The Society for the Diffusion of Knowledge P.O. Box 964, Kaunakakai, HI 96748

  The Induced Displacement Due to Interchange, abbreviated IDDI, is an instantaneous lateral displacement of surfaces as they adjust their position and curvature to the outward migrating ring of non-definity.
    The cause of this displacement, a movement above and beyond the normal motion of the surface is a matter of conflicting geometries at the exact location of the migrating ring of non-definity. Here, as the ring moves outward to meet the onrushing surfaces, two possible and mutually exclusive conditions are seen to exist, identified as system stress.
    First, the natural motion of the two surfaces will bring them together such that their orientation ends up being coplanar to each other where they join the ring, despite their orientations being quite different before (bold solid lines). In short, the surfaces, in respect to the orientation of their exterior regions, are caused to undergo swinging in order to accommodate their new orientations. Of course surfaces are not rigid, so how might they swing? This implies that they experience bending, in order to avoid swinging, in the accommodation of their new orientations. But how much bending? After all, the sharper the bend, the least amount they must swing, or, the more they are allowed to swing, the less the bending. Presumably the worse case scenarios would be maximum swinging without bending (dotted) or maximum bending with no swinging (light solid line), producing a cusp. Is a cusp bad? Well it is a place where a continuous and smooth function (such as a surface) is no longer smooth.

In the wholly imaginary and abstract, one can shrug their shoulders and walk away from this paradox, saving it for a rainy day. But if we are in fact (hypothetically in fact?) concerned about real issues, such as condition and variation of state, any cusp or sharp bending represents an infinite change in state, which to the average physicist represents, if not an alarming reality, an unlikely possibility: a condition representing virtually illimitable energy change within a finite context! There is no rule here, only the concern that extreme bending or extreme swinging over the entire length and breadth of both surfaces are undesirable options.

In addition to this, there is the question of manifest motion at the cusps. How is it directed, since all motion is directed orthogonal to a surface?

It seems, that a cusp is a bad thing; having no positive support nor affirmative value, whilst at the same time undermining specific premises essential to this study, leaving us with the problem of swinging.

Once interchange commences and surfaces experience the lateral displacement of IDDI caused by the outward migrating ring of non-definity, there is both the anthropomorphic assumption and perception that such motion is continuous. But how can this be so? If motion were continuous, anything which might be moving, in this case surfaces, would describe an extruded form: thus suddenly acquiring volume, whereas on the other hand, if motion is a series of discrete positions, no extrusion would occur. In short, if motion is a series of discrete steps, those forms which might be moving would remain unaltered in form, whereas motion is continuous, such forms might be sufficiently changed so as to be no longer the same. Case in point, a moving point, which, if motion is continuous, would describe a series of straight line segments, and thus no longer be a point.

In this light, imagine the ring of non-definity at instant t_n-1 causing an instantaneous IDDI of magnitude d_n-1, and then another at t_n+1 of magnitude d_n+1. Imagine also, a similar occurrence of IDDI produced by another ring of non-definity concurrent to the same two surfaces at t₀, of magnitude d₀, where t₀ falls between t_n-1 and t_n+1. Essentially, we have three discrete displacements, each occurring instantaneously and without duration, in this order: d_n-1, d₀ and d_n+1.

In the case of interchange occurring at two places along the same line or surface, the common exterior region belonging to both points or lines of non-definity, which are migrating towards each other, is subject to their original motions in free space as well as IDDI. If the IDDIs are in opposition (as shown following) their original motions will be offset, causing the disappearance of the points or lines of non-definity and then the subsequent cessation of interchange and all of its resultant effects.

In the case that IDDI is causing two surfaces to head towards each other and where both surfaces are participating in this combined action of IDDI, the combined magnitude of this displacement for participating surfaces cannot exceed the distance between two surfaces at the commencement of interchange for any given position where interchange, and hence IDDI, are occurring. Depending upon where interchange is occurring, and the original distance between surfaces, determines this maximum magnitude. Since both surfaces will move towards each other, in order to cover this distance, their combined IDDIs, generally but not necessarily equivalent, will equal this maximum magnitude.

Bear in mind though, that within a verdant field of surfaces stretching infinitely away from any given source of IDDI, each and every generated IDDI will encounter many other IDDIs neutralizing it, if they occur simultaneously to it. Thus, farther and farther away from the source, the possibility that a given IDDI emanating from this source might act, diminishes. Presumably, at an infinite distance, all IDDIs will diminish to nothing. This last statement should however be taken with a grain of salt since this has not been fully investigated nor have computer simulation attempts of the same produced good yields.

Given, say two radial configurations at some distance apart, virtually everywhere in the field their respective radials intermingle, cutting through each other at various angles, even at infinite distances away. Magnifying the microcosmic crossing or cutting through of two radials, actually a line of intersection, nothing can be expected to happen, since surfaces do not interact with each other where they intersect. But who is to say that these radials emerging from each radial configuration are perfectly and nearly flat? Such would not even be the case with ambient field surfaces, particularly in regions where auto-convolution is occurring: flatness being a unique condition, as would be perfect rest.

One can only imagine, that everywhere throughout the field, surfaces would normally be irregularly wavy, with few exceptions, and in constant motion; one of the more common motions being the bumping of surfaces into each other, which would be most likely to occur near where surfaces are the closest: where they intersect.

The irregular waviness of surfaces is similar to texture, in that there is a wide variety in the size of waves causing this texture, with presumably smaller waves more numerous than larger waves, except for very small waves which simply cannot propagate at all, simply because they are too small.

In close proximity to a line of intersection, because of this motion, both surfaces might collide, and if they do, the resultant IDDI will find itself translated along respective radials to the respective radial centers as a brief displacement; finite in both its magnitude and duration. No matter how the radials might be bent or twisted, IDDI can make this distance unattenuated to each radial core: the radial core being a fuzzy zone concentric to the radial center; a place where most radials associated with this configuration pass through, though not all, particularly those which happen to be entering or exiting the configuration. IDDI can be represented by a finite vector directed orthogonal to the surface, being that it is always the lateral translation of the surface. Of course this is happening everywhere surrounding both radials, designated M and N, resulting in impulses of a wide variety of magnitudes and directions.

What we have then is two radial configurations, M and N, some distance apart, mutually imposing discrete and different displacements on each other, causing these two radial configurations to achieve specific positions relative to each other, as the net effect of all displacements translated by all surfaces which are associated with the two radial configurations.

Not only then must one consider then the direct line of action between two surfaces serving as radials between two radial configurations, but as well take into account secondary surfaces, which are not radials, joining two such radials.

Illustrating this intermediate surface behavior are shown radial centers M and N with respective radials m and n intersecting at X. Interchange occurring around point X results in IDDIs denoted as x at M and N respectively. Passing through both radials and intersecting at points A and B is a third and intermediate surface (p), which because of IDDIs at A and B will complement IDDIs generated at X; IDDIs at all three locations, X, A and B causing triangle AXB to collapse so-to-speak. In other words, IDDIs occurring in the acute regions between these three surfaces will tend to cause them become more parallel.

Now already we know that IDDIs around X will manifest as discrete impulse at each M and N and directed as shown orthogonal to m and n. It is also apparent that if M and N are not allowed to move, being held at some fixed distance, surface p will tend to retain is original slope since IDDIs produced at A and B are in opposition. This is a general statement respecting the average of all conditions within an isotropic field. In this example the magnitude produced at B should exceed those produced at A since B embraces a larger acute angle between n and p than between m and p at A. Actually, in each specific case, because and due to these opposing IDDIs relative to secondary surfaces, the attitudes of the intermediate surfaces will change, though overall and in general, the sum of all IDDIs should be equivalently in opposition, the resulting in a net zero change.

The last remaining acute zone, at B and facing N, does generate IDDIs which can be translated to N along n. These are denoted as b and are directed orthogonal to n at N. Again the magnitude of these IDDIs varies as to the distance between p and n at their origin of commencement near B.