Potential contribution of symmetry group theory

Dynamics of Symmetry Group Theorizing (Part #13)

Symmetry and group theory, as explored by mathematics, derive initially from visual observations -- namely using the sense of vision. This has led to the recognition of the symmetry associated with forms such as the triangle, the square, the pentagon, the hexagon, etc. This understanding of symmetry in 2 dimensions has been extended into 3 dimensions through combining such forms into the tetrahedron, the cube, the octahedron, the icoshedron and the dodecahedron, to name only the Platonic polyhedra. Through group theory forms of symmetry have been explored in 4 dimensions and many more.

Whilst such forms are basic to patterns appreciated for their aesthetic qualities, most notably in architecture, they seem to be irrelevant to qualities that are appreciated in terms of the sense of sound, taste, smell, and touch. The point to be made in what follows can however be introduced through an aspect of vision, namely colour, through which the aesthetics of patterns may be distinguished.

Clearly it is possible to position 3 colours at the vertices of a triangle, or 4 at the vertices of a square, etc. Well chosen these may be appreciated as complementary, offering an aesthetic effect. The question is whether selections of colours offering an aesthetic benefit through their distribution onto more complex symmetrical forms in 2 dimensions or 3.

Generalizing from this case, consider use of the same approach to select and distribute:

separately: sounds, tastes, smells or textures to offer aesthetic experiences in each case. Clearly it is no longer a question of distributing them onto a visual form (a triangle, a square, etc) but rather to have three sounds (for example) together -- namely a chord. Or three tastes, etc.
together: a sound, a taste, a smell or a texture to form a complementary set (of 3 or 4, or more) -- namely engaging the different senses together

The examples given in each case are simple and rely to some degree on familiarity with the visual example by which they were introduced -- to give a sense of their complementarity through that symmetry. However, there is no reason why the argument should be limited to these simple cases. The qualitative descriptors used in the appreciation of wines (tastes) or perfumes (smells) are far more complex.

Group theory enables much more complex patterns of symmetry to be explored and distinguished by a suitasble notation system -- beyond the capacity for them to be visualized, for example. The questions to be explored are:

whether such a pattern description tool can be used to distinguish patterns of taste, etc -- or combinations of taste, smell, etc
whether the patterns so distinguished lend themselves to recognition as aesthetic experiences through the other senses, even though the pattern denoted cannot be visualized -- accepting that odour receptors are, for example, capable of distinguishing an extremely wide vartiety of odours
whether even more complex patterns, engaging the different senses, can be recognized aesthetically

Following from the preoccupations of group theory, associated questions (of possibly quite different degrees of significance) might include:

the nature of any cognitive relationship between the simple numbers (1 through 5) basic to the simplest polygons (and polyhedra), the similarly limited number of senses (1 through 5), and constraints of human memory ("7 plus or minus 2")
the types of structure emerging beyond the simple binary relationship, and its potential cognitive and qualitative analogues -- hot-cold, light-dark, etc, although fundmental to distinctions, are not especially interesting qualitatively in themselves
any special qualitative signifiance to the triangulation of qualities (namely beyond the binary) and the role of such triangulation in more complex polyhedra (of qualities)
degrees of symmetry in relation to asymmetry, notably from an aesthetic perspective (as illustrated by music) in which many forms of pure symmetry are uninteresting (or do not hold the attention); how does the complexity of design (Christopher Alexander) overcome such constraints through appropriate balance
the association of symmetry with a degree of "robustness" of structure that may well have its qualitative analogue
the qualitative implications of symmetries in the most challenging mathematical objects, as extolled with respect to the visual form of the Mandelbrot fractal and potentially implied by the Monster of group theory (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005; Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007)
what insight is offered into the possibilities and implicstions of multi-media experiences and synaesthesia

These questions aside, to what degree can group theory contribute to qualitative experience of subtler aesthetic experiences, notably in a web environment? Would enabling such possibilities dispose people to an understanding of psycho-social possibilities of relevance to the challenges of the times?

Main menu

Directions

You are here

Main menu

Search form

Directions

You are here

Potential contribution of symmetry group theory