Functional dynamics of a 16-fold configuration of strategic goals

Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (Part #10)

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The emphasis above is with respect to the challenge of representing a 16-fold complex in a memorable manner which enables enhanced comprehension of its coherence. Use of 3D polyhedral forms offers a valuable contrast to the conventions of 2D. The constraints of the printed page, most notably in academic and strategic documents, could be said to be an unfortunate reinforcement of the traditions of "Flat Earth" comprehension of what is otherwise upheld as being of a global nature. The point could be stressed more emphatically in that, if information is not presented "flat", it cannot currently be communicated, comprehended or considered to be of any relevance to global governance.

It is however one thing to achieve a more credible mapping, and another to shift beyond the static implications of such a mapping -- in a context which is inherently dynamic, as argued separately (Dynamic Transformation of Static Reporting of Global Processes: suggestions for process-.oriented titles of global issue reports, 2013; From Statics to Dynamics in Sustainable Community: navigating through chaos by playing on polarities as attitude correctors, 1998).

A dynamic perspective can notably be explored in two ways in relation to the dual of the simplest torus -- with respect to the faces, and with respect to the edges.

Using polyhedral edges as indicative of feedback loops: Use of Stella Polyhedron Navigator indicated that a pattern of edges was associated with an embedding of the star torus within a set of 6 great circles (shown below). Omitting the faces, four of these circles are slightly oriented to the principal plane of the torus, coloured below according to 4 rhombic "circuits" of edges (rendered more pronounced as cylinders in contrast to the representations above).

In the screen shots below, small spheres circulate around each such circuit -- coloured according to the circuit. Two additional circles (coloured green below), out of the plane of the form, are orthogonal to each other. What could be considered a 7th great circle would lie in the plane of the form, passing through the four points most distant from its centre.

Movement of spheres on rhombic circuits of edge cylinders of dual of simplest torus -- a "star torus" (with indication of great circles)
Screen shot	Rotation	Screen shot
Animations adapted from Stella Polyhedron Navigator using X3D-Edit

Such dynamics then raise the question as to what circulates in this way around the polyhedral form, and what speeds of movement of the spheres would be most suggestive of that (Circulation of the Light Essential metaphor of global sustainability? 2010). How should those movement be phased in relation to one another -- possibly with multiple spheres for each circuit? Clearly the aesthetics of the representation can be variously adjusted (preferably interactively) to suggest different possibilities.

What feedback processes are required to ensure the integrity and viability of a 16-fold set of strategies? How do these relate to the functional considerations of the viable system model? The image above restricts the movement of each sphere to the rhombic circuit with which its colour is associated. An alternative possibility would be to allow the spheres to move between those rhombic forms.

Of some interest, is the manner in which the 4-edged rhombic forms are paired, therefore together constituting two sets of 8 edges. The emergence of each rhombic form resulted however from treating together the two colinear edges composing each of the rhombic edges in the original form (one shorter than the other). Understood in that way, the two sets of 8 edges imply double that number, namely 32 (as 4x8), which contrasts with the geometrical definition where such discontinuity is not be a feature of the base polyhedron (as noted above). This approach ignores the edges passing around the narrow radius of the torus, and aligned with the great circles coloured green.

Using polyhedral faces as suggestive of flow processes: Distinctive flow processes can be envisaged, of which those presented in the animations below are some examples. Note, as mentioned above, the separation marked by the white vertex is an effect of the first stellation of the form and (although indicative) can be considered as absent from the base polyhedron.

In the animation on the left, a flow is envisaged as emerging from the point of convergence of the faces at the point marked by the white vertex -- or sinking into that point. On the other hand, it is assumed that the flow is continuous across the edges at the junction of the faces between the yellow vertexes. The same would occur on the lower set of faces (not visible in the image on the left).

In the animation on the right, it should be recalled that the face visible on the inner side is continuous with that on the outer side, and that parallel faces are similarly coloured. There is therefore assumed to be a continuous flow from inner side to outer side (or vice versa) through the intersection between the two white vertexes. Consideration could be given to the source of that flow at the other end of the face, on the inner or outer side.

Animations suggestive of flow processes over the surface of the dual of the simplest torus -- a star torus
Possibilities on "upper" and "lower" set of faces?	Possibilities on "outer" and "inner" sets of side faces?
Animations adapted from Stella Polyhedron Navigator using X3D-Edit

One indication of such a hypothetical flow process, at the point of convergence, is offered by the following animations (left and centre) using a horn torus. (Further programming would be required to indicate the process as suggested by the arrows above). In relation to the horn torus, Wolfgang Daeumler comments on the presentation of Lissajous curves on the surface (below right), that the visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. The animation below is somewhat reminiscent of the hypersphere animations suggestive of higher-dimensional brain functioning, as discussed separately (Transforming vehicles of identity between global and toroidal forms, 2016).

Horn torus flow-related animations
Flow sinking into convergent point	Flow emerging from convergent point	Animation of Lissajous curve
Animations developed with Antiprism by Adrian Rossiter (reproduced with permission)		Reproduced, with permission, from Wolfgang Daeumler (Horn Torus)

Of further interest is how the various flows might be understood as continuous across the torus as a whole. Other colour effects could be used to suggest shifts in phase at the various "source" and "sink" positions as points of inflection. From a strategic perspective, what indeed flows to engender strategic integrity? Do the various points of "inflection" represent zones of "concentration" or "compression" (through "pumping") or filtering -- of information or energy?

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus. This is related to understanding of a completely integrable system.

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