Eliciting the beauty of the fine structure of the Monster

Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks (Part #14)

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Another set of clues might be provided through alternative understanding of the operational "slices" that make up the structure of the Monster and through which its structure was determined. If the Monster is typically only understood partially, whether intuitively or through particular ways of knowing, then it might be expected that these would be more attuned to particular slices. This suggests that slices might be understood as forms of engagement with reality -- the categories or modalities through which reality is articulated within a particular cognitive framework. In this sense the various examples of "concept sets" indicated in Table 2 suggest how slices might be determined in terms of factors -- given the crude comparability with Table 1.

The coherence of such "slices" may be usefully explored in relation to the Poincaré sections common to nonlinear dynamics, as in the work of Helwig Löffelmann (Visualizing Local Properties and Characteristic Structures of Dynamical Systems, 1998; Visualizing Poincaré Maps together with the Underlying Flow, 1998).

Much is made of the skill of number theorists in recognizing "interesting numbers" -- and deriving considerable pleasure from such recognition. Less publicized is the analogous pleasure in "interesting shapes", or "interesting patterns" -- presumably derived by topologists and those mathematicians with skills in spatial representation and complex geometries. To the extent that the relations of number and group theory can be transformed into such representations, there is clearly scope for seeking some form of geometrical analogue to the different sporadic groups in terms of axes of symmetry. Ronan describes the Monster as a snowflake in 196,884 dimensions. A difficulty is that any "geometry" may be described analytically rather than graphically (cf Alexander V. Ivanov and S. V. Shpectorov, Geometry of Sporadic Groups, 1999). An, noted earlier, exception is the visual representation of a hyperbolic plane.

Earlier reference was made to the discussion elsewhere (Systematic Visual Representation of Musical Possibilities on an Orbifold, 2007) of the use of an orbifold (by Dmitri Tymoczko) as a means of ordering musics, especially in the light of a recognized relation of orbifolds to the Monster (Michael P. Tuite, Monstrous Moonshine from Orbifolds, 1992) and Conway's own involvement in orbifold notation.

Of interest therefore is the possibility of generating music -- as "interesting sounds" -- from the factors describing individual sporadic groups as a method of obtaining another form of insight into them. As noted above, this would be consistent with the work of Ernest G McClain (The Myth of Invariance: the origins of the Gods, Mathematics and Music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978). who notably focuses on the implication of such factors. His work was used as one of the examples in Table 2 (Tonal patterns of Rg Veda poetry). As noted by McLain (The Myth of Invariance, 1976) with respect to the study by Antonio de Nicolas (Meditations through the Rg Veda: four-dimensional man, 1978) :

The four Rgvedic "languages" de Nicolas defines have their counterparts in the foundation of all theories of music. His "language of Non-Existence" (Asat) is exemplified by the pitch continuum within each musical interval as well as by the whole undifferentiated gamut -- chaos - - from low to high. His "language of Existence" (Sat) is exemplified by every tone, by every distinction of pitch, thus ultimately by every number which defines an interval, a scale, a tuning system, or the associated metric schemes of the poets, which are quite elaborate in the Rg Veda.

The "language of Images and Sacrifice" (Yajna) is exemplified by the multitude of alternate tone-sets and the conflict of alternate values which always results in some accuracy being "sacrificed" to keep the system within manageable limits. The "language of Embodied Vision" is required to protect the validity of alternate tuning systems and alternate metric schemes by refusing to grant dominion to any one of them". (21, p. 3). The embodiment of Rg Vedic man was understood... as an effort at integrating the languages of Asat, Sat and Yajna to reach the dhih, the effective viewpoint, which would make these worlds continue in their efficient embodiment (17, p. 136).

In the light of the work by Dmitri Tymoczko (The Geometry of Musical Chords, Science, 7 July 2007), might sporadic groups correspond to quite different styles of music or tuning system? In the light of the role of the "organ" as a musical instrument that metaphorically inspired conventional approaches to"organization", is there a possibility that the orbifold approach might lead to an "organ-ization of knowledge" sensitive to musical harmony? Such was indeed the implication of the magnum opus of Nobel Laureate Herman Hesse (The Glass Bead Game, 1943).

Given the recognized potential of sonification (discussed elsewhere) in enabling the human mind to recognize patterns that are otherwise challenging, it is therefore interesting to consider how factorized sporadic groups could be represented through parameters of sound and music beyond the indications of Tymoczko and McLain. McLain explores a number of possibilities. Can particular musical properties (tone, rhythm, beat, etc) be significantly associated with:

• the distinct prime number factors?
• the exponents of these factors?

Is it to such patterns, implicitly associated with sporadic groups, that music enthusiasts worldwide have long been attracted?

A classic approach to such matters is through number symbolism (cf Marie-Louise von Franz, Number and Time, 1974) which despite numerous reservations (regarding numerology) remains a major factor, even in stock market trading. One effort to integrate the implications of such insights is reflected in a study associated with the work on Table 2 (Distinguishing Levels of Declarations of Principles, 1980) which endeavoured to highlight the comprehension challenges and possibilities associated with each of the numbers from 1 to 20.

Especially interesting in the technical distinctions between the sporadic groups is the concept of a "cycle" which has many different connotations in mathematics. But from the perspective stressed here, of great interest is how an individual may comprehend and identify with sets of interlocking cycles that might be expressed musically (cf Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007). Extreme examples of efforts to express complex integrated wholes through music are works like:

• Richard Wagner's The Ring of the Nibelung (1848-1874)
• Johan Sebastian Bach's Goldberg Variations (1741)
• International Alphabet of Sanskrit Transliteration Mahabharata and Ramayana

Given the central role they have played in a culture over an extended period, and the mathematical interpretations to which they have led, several classical Chinese texts explicitly concerned with a representation of the whole might also be considered as offering insights into how the Monster can be comprehended (9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2007; Mapping Songlines of the Noosphere: use of hypergraphs in presentation of the I Ching and the Tao te Ching, 2006; Hyperspace Clues to the Psychology of the Pattern that Connects: in the light of the 81 Tao Te Ching insights, 2003; 9-fold Higher Order Patterning of Tao Te Ching Insights, 2003).

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