Generic possibilities in the topology of cultural symbols?

Global Psychosocial Implication in the Pentagramma Mirificum (Part #5)

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Such exercises then frame the question of how seemingly disparate cultural symbols might be compared and interrelated through their experience in virtual reality. In this case, how different is the structure of the Pentagramma Mirificum from the Kabbalah in topological terms? Can the one be fruitfully mapped onto the other? How are differences to be interpreted as misrepresentation (and loss of vital distinctions) or suggesting further insight?

The possibilities of exploration can be further extended, using common morphing applications, by considering how different are disparate symbols in the light of how many topological operations are involved in morphing one into the other as part of the virtual experience. How can such structures be "collapsed" and "expanded", as is now familiar with use of map zooming facilities?

The Kabbalah also provides a valuable example because of the amount of work done on it, or inspired by it, from a mathematical perspective. Thus there are many references combining Kabalah and topology. Of particular relevance, in the light of the cybernetic arguments above, is the work of Gabriel Burstein and Constantin Virgil Negoita (Foundations of a postmodern cybernetics based on Kabbalah, Kybernetes, 2011; A Kabbalah System Theory of Ontological a nd Knowledge Engineering for Knowledge Based Systems, International Journal of Advanced Research in Artificial Intelligence, 2013; Kabbalah Logic and Semantic Foundations for a Postmodern Fuzzy Set and Fuzzy Logic Theory, Applied Mathematics, 2014).

To what extent does such thinking apply to the Pentagramma Mirificum? More intriguing is the possibility that the class of symbolic structures of this kind -- purporting to organize fundamental knowledge -- may lend themselves to even more generic exploration. Consideration could for example be given to the nature of the relationship between the following, especially when understood as potentially indicative of dynamic cognitive processes:

• icosahedral and related patterns, as notably explored by Stafford Beer to dialogue processes, and applied to Christopher Alexander's transformations
• topological approaches, as notably applied to the Kabbalah, and to depictions by various cultures of the Tree of Life and the Flower of Life (Andrew Monkman, Sacred Geometry - Flower of Life, 2007), as discussed with respect to Recognition of cultural icons within a pattern of nested circles? (Eightfold Configuration of Nested Cycles of Cognitive Transformations: meta-pattern of connectivity through a hypersphere? 2012)
• patterns of systemic integration implied by the genetic code and the associated set of 20 amino acids, as notably explored by Chi Ming Yang (The naturally designed spherical symmetry in the genetic code, 2003; On the 20 canonical amino acids by a cooperative vector-addition principle based on the quasi-20-gon symmetry of the genetic code. 2003). The 16 genetic code doublets and their 20 cognate amino acids are fitted into a polyhedron model. Based on the structural regularity in nucleobases, and by using a series of common-sense topological approaches to rearranging the Hamiltonian-type graph of the codon map, it is identified that the degeneracy of codons and the internal relation of the 20 amino acids within the genetic code are in agreement with the spherical and polyhedral symmetry of a quasi-28-gon (an icosikaioctagon). Two evolutionary axes within the 20 standard amino acids are suggested. Further relationships have been noted by Tidjani Négadi (A Connection between Shcherbak's arithmetical and Yang's 28-gon polyhedral "views" of the genetic code, Internet Electronic Journal of Molecular Design 2003)

The question meriting exploration is why the patterns of numbers cited seem to prove so valuable to systemic integration. The recognized significance of 20-fold patterning are exemplified, for example, by the "groups of 20" of global governance, and the "twenty-fold way" in combinatorics (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). There is the implication that the amino acids are better understood as implying a mode of process organization intimately associated with metabilic pathways for which cognitive analogues are yet to be fully recognized. The fundamental issue is how "cognitive metabolism" is embodied and how "cognitive metamorphosis" is enabled -- issues explored by René Thom (Structural Stability and Morphogenesis, 1972; Semio Physics: A Sketch, 1990).

A generic approach to global organization of knowledge and cognition would seem to follow from references to generalization of Napier cycles and Hamiltonian cycles in relation to the particular instances cited (Eric W. Weisstein, Hamiltonian Cycle, MathWorld; Johannes Böhm, Generalized hyperbolic Napier cycles and their hyperbolic kernels, Jenaer Schriften zur Mathematik und Informatik. 2008; Hans-Christoph Im Hof, Napier cycles and hyperbolic Coxeter groups, Bulletin de la Société Mathématique de Belgique, 1990). The concern of the latter with "negative curvature" might, for example, prove of value in addressing the strategic issues of so-called wicked problems (Jon Kolko, Wicked Problems: Problems Worth Solving, 2012). Of particular value to psychosocial application is the cited work work of Stafford Beer (1994) and Christopher Alexander (2009).

In this respect, given the bloody conflicts engendered by what may be "superficial" understanding of elusive insights, the main paper indicated a case for exploration of mathematical theology, as separately argued (Mathematical Theology: Future Science of Confidence in Belief: self-reflexive global reframing to enable faith-based governance, 2011).

Given the nature of the immersive experience in virtual reality, how might belief be evoked and sustained by experiencing symbol structures in this way? How do the elements of the structure serve as vehicles (containers) and conduits (pathways) for belief and conceptual processes -- especially when framed by metaphor?

Indicative possibilities are offered by the work of Steven M. Rosen (Topologies of the Flesh: a multidimensional exploration of the lifeworld, 2006; Dimensions of Apeiron: a topological phenomenology of space, time, and individuation, 2004; Dreams, Death, Rebirth: a topological odyssey into alchemy's hidden dimensions. 2014).

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