International Encyclopedia of Systems and Cybernetics

A possible general representation of 3-dimensional phenomena produced by field dynamics (Adapted from D. Mc NEIL).

Helical dynamics are the result of interactions between fields, as in vortexes. In abstract, a torus should be a specific topological attractor for this kind of movements. However, in many complex systems, the imbricated character of their internal organization needs be represented by more complex forms, including toroids containing toroids.

Mc NEIL generalizes the topological torus, using fully its dynamic character, that results from the complementary opposition of two types of displacements.

And: "… toroidal contours represent dynamic, connected volutions rather than the separate, rigid structures or the hermetically encapsulated nuggets found in so many conventional worldviews" (1993 a, p.15).

Or: "Toroids map the conceptual space into comprehensible orders without resorting to artificial linear dichotomies or rigid arrays of pigeon holes" (1993 b, p.14).

Mc NEIL states: "Natural as well as artificial toroids can be generated in many ways other than by the rotation of a plane figure… Toroidal forms can be traced by continuous spirals of helical dynamics" (1993 b, p.9).

As to the toroidal dynamics: "In the context of a simple toroidal whole, the center is the region closest to equilibrium in the sense that the meridial as well as the annular fluxes are most concentrated there, much as a whirlpool captures and holds what would otherwise drift along in a stream. The periphery of the toroid is farthest from equilibrium, with relatively diffuse currents and cross-currents over relatively broad surfaces" (1993 b, p.16).

Mc NEIL offers the tree, as a general graphical and dynamic representation of a toroid. This is at the same time a good homomorphic model of real trees and a metaphorical model of any system which obtains inputs from an environment, metabolizes them and produces outputs. This tree model is widely different from the classical tree model of hierarchical organizations, or a computer program: It is not merely the representation of a structure, but altogether one of the ongoing process or processes.

Any living system seems to be basically a toroid, and this model also fits stellar systems, hurricanes and possibly ecosystems and social systems.

It should be taken into account that toroids become easily unstable.

The now rediscovered toroidal model seems to have been first introduced by C.L. WEYHER as early as 1887! (as quoted by C. LAVILLE, 1950, p.54).

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