Possible resonant insight into the gross structure of the Monster

Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks (Part #12)

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On the assumption that insight into the structure of the Monster is both possible and potentially even a daily reality -- embodied (as it is thought to be) into the very structure of the universe -- how might traces of it be recognized? Is there a sense, intuitive or otherwise, through which the gross structure of the sporadic groups is recognized?

One set of clues would naturally be expected to be associated in some way with the gross structure of the Monster in relation to the 25 other sporadic groups. The set of such groups might, for example, be suggestively reclustered in various ways:

• Monster plus six other groups. As noted, six of the groups do not form part of the Monster. The others are seen by some mathematicians, including Robert Greiss, as constituting a single "Happy Family" (Griess would like the Monster to have been termed the "Friendly Giant"). The six are considered to be "pariahs". If only from a mnemonic point of view, this group of seven can then be variously associated with different 7-fold symbol systems.
  • Chakras: Potentially of great interest for example, is the Hindu chakra system where the distinct multi-petaled representation of each chakra might be seen as a collapsed representation of each of those 7 sporadic groups -- with the Monster represented by the crown chakra. The latter has 1000 petals which are arranged in 20 layers, each of them with 50 petals. Just as efforts to represent the Monster have been considered in relation to the hyperbolic plane, the various visual models of the hyperbolic plane bear a striking resemblance to those of the crown chakra. Any chakra equivalence would raise interesting questions regarding the associated challenge to comprehension in disciplines of yoga.
  • Cosmological pantheons: the embedding of 19 groups within the Monster, and the 6 associated deities, might be understood in terms of a pantheon engendered from a supreme deity (such as Zeus)
  • Fallen angels: Given the notion of "pariah", there are possible associations with the set of fallen angels common to the Abrahamic religions. Intriguingly, given the size of the sporadic groups, the term derives from the translation of "giants" from biblical Hebrew. There were giants in the earth in those days... (Genesis 6:4); And the angels which kept not their first estate, but left their own habitation... (Jude 1:6). It would be most amusing if the set of such angels suggested some kind of mapping of the sporadic groups -- as mathematical theology might expect.
• Metabolism: Assuming an intimate correspondence of the sporadic groups to life processes, it is therefore intriguing to consider their possible relationship to:
  • the set of 20 standard amino acids, as directly encoded for protein synthesis by the standard genetic code (Examples of Integrated, Multi-set Concept Schemes: Genetic code and amino acids, 1984). They are used in building proteins by living cells. 9 are considered essential and required in the diet.
  • the set of vitamins required by humans from external sources, now recognized as being composed of 13 (4 fat-soluble and 9 water-soluble). However a further 13, previously included, have been reclassified over time, discarded as false leads, or renamed because of their relationship to "vitamin B", which became a "complex" of vitamins. The term vitamin does not include other essential nutrients such as dietary minerals, essential fatty acids, or essential amino acids.
• X-ray diffraction: Given that simple finite groups can be understood metaphorically as "atoms of symmetry" (as suggested by Mark Ronan), especially interesting is a possible analogue to the X-ray crystallography techniques through which complex molecular structure is determined. The technique of X-ray crystallography has three basic steps.
  • obtaining an adequate crystal of the material under study -- sufficiently large, pure in composition and regular in structure, with no large internal imperfections.
  • placing the crystal in an intense beam of X-rays, usually of a single wavelength (monochromatic X-rays), such as to produce a configuration of spots called reflections. As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded at every orientation. Multiple data sets may be collected, with each covering a full rotation of the crystal and containing tens of thousands of reflection intensities.
  • combining these data sets computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal.
  The detection of principles, and other organizing fine structure in concept schemes, may possibly occur through the (intuitive) application of some such "X-ray" "brilliance". In effect the challenge is to "X-ray" one's own cognitive structure -- as the crystal -- and to combine the pattern of insights from different angles (to the extent that diffraction patterns have been adequately resolved). This degree of self-reference is consistent with that indicated by Douglas Hofstadter (I Am a Strange Loop, 2007).
  
  Alternatively the "amino acids" and "vitamins" on which psychocultural life is dependent may be derived by exposure to the kind of variety of perspectives implied by diffraction patterns in the X-ray metaphor. Expressed otherwise, there is a "psychic" need for variety greater than that associated with a simple grid pattern (characteristic of conditions of incarceration).

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