The Soliton Web

The concept of the Soltion appears in many different guises. In a chaotic system, at the point of bifurcation, new components emerge and are revealed. If we think of each component as a separate whole unto itself, we obtain a complex dynamic structure, non-linear in nature, enjoying negative feedback, and geometrically modeling both a phase-state snapshot across scale (fractalized composition series [Group Theory]), and, taking a vertical slice, a model of continuous cooperative evolution.

What happens at the boundary or membrane of the Soliton may be the most important focus for examination. This boundary is chaotic in the sense that both order and disorder occupy, reside in, the same spacetime at the interface between dimensions of scale.

Probably the most significant characteristic of the Soliton is that it contains an 'infinite' number of sub-components (frequencies) interdependently relating; and, if a single frequency were to suddenly be removed, the entire edifice, the wave itself, would lose cohesion and integrity and dissipate. They form, in a sense, a nondivisible set of basis elements for the Soliton.

Forming larger groups from the products of simple groups (discrete, separate, nondivisible) covers completely the factorization of groups of all possible permutations. Accordingly, we can relate across dimensions of varying scale however we may wish for purposes of scrutiny. Similarly, nonlinear dynamic systems and structures are the fundamental constituents and building blocks producing natural processes, pattern is basic. Such systems can also model social, economic and psychological processes, also across dimensions of scale.

Mechanistic, Newtonian images still relied on to a great extent in order to base predictions and describe events; apperceptions of how we, living in absolute time coupled with homogeneous and empirical space, express and by which we understand the dynamics of world; ignore spontaneity and the natural propensity for disorder.

Does chaos order disorder or does order break down into disorder at a certain critical level or point? Which came first, the chicken or the egg?

In a totally symmetric system, all dimensions of phase-space (which ordinarily consists of six, three of position and three of momentum) are reduced through slices or shells to a single point. From a topological perspective, considering the theory that speculates the Universe to be composed of ten dimensions, these six are compacted into what is called a ball. At infinite density, maximum compactification, this ball shrinks to a single point, a singularity, dimensionless, until the tendency towards chaotic behavior effects asymmetry.

Spontaneous symmetry breaking creates the stage for a categorizing of previously unrelated phenomena by affinity, predisposed orientation, and the magic of self-organization. A singular, ten-dimensional ball may very well be the well-spring of all that is manifest.

The concept, Infinity, contains the sub-idea of density. Density sub-divides infinity into something resembling a composition series of fractals, a dimensionally scaled hierarchy of self-contained, nondivisible and nested universes, each, in series, a rearrangement of the elements of its parent. The identity integrity of these separate layers orients, like an axis, an infinite regression [progression?] of overlaying homomorphic images. This factoring is what gives infinity its spacio-temporal characteristic.

At the sound-to-noise ratio of S/0, all phase-space points of information merge into one. On closer inspection however, increasing magnification, this 'ONE' reveals 'holes' of noise contained throughout (like the irrational numbers scattered through the reals). Chinese box-like, within these holes we find signals or forms of pure sound. As we plumb more deeply, we find that the nested structures continue, resembling a fractalized composition series construction elucidating areas of irrationality and disorder, of 'quantumly' differentiable dimensions.

We 'move,' consciousness-wise, from unity through form/other to integration; Or, from Self to self to Self/self. Paralleling this progression [in fact, an identity of aspects] dimensionless ball expands to boundary/environment, then to the identification with this boundary or surface (chaotic interface) of the sphere containing the ball. As the end result of this process, the boundary, border, membrane transforms its definition and intent from static 3-D barrier to living, dynamic, four-dimensional relationship, the very beginning and end of time and infinity as 'now-here.'

At zero density the self has no substance or form. In the third dimension there is a sense of self and of other; there are definite boundaries of which we can be and usually are aware. But at 'infinite density' there is a breakthrough to a sense of self defined by this mutual blending of 'self' and 'other' from one point of view, and an awareness of being transformed and seamlessly connected to the whole from another. Global topological properties or features common to the whole induce, foster, a mapping to the kernel, the singularity, the axis, from the overall group complex. This is a nonlinear Soliton.

But what does 'infinite density' mean in this context? It could represent the mapping between self and other; so complete that there is no 'discernable' boundary. Or, by compactification (the finite covering by a set of open [closed] point-sets contained in the topology), the six dimensional ball as symbol of some unformed self loses all specificity and becomes a collective 'point' of or in a four-dimensional psychic field.

Why is the idea 'density' so important here? When I think of the density of something I imagine the closeness of separate geometric points. At infinite density the points are squeezed together until the possibility that there may have been holes lurking (local neighborhoods are not allowed) in the respective space is eliminated; as is any sense or definition of separateness. If this space is a phase-space, then the melding of all content/border will embody the life of the system under examination, dynamically apprehended.

Furthermore, what does it mean for a whole of a life, in relation to its parts, to be a dimensionless point of 4-D phase-space? Statically, it means little to nothing, there is no life. But, as a dynamic, this life is expandable to the further reaches of its potential limited only by the demands of nonlinearity. Without the inherent tendency for asymmetry, all naturally occurring systems, and some not so natural, would not and could not be. In the act of asymmetry, we have the generator of phase-space.

A topological open-ball has, at least, one significant aspect which is counter intuitive-- an open-ball, in topology, has no surface. It consists of simply-connected points comprising the interior of a sphere, but not the surface or any of the boundary points; this would necessitate the closing of the ball to include what is called the 'frontier.'

Imagining a solid ball without a surface is not possible, for me anyhow. What we have to do in such cases is to bring in 'set theory.' Set theory possesses the tools for defining, by the use of boolean operations, whether or not a particular set of points lies in the interior, surface, or exterior of a specific solid.

The interior of a simply-connected open curve is the set of all points which make it up, excluding the two endpoints which represent the boundary. A closed curve, for example, a circle, therefore, has no boundary points. Similarly, the closed connected surface of a sphere has no boundary. The surface, a two dimensional creature demanding three dimensions to define itself topologically, bounds the interior ball. As a three-dimensional shape in space, it can be represented as a particular solution of a four-dimensional differential equation describing and depicting the state of this phase space in time. It generates the interior of the fourth from a singular point of view. And, from the subsequent 4-D 'boundary' of a hypersphere, both interiors, of the third and fourth dimensions, are 'seen' to merge. Equivalently, it might be said that together they generate this chaotic boundary.

Bringing the argument down a notch may make it easier to see. The make-up of the interior of a bounded three-dimensional sphere can be thought of as an infinite set of 2-D discs (any one of which is sufficient to generate that interior). These two interiors, that of the solid sphere's and the discs taken as a set, from the point of view of the boundary, merge and interpenetrate, becoming indistinguishable.

In terms of the fourth dimension, identification with this extended domain necessarily and by definition localizes three dimensional solids allowing them to be considered only as surface or form.

In other words, from this point of view, three-dimensionality itself is 'seen' objectively. As the fourth dimension is one of time, the vector normal, or orthogonal, to the third dimension corresponds to the integration of the fourth's interior ball with time point-sets, the domain of the fourth dimension.

The solid 3-D object, now viewed only as form, takes on substance and meaning by virtue of its 'global' properties, while yet maintaining a 'local' identity peculiar to its uniqueness in time and space. It is as though the object is a nondetachable aspect of the foreground against a background [a 'background' that penetrates and vitalizes the 'foreground'] of universal geometric features, and by 'geometric' we mean 'invariant,' defining the kernel, the identity. These local properties are now understood in terms of the fourth dimension, through this lens, orientation across scale being complete.

What does all this have to do with the soliton dynamic, one might ask? A soliton wave resists dispersive disequilibrium, its asymmetric proclivity to break into individual frequency constitutents, through internal negative feedback and surface tension. Instead of considering it in terms of these interactive and interdependent components, we need to look at it through a collective lens as a single integrated entity.

For there to be growth, however, there must be this asymmetric urge towards a break-up of existing conditions, the essential kernel remaining intact through the transition, resolving to a reorganization and reconfiguration more complex and yet simpler than before, pruned and strengthened, invigorated, quickly regaining a soliton dynamic and form.

With organics these two processes, disintegration and reintegration, happen 'almost' simultaneously and continuously, asymmetry being the engine and generator, symmetry responding to coalesce and give intelligible and cohesive shape and meaning, Disorder/Order at the border.

If we take the collective to be represented by the third dimension, and the dynamic of the whole by the fourth, we have a fairly reasonable model of the Web. A four-dimensional hypersurface, internally interdependent and self-organizing, each component of which essential at any given moment in time, yet subject to asymmetric spurts of growth and reorganization to a new and more complex form. A network of networks, pattern being basic, each network emulating a soliton wave, self-similar across varying scales, interconnected, enjoying negative feedback and undergoing positive amplification, symmetry breaking, growth and reformation, this is the Web of elements, a spider's web, as 'seen' from the perspective of an organic whole, and defined in terms of a 'four-dimensional hypersurface.'

The Web is not just an assemblage of linked objects located at myriad points on the Net, it's a true worldwide community of in-dividuals, a Soliton Web.

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Owner and Operator: Adrian T. Dorn;
copyright: ©

February, 1995,