In quest of number 37 through the pattern of spherical polyhedra

Memetic Analogue to the 20 Amino Acids as vital to psychosocial life? (Part #6)

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]

The main paper associated patterns of transformation with a particular polyhedron, namely the drilled truncated cube. The question meriting exploration is whether the spherically symmetrical polyhedra commonly recognized -- Platonic and Archimedean -- offer a sense of pattern which can be depicted as suggestive of transformation between different forms of order. The exercise here is designed to determine whether the prime number 37 figures significantly in the pattern -- as potentially implied by interpretation of the Kazakh report (in the main paper, as noted above).

The neglected psychosocial ordering role of polyhedra has been variously considered previously (Spherical Configuration of Categories -- to reflect systemic patterns of environmental checks and balances, 1994; Spherical Configuration of Interlocking Roundtables: Internet enhancement of global self-organization through patterns of dialogue, 1998; Representation of Interlocking Elements for a Sustainable Global System: configuring strategic dilemmas in intersectoral dialogue, 1992; Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008). These followed an earlier exercise in endeavouring to map the transformations associated with the cuboctahedron alone (Vector Equilibrium and its Transformation Pathways, 1980).

The focus here is on how the relationships between the selected polyhedra could be depicted in the light of the numbers associated with their faces, edges and vertices. Where any such numbers are identical, irrespective of whether they are associated with faces, edges or vertices, this suggests the possibility of transformation -- a pathway. This may be understood in geometrical terms (duality, etc) or through associated categories (namely collapsing or expanding sets of distinguished elements). This suggests a form of cognitive route map between different patterns of order (as highlighted further below). Polyhedral "stations" on the map may then offer the possibility of switching to another "route" associated with a different number. The polyhedron at each node in the network is then usefully understood as a mode of organization -- a potential configuration of categories or experiences.

In the following map the simpler 5 Platonic polyhedra are positioned in relation to a central circle with the 13 Archimedean (semi-regular) polyhedra positioned around a larger circle. The relative complexity of the polyhedra -- indicated by indiscriminate totalling of their faces, edges or vertices -- is used as a guide to vertical positioning. This enables indication with:

• solid horizontal lines of an equivalence between three pairs of Archimedean polyhedra, linked in the map: truncated octahedron / truncated cube (74); icosidodecahedron / snub cube (122); truncated icosahedron / truncated dodecahedron (182).
• dotted lines of various relationships between polyhedra (based on faces, edges and vertices) -- duly marked with the numerical commonality between them (irrespective of whether it relates to one or more such features).

Tentative map of relationships between spherically symmetrical polyhedra (regular and semi-regular) (numbers indicate: F=faces, E=edges, V=vertices; total of these in parenthesis, with indication of prime factors) [Product in square brackets with indication of prime number factors]

This schematic acquires greater meaning when used (below) to identify "route maps" of possible change between modes of organization based on sets of a given size. It is however noteworthy how the psychosocial proclivity for 12-fold organization is "nested" within the inner Platonic circle of the schematic (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011; Eliciting a 12-fold Pattern of Generic Operational Insights, 2011; Implication of the 12 Knights in any Strategic Round Table, 2014). Such exercises associate 12-foldness with a variety of qualities, notably recognizable through characteristic "languages" variously considered appropriate to strategic articulation (12 Complementary Languages for Sustainable Governance, 2003).

With respect to the nature of 20-foldness, again this is to be noted within the central circle, closely associated with the 30-foldness which notably figures in the cybernetic preoccupations of Stafford Beer (Beyond Dispute: The Invention of Team Syntegrity, 1994)

Simplified version of above map of relationships between spherically symmetrical polyhedra (regular and semi-regular) (numbers indicate: F=faces, E=edges, V=vertices; total of these features in parenthesis) [Total reduced to prime number, other than 2, in square brackets]

The following schematic takes the prime numbers of the map above and emphasizes to a higher degree the recurrence of particular numbers. Especially interesting, and potentially significant to any intuitive sense of the integration of the pattern as a whole, is the degree of patterning amongst those numbers. Prime numbers derived from polyhedral feature totals and from feature products are distinguished.

Within the set of numbers indicated, of the complete set of the first 12 primes (2, 3, 5, 7, 11, 13, 17, 19, 23 29, 31 and 37). The first four are especially evident with respect to the tetrahedral pattern of Platonic polyhedra. Only 29 is not immediately evident, although 17 is seemingly relatively rare. Perhaps 29 is to be understood as derived from selected tetrahedral relationships: 37-8, 23+6. As suggested by the Kazakh research noted in the main paper, the number 37 is indeed evident and significantly so.

Schematic emphasizing pattern of prime numbers in above maps of relationship between spherically symmetrical polyhedra Prime numbers in parentheses are from feature totals; those in square brackets from feature products] This version excludes some details from the above maps, notably factor 2 in feature totals. Various related totals and products are indicated below the schematic as being of relevance to its improvement. Some degrees of correspondence across the pattern may be of interest, including that between 19 at the top and 91 at the bottom.

Of particular interest is the interlocking of the primes associated with the central set of Platonic polyhedra. There is of course some probability that the interlocking results from more general implications of the Euler characteristic. A similar approach could be taken to the primes in the outer circle of Archimedean polyhedra.

The method can of course be challenged, whether as exemplifying deprecated "number games" (however these may be associated with a legitimate sense of integrative satisfaction), or because of use of totals of faces, edges and vertices (and removing the factor 2 to isolate any other prime). This does however bear some resemblance to the Kazakh use of the "molecular core", and their mathematical verification of the statistical probability of their result -- including their highlighting of 37. A further step would be to switch to a number base other than 10. Twenty might be especially interesting. Note that the emirps are base dependent, whereas primes are not.

Many of the apparent correspondences may well be trivial, including the curious relationship (by reversal) between 26 and 62 (namely 13x2 and 31x2), and between 37 and 73; this may also hold for that implied between 16 and 61, and between 19 and 91, although one of each is not a prime (as noted below with respect to emirps). As indicated by lists of the characteristics of 37, many are associated with operations in addition to summation and multiplication. This suggests the need for a critical approach to the emergence of patterns as a result of operations such as 61+12=73, 61-12=7², or 13+7=20

In this respect, as discussed further below, of particular potential interest is a (questionable) operation of the form (2²x5²)+37=137, given the special significance attached to 137. With respect to the seeming rarity of 17, in the above pattern, it should be noted that the prime factors of the reversal of 137 (namely 731) include 17 and 43.

How the method might be related to (or extended by) more conventional distinctions amongst polyhedra -- such as axes of symmetry, the Euler characteristic, the Schläfli symbol, or the number of cells -- justifies future exploration. A particular reservation to be borne in mind is a degree of recognition that patterns can be found anywhere -- if investigation is sufficiently assiduous and criteria are shifted accordingly.

Deprecation of the detection of such patterns may obscure the question of the factors that may well be associated with a vital sense of psychosocial coherence lasting centuries -- whether illusory or otherwsise.

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]