formed from the truncations of the icosahedron. The 12 are presented as a "closest packing" configuration forming a cuboctahedron as clarified in the 3D rotation around the truncated tetrahedron -- with each of the 12 on one of the 12 axes unique to the truncated tetrahedron as degrees of freedom. This offers indications for both the unexplained preference for 12-fold strategic organization and its operational implications (Checklist of 12-fold Principles, Plans, Symbols and Concepts, 2011; Â Eliciting a 12-fold Pattern of Generic Operational Insights, 2011).

Indications of role of truncated tetrahedron as cognitive keystone
Interrelating 12 semi-regular polyhedra (presented on right as a rotating cuboctahedral array, omitting the truncated tetrahedron at centre)		Distinctive "route maps" between spherically symmetrical polyhedra	Animation augmenting / excavating selected faces
Adapted from Critchlow (1969)	Interactive 3D version	Reproduced from "route maps" (2015)	Animation prepared with Stella4D

Coherent organization of individual or collective memory -- memorability -- merits particular attention with respect to closest packing. Unfortunately this is now primarily explored with respect to computer memory efficiency (Higher dimensionality, polyhedral packing and transformation, 2019).

Given the systemically problematic efforts to integrate strategies with respect to the physical characteristics of the environment, as with the various UN sets of goals, there is a case for revisiting Plato's recognition of the 5 Platonic polyhedra as archetypal forms: cube (earth), octahedron (air), tetrahedron (fire), icosahedron (water), and dodecahedron (universe) (Cognitive Navigation of the Elements as Indicative Strategic Metaphors, 2023).

The pattern recalls the importance attached to the 5-sided square pyramid and its potential significance for the "5 turnarounds" of the Earth4All strategic initiative (as noted above). The "5 turnarounds" of Earth4All could well be understood as an unconscious sensitivity to the articulation of a form of higher order thinking relative to the 3-fold and 4-fold patterns more readily presented (Requisite fifth-order thinking, 2019; Eliciting systemic answers from a 5-fold web of meaning? 2019).

"Twenty-fold way" to coherent memory organization? Clearly of potential strategic interest are the transformational pathways between distinctive patterns of organization and any coherent mapping of them. One indication is presented above (Pathway "route maps" of potential psychosocial transformation? 2015). Of further potential relevance is a 20-fold pattern, indicated in the table above.

A point of departure is the variety of domains in which a 20-fold ordering has acquired significance, as discussed separately (Global strategic significance of 20-fold configurations, 2015; Indicative examples of the recognized significance of 20-fold patterning, 2015). A checklist of web resources on 20 strategies, rules, methods and insights is presented separately (Requisite 20-fold Articulation of Operative Insights? 2018).

As a distribution challenge, it offers one indication of a constraint on the successful implementation of the UN's 16 SDGs (" plus 1") in relation to the "Twenty-fold Way" distinguished by combinatorics. In that respect Kenneth P. Bogart notes:

When we are passing out objects to recipients, we may think of the objects as being either identical or distinct... If we ignore the possibility that the order in which objects are received matters, we have created 2 x 2 x 4 = 16 distribution problems. In the cases where a recipient can receive more than one distinct object, we also have four more problems when the order objects are received matters. Thus we have 20 possible distribution problems. (Enumerative Combinatorics Through Guided Discovery. 2005, pp. 86-89; The Idea of Distribution)

A fundamental set of 5 hindrances is recognized by Buddhism, potentially to be related to the 5-fold set of Platoinic polyhedra. Separately Buddhism recognizes a 20-fold set of "secondary" hindrances to mental development (Lama Geshey Ngawang Dhargyey, Tibetan Tradition of Mental Development, Library of Tibetan Works and Archives, 1976, pp. 34-35). In Biblical symbolism, twenty denotes that which is unholy, unclean, and profane, namely "remains".Â

As variously indicated above, there is a curious confusion relating to "16 plus or minus 2" (or 3) in distinguishing logical systems as they may be inform speech modalities. One configuration as a menmonic aid could be explored in terms of the set of 20 regular and semiregular polyhedra, namely the 5 Platonic solids, and 15 Archimedean solids (including the chiral forms of the snub cube and snub dodecahedron).

Exploratory animations of 20-fold mappings onto polyhedra
Mapping of 20 Platonic and Archimedean polyhedra onto 20 dodecahedral vertices (solid central dodecahedron)	20 Buddhist secondary hindrances mapped onto Platonic and Archimedean polyhedra (central dodecahedron transparent)
Animations prepared with Stella4D Interactive 3D version (without labels)

Such a set could imply an unexplored relationship to the set of 20 standard amino acids used by living cells to build proteins -- the so-called proteinogenic amino acids. From a systemic perspective it could be asked whether correspondences are to be found of cognitive relevance (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life?