Complementarity of pattern languages -- an all-encompassing meta-pattern?

Global Insight from Crown Chakra Dynamics in 3D? (Part #11)

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Contrasting patterns: As noted above, as a potentially comprehensible pattern, the "1,000-petalled lotus" could be contrasted with that celebrated in the mathematics of group theory as the so-called "monster group".Its complexity is indicated by its order, namely the number of elements in its set:

246 x 320 x 59 x 76 x 112 x 133 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71 namely 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 or approximately 8Ã.1053

The group contains 20 sporadic groups (including itself). It is the biggest of the sporadic groups and is equipped with the highest known number of dimensions and symmetries. The set of groups has been termed the "happy family" -- and hence the alternative name for the "monster" as the "friendly giant".  A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

The monster group is regarded by many mathematicians as a mysteriously beautiful object -- intriguingly framed by the so-called  monstrous moonshine conjecture. As seemingly beyond conventional human comprehension, it might then be asked how that "mathematical object" relates to the significance otherwise attributed to the "1,000-petalled lotus" (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Dynamics of Symmetry Group Theorizing: comprehension of psycho-social implication, 2008).

Is the "1,000-petalled lotus" to be understood as another form of "monstrous moonshine".

Correspondences: The existence of the monster groups resulted from the recognition of strange and unexpected correspondences, an odd coincidence, as described by Peter Diamond (Mathematicians Chase Moonshine's Shadow, Quanta, 12 March 2015), noting that its 10⁵³ elements were more than the number of atoms in a thousand Earths. This was originally reported by John Conway and Simon Norton who conjectured that these relationships must result from some deep connection between the monster group and the  j-function (Monstrous Moonshine, Bulletin of the London Mathematical Society, 11, 1979, 3). The deliberate reference to "moonshine" was made because the connection appeared so far-fetched -- beyond any capacity to ever prove it.

Since that time, a numerical proof of the so-called Umbral Moonshine Conjecture proposes that, as summarized by Diamond:

... in addition to monstrous moonshine, there are 23 other moonshines: mysterious correspondences between the dimensions of a symmetry group on the one hand, and the coefficients of a special function on the other.... The 23 new moonshines appear to be intertwined with some of the most central structures in string theory, four-dimensional objects known as K3 surfaces. The connection with umbral moonshine hints at hidden symmetries in these surfaces.Â

Aside from this particular focus of mathematicians, a more general question is the nature of "mysterious correspondences" and "deep connections" -- readily to be framed as "moonshine" -- and the manner whereby they may be recognized. As discussed separately, the question can be explored in terms of contrasting understandings of correspondences (Theories of Correspondences and potential equivalences between them in correlative thinking, 2007). The latter was produced in the light of the above-mentioned discussion of the potential psychosocial significance of "monstrous moonshine". It focused on the contrast between the "algebraic" theory of correspondences and the "symbolist" theory of correspondences -- neither of which has the remotest appreciation of the other, although both can be considered fundamental to correlative thinking.Â

Curiously however both function as strange attractors -- "mysteriously beautiful" in some way. This is indicative of an aesthetic dimension, readily associated with an appreciation of symmetry, perhaps to be understood as appropriate connectivity..

Fundamental patterns and their correspondences: The point to be emphasized is the future possibility of recognizing other patterns of correspondences and the complementarity between them. Such an exploration would be consistent with arguments made by Susantha Goonatilake (Toward a Global Science: mining civilizational knowledge, 1999) and George Lakoff and Rafael E. Nunez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

Provocatively it might be asked, for example, whether the relation between the 20-fold set of amino acids constitutes another form of moonshine, given its importance to the mystery of life -- and the complexity of the genetic code. Does that 20-fold pattern bear any relation to the 20 sporadic groups recognized by mathematics -- or the 20 rings of the crown chakra -- if only as a pattern of memetic organization? Why do the dodecahedron and the icosahedron offer complementary means of ordering such a pattern -- with their particular symbolic appeal to the human mind?

Pattern language: A major contribution to further exploration is the seminal work of  Christopher Alexander (A Pattern Language, 1977), following his earlier work (Notes on the Synthesis of Form, 1964), and followed by a major study (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-4). From this he drew further insights (Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole, International Journal for Unconventional Computing (IJUC), 5, 2009).

The patterns in Alexander's focus on nature and the built environment lend themselves to generalization extending to to the psychosocial environment (5-fold Pattern Language, 1984). That with respect to "harmony-seeking", also invites further relfection (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).

In confronting seemingly disparate patterns, there is a case for drawing on the insights highlighted by Alexander with respect to 15 transformational principles as summarized and discussed in the latter -- and the extension of their interpretation, as discussed in the following sections:

Tentative adaptation of Alexander's 15 transformations to the psychosocial realm Systemic comprehensiveness of sets

Geometrical configuration of Alexander's 15 transformations Relevance to global governance in the psychosocial realm

There is a case for comparing such understanding of "transformation" with the set of supersingular primes dividing the order of the monster group  There are precisely fifteen such prime numbers. It could then be asked whether any such 15-fold pattern lends itself in polyhedral form such as to be memorable as a whole, as discussed with respect to their geometrical configuration. This noted that:

• three golden rectangles at right angles to each other determine the vertices of an icosahedron.
• there are 15 different circles through the pairs of opposite edges of the icosahedron.
• these circles are the great circles of a sphere circumscribing the icosahedron, since they are in the plane of the golden rectangles, and by one definition, a great circle is the intersection of a sphere and a plane passing through the center of the sphere.

There are therefore 15 intersecting golden rectangles, each edge of the icosahedron being defined by an edge of a golden rectangle. The 15 golden rectangles span the interior of the icosahedron. These rectangles have 30 edges, and each edge pairs up with its opposite edge to form a golden rectangle.

Indication of patterns of 15-foldness
Icosahedron showing single golden rectangle (made with Stella Polyhedron Navigator)	Icosahedron showing all 15 golden rectangles (made with Stella Polyhedron Navigator)	Simplest magic square (of order 3)

With respect to use of a magic square, as previously noted (Magic Carpets as Psychoactive System Diagrams, 2010), if any underlying systemic pattern is to be found in Alexander's 15 transformations, a mathematical curiosity of possible relevance is that all the dimensions of the smallest non-trivial magic square total to 15. More subtle understandings of the relevance of magix squares to coherence in governance is evident in their cultivation by Benjamin Franklin (Magic square integrity and implications for the US Constitution, 2015).

Rather than associating any such transformations with 15 golden rectangles as internal features of an icosahedron, another potentially valuable approach is to associate them with the edges of a polyhedron, of which one interesting candidage is the tridiminished icosahedron, as presented below. This has 8 faces (of 4 types), 15 edges (of 5 types), and 9 vertices (of 3 types). Reflection planes are shown in several animations; one shows the stages in (un)folding.

Indicative mapping of supersingular primes onto 15 edges of tridiminished icosahedron
Faces shown	Faces transparent	Folding	Dual
Made with Stella Polyhedron Navigator

In the quest of memorable mappings, the same exercise can be performed with a star polyhedron of 15 vertices (of 3 types), 22 faces (of 6 types), and 35 edges (of 7 types)

Indicative mapping of supersingular primes onto the 15 vertices of a star polyhedron
Faces shown	Faces transparent	Folding	Dual
Made with Stella Polyhedron Navigator

Power laws: Considerable importance is accorded in physics to power laws, namely a functional relationship between two quantitites. with one quantity varying as the power of another -- as understood in terms of exponentiation. The exponents defining the monster group and its components are indicative of this. The square-cube law is applied in a number of scientific fields dealing with the natural world. In psychosocial systems, it is of notable importance in proxemics (Scale: power laws exemplified by the square-cube law, 2019).

The latter notes the eexistence of 80 kinds of power laws ranging from atoms to galaxies, DNA to species, and networks to war highlighted by Pierpaolo Andriani and Bill McKelvey (Beyond Gaussian Averages: redirecting international business and management research toward extreme events and power laws, Journal of International Business Studies, 38, 2007, pp. 1212-1230). The authors apply their insights to organization, most obviously to corporations (Perspective: From Gaussian to Paretian Thinking: causes and implications of power laws in organizations, Organization Science, 20, 2009, 6; From Skew Distributions to Power-law Science, In: Peter Allen (Ed.), The Sage Handbook of Complexity and Management, 2011).

It is therefore appropriate to explore the organization of the crown chakra from a power law perspective, especially given the role of exponentiation in the case of the monster group. The relation to the "10,000 things" of the Tao Te Ching invites similar consideration -- given reference above to the chiliahedron and the myriahedron. With the former as a framing of high order of subjectivity, and the latter of the objectivity of the real world, their entanglement offers a provocative visual mnemonic. This is potentially in the spirit of the preoccupations of the Global Sensemaking Network.

Suggestive pattern of power law relations between subjectivity and objectivity?
Chiliahedron "crown chakra"	Correspondence?	Myriahedron "10000 things"

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