From "periodic table" to "periodic container"

Eightfold Configuration of Nested Cycles of Cognitive Transformations (Part #3)

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Given the above argument with respect to metaphors derived from the electricity and electronics on which the current knowledge-based global society is dependent, it is appropriate to recognize that electrons have been understood as "organized" by that society into a variety of "elements". Each element is a complex configuration of electrons represented simplistically as orbiting a nucleus -- with such orbits better understood as forming "electron shells" whose spherical nature is a challenge to the conventional imagination. The elements have of course been conceived as organized into a periodic table.

In a world focused on tangibles, none has seen an electron, however their nature and existence is inferred. Few can be said to have seen many elements in their pure state. As noted above, the periodic table is an inspiration in the quest for a meaningful ordering of transformations -- perhaps to be recognized as being as intangible as electrons. It is especially significant in that it is considered to be one of the most comprehensive generalizations of science. In more general terms, however the focus here is on a suitable "pattern container" -- of which a table is but one possibility (and potentially problematic for reasons noted below).

Also of interest is the quality of representational coherence commonly associated with readily remembered symbols. It was for this reason that the geometric combination of the Lauburu and Tao symbols was explored in a section of the preceding paper (Modulating cognitive transformations: electrical metaphors and semiconduction, 2012). This gave rise to the following images illustrating their integration.

The approach there was related to mappings onto a hyperbolic sphere, whether represented in 2D (as the Smith Chart) or the generalized 3D variant (cf. Madhu S. Gupta, Escher's Art, Smith Chart, and Hyperbolic Geometry, IEEE Microwave Magazine, October 2006). The 2D version is one of the most widely used charts in electrical and electronic engineering. The chart is most frequently used at or within the unity radius region. The remainder is however mathematically relevant, as recognized in oscillator design and stability analysis, for example. It can be used to represent many parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The chart is plotted in 2D on the complex reflection coefficient plane and is scaled in normalised impedance, normalised admittance or both, using different colours to distinguish between them.

The complexification here of the image (as explored in relation to electrical engineering) is a consequence of mapping a smaller version of the larger image into the branches of the larger image -- recursively engendering a fractal form. Two such recursive mappings are illustrated in the following

Superposition of a reduced version of the geometrical construction of the Lauburu as a whole within each branch of the Lauburu (as with a fractal)	Shading of the geometry of the Lauburu (left-hand image) to highlight one orientation of the Tao symbol and emergence of its traditional "eyes" from the geometry

As the shaded image on the right illustrates, where a figure of eight pattern can be recognized, the Tao symbol can be recognized -- although at different scales.

The speculative exploration here derives from the assumption that a pattern of eight cognitive transformations can be recognized -- at different "scales" -- as a form of "eightfold way". The focus is on the suggestion that, through the cognitive implications of metaphor, the basic electrical functions above constitute such a pattern. This is informed by the further assumption that these are more tangible expressions of cybernetic functions of greater abstraction -- as required by the Viable System Model. The assumption derives from the question: how many kinds of component are required for a viable electric circuit?

Such a question relates to that with respect to team building, as with the Belbin Team Inventory developed by Meredith Belbin. The team roles so identified are not equivalent to personality types, and unlike the Myers-Briggs Type Indicator (which is a psychometric instrument used to sort people into one of 16 personality types), the Belbin Inventory scores people on how strongly they express behavioural traits from 9 different team roles. A person may and often does exhibit strong tendencies towards multiple roles.

To the extent that these assumptions are credible, the question is then how the above image might serve to "hold" this eightfold pattern -- and how it might indicate how any articulation of these cognitive functions might also be held together with(in) them. As a "container", the issue is how these more specific functions might be "packed" within the geometry, namely how the geometry might be filled. In this connection, as stressed in the preceding paper, the concern is with cognitive transformational dynamics (as experienced) -- rather than classification of the functions as categories or styles (as descriptions).

In the following discussion, the patterning constraints of the standard periodic table -- conventionally conceived as blocks of variously completed orbital electron shells -- are used to guide eightfold "cognitive" attributions in the mapping process. It is appropriate to note that an eightfold pattern has long been recognized as fundamental to the organization of the periodic table.

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