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DYNAMIC GEOMETRY
INTRODUCTION

    Euclidean geometry affixes markings on the flat two dimensional surface of paper.  With these imaginary limitations, Euclid formalized certain rules, such as the shortest distance between two points is a straight line.
    This same rule applies to three dimensional non static geometry, with modification:  that  points are not confined to two dimensions, but only to a volumetric region and that points are not fixed, but able to change position, a principal difference from static Euclidean geometry.
    This change of position is the critical difference between three-dimensional non static geometry and all other geometries, inclusive of analysis situs, topology, and the open and closed geometries of Georg Friedrich Bernhard Riemann and Nikolai Ivanovich Lobachevsky.
    It should be clear, that change of position may be abstracted to be confined to one degree of freedom; such movement confined to the boundary of a line, or to two degrees of freedom; such movement being confined to a surface, as in Euclidean and non-Euclidean geometries, or it may have three degrees of freedom, such movement confined to a volumetric region.
    In regard to the Euclidean rule that the shortest distance between two points is a straight line, this would remain true, in the understanding that this distance would not necessarily remain constant, but change as the points moved.
    Another variation to pure Euclidean geometry found in this study;  points are not unique and fundamental elements, but belong to a class of elements of a broader imaginary system.
    Unlike Euclidean geometry, where lines and surfaces are defined as being a locus of points, such definition is not applied in this imaginary system;  in this case, points, lines and surfaces, along with what is called an infinite volume, are intrinsic unto themselves.
    As a useful definition, any element comprising at least one infinitely small dimension, is also referred to as system object.  An infinite volume is not a system object.
    Dynamic Geometry started as a study of ontology (Science), but it was soon recognized that it belonged in the category of mathematics.  Pragmatically speaking, there is only a fine line between the two.  In science, it predominates as the study of Surface Mechanics, where surfaces are considered to be the fundamental entities of the real world (Cosmos).
    However, long before surface behavior was tackled, the behavior of points and lines was investigated in that order.  There is a reason for this, ontologically speaking.
    What if the underlying fabric, the field so-to-speak, consists of a whole system of criss-crossing strings, and wherever they meet, a unique condition describes a speck or a globule? Where two strings meet, there might be a speck, and where three or more meet, would be globules; the more strings meeting at this junction, the more complex the globule. Though not at all a good idea, it fulfilled several requirements: a certain level of simplicity, the potential of interaction and dynamics, a universal underlying field, and of course, globules and specks as the manifest presence of real, phenomenal particles.
    As a possible model, it had several problems. What kept the strings "glued" together at these locations? What was the nature of interaction? And, did not the differential in terms of the number of lines joined together in the determination of the behavioral nature of these junctions, require counting skills, perception and the ability to think?
  It was a knotty problem with no solution, other than abandonment.
    One positive outcome was the accidental recognition of locations which might occur if three surfaces, rather than lines or strings in space, were considered. Given an imaginary three-dimensional volumetric region of space where within resided numerous surfaces, each floating at some angle to one another, there would be many places where three surfaces would coincide. The idea of course was, that at the exact location where three surfaces intersected, an anomaly might be formed, representative of some sort of particle or proto-particle. As the string model before proved meaningless, this system of surfaces proved the same. It did however lead to the notion that space is filled with surfaces, myriads of them, whose dynamics and interaction might provide support as premise to a universal field.
    This was only the beginning of a great journey.   Along the way, one thing led to the next: from the study of Dynamic Geometry and Surface Mechanics, to the recognition of an infinite and random field of surfaces, field configurations, geometric stress and the quantification of these stresses in conjunction with field configurations, to a new cosmology called the Mass Decrement Cosmological Model (aka:  the Contracting-Universe hypothesis).

ELEMENTS

    Within this study, there are only four classes of elements, three of form and one of volume.  Each class exclusively consists of one or more points, or one or more lines, or one or more surfaces or one or more volumes.
    Any element may be defined as the adjacent non-finite extension of itself along any of the three orthogonal vectors such that such extension along any vector is, consistently, infinitesimally small or infinitely great, as the case may be for each vector.
    These four classes may be represented by the following:

        (0,0,0) representing any point thats volume is zero),
        (0,0,a) representing any line thats volume is zero)
        (0,a,a) representing any surface thats volume is zero)
        (a,a,a) representing any volume thats volume is infinitely great).

Space and Time
    Within this system there are no concepts nor constructs of space or time.

Distance
    Within this system there are neither concepts or constructs of distance nor mensuration;  all forms are either touching (coincident) or not touching (apart).

Variation of Position
    Relative to any form, any other form may be congruent to it at any discrete location common to both forms at any instant, or they may be separate at this same instant.

Orthogonal Vectors
    In order to provide consistent definition of these four elements, one set of three orthogonal vectors of intersection may be assigned at any discrete location coincident to any one of these elements.
    These vectors are purely imaginary convention and may be derived from the arbitrary swinging of a single ray emanating from a single origin.
    As imaginary convention, they may be considered equivalent to left- or right-hand Descartian rigid frame of references.
    These are shown as follows:
 

ORTHOGONAL VECTORS
A point with its corresponding orthogonal vectors.
A line with its corresponding orthogonal vectors.
A surface with its corresponding orthogonal vectors.

Definition of Points (0,0,0)
    Within the class of points, there are no intrinsic differences in points and each is defined in the same manner.
    Each point is defined as the adjacent non-finite extension of itself along each of the three orthogonal vectors such that such extension along any vector is consistently infinitesimally small.
    From one point to the next, there is no common orientation of these vectors, nor left- or right-handedness.  Accordingly, the order of the three variables (x, y, z) belonging to the three orthogonal vectors representing these forms may be in any order.

Definition of Lines (0,0,a)
    Within the class of lines, there are no intrinsic differences in lines and each is defined in the same manner.
    Each line is defined as the adjacent non-finite extension of itself along each of the three orthogonal vectors such that such extension along any two vectors is consistently infinitesimally small (0) and along the third, infinitesimally great (a).
    From one line to the next, there is no common orientation of these vectors, nor left- or right-handedness.  Accordingly, the order of the three variables (x, y, z) belonging to the three orthogonal vectors representing a line may be in any order.

Definition of Surfaces (0,a,a)
    Within the class of surfaces, there are no intrinsic differences in surfaces and each is defined in the same manner.
    Each surface is defined as the adjacent non-finite extension of itself along each of the three orthogonal vectors such that such extension along any two vectors is consistently infinitesimally great (a) and along the third infinitesimally small (0).
    From one surface to the next, there is no common orientation of these vectors, nor left- or right-handedness.  Accordingly, the order of the three variables (x, y, z) belonging to the three orthogonal vectors representing a surface may be in any order.

Definition of Volumes (a,a,a)
    Within the class of volumes, there are no intrinsic differences in volumes and each is defined in the same manner.
    Each volume is defined as the adjacent non-finite extension of itself along each of the three orthogonal vectors such that such extension along any three vectors is consistently infinitesimally great (a).
    From one volume to the next, there is no common orientation of these vectors, nor left- or right-handedness.  Accordingly, the order of the three variables (x, y, z) belonging to the three orthogonal vectors representing a volume may be in any order.
    In this geometric model, there is no distinction between one such volume or many, by virtue of the intrinsic characteristic of the region which they might occupy is invariant to their quantity, such a region being referred to as an Infinite Volume.
 
 

POINT BEHAVIOR

Point Relationships
    Given two points spatially coincident undergoing change.  Since there are two points, their positions are relative. In the absence of any intrinsic value for either point, values such as those found in physical studies, i.e., physical characteristics, like momentum or color, the only change possible is position.   If such change occurs, the points will no longer be coincident, but apart.  This is called divergence.  The reverse of this, when to points apart fall into coincidence, is called convergence.  To the observer, convergence and divergence are opposite actions of this type of change.
    In convergence, the observer would see points apart come together, and in divergence, points together come apart.
    Since neither time, velocity nor space have any meaning at this level of discussion,  two points may diverge in any direction at any distance.  However, though they may be depicted this way, as in this first illustration, they may as well be depicted as nearly coincident, as though they were just touching, as in the second illustration. Though this provides some sort of ontological basis, the perception of the condition of grazing, ordinarily associated with the touching of real objects, is not permitted, because the points are not finite spheres as depicted, but infinitesimal.
    In either case, the observer may arbitrarily denote the start of convergence as t-1 and its end as t0, and the beginning of divergence as t0 and its end as t+1.  Since in both cases t0 is an identical state, the sequence of change t-1, t0 and t+1 is possible.
    If the observer declares these terminal conditions between the actions of convergence and divergence as moments, then the potential for bi-directional time is forbidden, and in this limited sense, the notion of time is established as a reasonably permitted possibility.
If the observer declares these intervals of time to be constant, then the period of time to execute these respective actions establishes the notion of speed.  If the discrete positions of both points corresponding to t-1, t0 and t+1 fall upon a straight line relative to the observer, and if the convergent and divergent intervals are of equivalent duration, then the notions of momentum, conservation of momentum, and Newton's First Principle are established.

Collision Of Points
    Given two points within any region, which are apart and moving on a collision course, they will momentarily coincide.  Given that these two points are infinitesimally small, their momentary passage through coincidence will occur instantaneously.  Let us call this instant, t0.  Because this instant has no duration, it is not a process, but rather an event.
    Just prior to coincidence at t0, each point is apart, since the condition of points grazing is impossible, because they are infinitesimally small.  This instant may be referred to as t-1.  There is no duration between t-1 and t0.
    In the adjacent illustration, the two points are pictured at t-1.  Here, we see two vectors, vector V1 associated with point p1 and vector V2 associated with point p2.
    The point mp is an imaginary midpoint resting on an imaginary straight line (not shown) passing through points p1 and p2, illustrating the difference between imaginary Euclidean elements, such as the point mp, and the elements of this model, such as points p1 and p2.
    The volume defined by twelve lines, called a frame of reference, is also imaginary, serving only to demonstrate that this example, concerning the motion of points p1 and p2, occurs within a finite volumetric region of undisclosed dimension or shape.
    Though these imaginary elements are shown, neither the frame of reference nor the midpoint mp should be construed to share the same null rest, or in any way associated with principle elements points p1 and p2.
    In similar respect, the distance between p1 and p2, is indeterminate as are the magnitudes of the vectors of motion V1 and V2.
    Conversely, it is axiomatic that these two vectors are directed towards each other at t-1, bearing on what might be described as a straight line passing through these two points.
    At coincidence, there is no difference between points.  Accordingly, it is called a point of non-definity; a location where both points may not be distinguished from each other.
    However, any subsequent change of position of either point relative to the other, after t0, may depend upon the specific changes affecting both, those which brought them into coincidence in the first place.
    In this accord, the only potential distinguishing change of these points prior to coincidence would be their direction of change an instant before coincidence, each point changing 180o opposite to the other.  This may be represented by directional vectors emanating from each point, whose magnitudes are indeterminate.
    Upon subsequent departure from coincidence, where both points are indistinguishable from each other, their correspondence to their difference regarding this change is as well indeterminate.
    One might ask as to how these changes are appropriately reassigned to their respective points, or if they are reassigned at all, if not disappearing altogether from consideration and affect, clearly nullifying any prospects for their mutual departure from each other at instant t0.
    It would seem then, that several outcomes are possible:
    (1) a retention of their original relationships allowing both points to depart from coincidence with their own original directions,
    (2) a loss of association allowing both points to reverse direction or
    (3) a dissolution of the prospects of change, causing both points to stick together at coincidence for an indefinite duration, essentially for all practical consideration reducing the two elements to only one in any given finite region of study.
    Depending upon these outcomes, t0 may subsequently lead to:
    (1) no exchange
    (2) exchange
    (3) points stuck together.
    In keeping with the tradition of geometry, it would seem that there should be a rule for this, a means to at least deliberate these outcomes.  But there is not.
    Having said this, there is the understanding that since these choices must occur instantaneously there is no time for a decision making process.
    In the cases of outcomes (1) or (2), either (a) an exchange of motion (as directed change of position) between points occurs, or (b) an exchange of points between respective motions occurs.
    In the case of (b), motion would be absolute.  In the case of (a), points would be absolute.  However, since change of position is not exclusively restricted to point forms of this model, lines and surface being both capable of displaying change of position, motion is universal within this model, whereas points are secondary objects.  Accordingly, motion (as change of position) supersedes form.
    Putting it differently, one may say that this model provides for change of position of anything in any finite volume.
    In the case of option (3), points stuck together, this would not be permissible simply because the presence of change of position is not subservient to form and the relationships between form.  Thus only the two outcomes (1) and (2) are thought to occur.
    This being true, the chances of their occurrences can only be the same;  the probability of occurrence of either outcome (1) or outcome (2) being 0.5 for each.

    This event at t0 is termed interchange.  Though it may be referred to as motion being exchanged between points, it is most accurately explained as points being exchanged between motions.
    If only two points are involved, these motions are directed opposite.
    If more than two points are involved, these motions are not necessarily directed opposite, but directed in accordance to the respective directions prior to coincidence.
    At this level of study, time has not been introduced, thus the magnitude of any motion is indeterminate.  In each case, motion is change of position of least increment, therefor the vector magnitudes representing these motions approach zero and are not dissimilar.
    The projection of points on a surface, not necessarily flat, which undergo interchange, will always appear in such projections as coincident at t0.  In other projections of points moving at dissimilar elevations to the surface of projection, though they might appear to be coincident, they are not, but merely passing by each other.
    In this accord, projections must always be considered as imaginary abstractions to this abstract model of study, and if used, only used as necessary as expository tools.

SURFACE AND LINE BEHAVIOR

    Traditionally line behavior was taught prior to and separately from surface behavior, somewhat as its prerequisite.
    However, in the teaching of surface behavior this way, there seemed to be too much cross-referencing between the two, leading to some confusion as to whether lines or  surfaces were being referenced.
    Further, the study of surface behavior leads to a great deal more interesting conditional postulates than does the study of line behavior, for example, self convolution.
    Remember, the study of both lines and surfaces is confined to the same volumetric region, finite of course, leading to considerably different potential outcomes.
    For example, two surfaces somewhat parallel to each other and moving towards each other, will invariably meet at a common location where they are coplanar and just like points, lines can demonstrate similar behavior, but with limitation. To understand this, we need to know a bit more about lines.
    Just like points, lines are intrinsic forms, rather than a Euclidean locus of points. The lines with which we will deal, are not finite line segments, but lines whose lengths are infinite.
    As a matter of simplification of study, lines are generally confined to a flat plane of projection, though as mentioned, they can also be imagined and illustrated to reside in three dimensions, though this latter presentation may get somewhat visually confusing.
    In addition, in order to be able to move, lines are flexible and able to undergo bending, or what we might more exactly referred to as change in curvature.  This is particularly true because they possess no structure affording rigidity.
    In regard to line motion, one may express motion as the motion of a line at a given location on it, say at an imaginary point p. This motion may be represented by an imaginary vector emanating from point p, whose length of the line at point p relative to either the line itself represents the scalar magnitude of the speed at other locations (such as point r), to an imaginary and arbitrary frame of reference imposed by the observer or to the system null rest.
    Since the line is continuous form, without discrete parts, any movement of a line along itself, viz., tangential to itself at a given point, is motion which is unobservable to the observer, and as far as we can imagine, meaningless to ourselves as well as to the line.  Computational or analytical procedures are disallowed concerning tangential motion.
    As convention, all motion of a line is resolved into orthogonal vectors without its tangential component.
    In this accord, all motion of a line may be represented by a series of imaginary and orthogonal vectors, which may be arbitrarily placed by the observer, ideally at uniform and regular intervals along the line as illustrated following.  Here we see permitted orthogonal vectors (normals) emanating from line l (solid).  Running through the tip of each vector is a dashed line representing the lateral motion associated with the line at the base of each vector.
    This motion is not measured against the empty space (Being) surrounding the line, since such space is not geometrically stable.  Motion under these conditions is called motion in free space.  Such motion is very significant in the transmission of lateral surface activity through the Cosmos, contributing in field noise everywhere.
    Since by the theory of relativity, and just good common sense, all motion must be relative to something, all motion depicted here relative to this line segment as shown below is motion relative in scalar magnitude (speed) to adjacent motion.  For this reason, it is called relative contiguous motion.


    Using these conventions, several different types of line motions may be easily recognized, such as a bell-shaped distribution of motion, a transcendental distribution, or damped motion.

    Given a bell-shaped distribution of motion over a finite segment of a line shown as being straight at t0 we will illustrate the motion and changing curvature of this line over successive instants (tn).

    As a matter of convention in order to simplify this illustration, all vectors shown are unit vectors; the dashed line therefor representing the bell-shaped distribution of motion associated with each vector at t0 as well as the next successive position of the line at t1. Vectors emanating at locations p0 and p4 are of zero magnitude; the line not moving at these two locations. Vector p2, is a vector of maximum amplitude and vectors p1 and p3 are intermediate vectors. All vectors and locations are imaginary, of which any number, ideally at uniform intervals, may be shown.
    By careful plotting, maintaining vectors which are always orthogonal to the line, the line can be moved through any number of successive instants.
    Eventually, portions of the line (t8) just outside the null vectors (p0 and p4) will begin to convolute back upon itself. When this happens, the line is said to be no longer moving in free space. 


    If the distance measured between locations p0 and p4 at t0 is a unit distance, the line at position p2 at t0 will have traveled approximately three times this unit distance (3h)  when the line finally convolutes upon itself.
    At the location where the line undergoes auto-convolution, a location denoted as O, both segments of the same line will have converged together at point O. As with the two points that converge and can undergo interchange of their respective motions, because of loss of their identity at point O, one can only ask if the same is true for a line.

    Though we are dealing with the imaginary and the abstract, there seems to me a correlation to the physical in terms of the absolute characteristics of something, what might be referred to as the condition of state. Thus, though given two intrinsically identical things, if their conditions of state are different, they are unquestionably two different things. If on the other hand, these two identical things, say two electrons, share the same conditions of state, such as velocity, momentum and position, then they are indistinguishable from each other. However, in physics, because of the Principle of Uncertainty this cannot be demonstrated.
    Already, in this abstract domain, because of their conditions of state, namely position. when two points coincide, they are one-in-the-same. Can we not as well imagine the same possibility for two distinct and separate line segments when they coincide at point O?
    Besides position, both lines share in the same exact orientation in space at point O; they of course being collinear. But what else might we imagine to be a condition of state. What about curvature?

INSTANTANEOUS CURVATURE

    In modern analytic geometry and calculus, curvature is said to be the only absolute attribute; position, the other, always being relative to an imaginary frame of reference. Accordingly, without any frame of reference, the instantaneous curvature of a smooth function, i.e., a line or a circle, can be assigned to any given function at any location on that function, providing it is not discontinuous at that point.  Significantly, in physical studies concerning real things, any conditions of state associated with these things should be absolute. For example, color, though often used to describe something's attributes is not absolute, but relative, because it is dependent upon the ambient color spectrum illuminating such an object; a red ball appearing black if no red is present in this spectrum.  Because of this absolute quality, is it not reasonable to suspect that curvature is in parity to position and orientation in the determination of the complete congruency of the conditions of state for a two line segments?
    By simple comparison of the two convergent line segments coinciding at point O,  their curvatures seem to be directed opposite; and indeed, if express by conventional calculus, they would not be the same; their centers of curvature (an expression of this) not being coincident. But is instantaneous curvature the only way to express the exact curvature at a point?
    Bearing in mind that the development of calculus, along with its many presumptions, is hardly infallible; which includes instantaneous curvature as being the interpolation of the difference of curvature of two distinctly different and separate locations along any given smooth function, rather than the exact curvature at a singular and infinitesimal location, which it is suppose to be.
    This is done through the practice of limits, which in the case of instantaneous curvature is the limit of the ratio between the difference in angles between two tangents at these two locations which never quite meet, to the distance between these two locations as measured along the function itself.  But is this really the curvature at point O?















INFINITE MAGNIFICATION
    Another way to look at it is to infinitely magnify point O so that the actual curvature at exactly this location may be seen, or in the least evaluated through logical tools, aside from limits.
    In order to do this, magnify the conditions surrounding point O and draw a circle concentric to this point, which represents the next, subsequent, magnified field of view (frame). In the next frame or drawing, first draw in the circle representing the next field of view, and then carefully the two line segments, being careful to match their curvatures relative to the new circle. As you do this over and over again, what begins to happen to the lines from one frame to the next? After several frames, they begin to appear straight, and no longer curved, indicating that after an infinite number of frames (infinite magnification), the instantaneous curvature of any line at a discrete location is always flat!

    Thus through an alternatively valid approach, we find that the third condition of state, namely curvature is equivalent for both line segments at point O; being in both cases flat. This is of course true for any given point location along a line.
    What this means is, that because all three conditions, namely position, orientation and curvature, are identical, the two lines are indistinguishable from each other at point O.  Accordingly, just as with discrete points which converge to coincidence, lines will also experience the same potential dissociation between their respective motions.
    Again, coincidence at point O is called an event, and interchange can occur, being that there is a fifty-fifty chance of exchange.
    It is not an essential condition that the two line segments belong to the same line. Interchange can occur between different lines as well.

TWO SURFACES BULGING TOWARDS EACH OTHER

     Since neither a surface or line have any volume, they may not possess any structure.  Not having any structure, they may not be rigid and are permitted to bend.
    In the accompanying illustration, we see two surfaces bulging towards each other.
    In contrast, two lines under these same conditions, being nearly collinear, may never arrive at any common location where they might be collinear, and as you will shortly learn, unlike surfaces which might sustain a process called interchange, lines cannot do the same.
    Here, in the accompanying illustration, we see two cork screwed lines just happening to touch at a common location where they are collinear (in blue).  Since they are moving, moments later they will shift to another position where such collinearity is unlikely to be maintained nor ever again achieved, being highly unique in any substantial finite volumetric region.
    In the case of motion in association to surfaces (s1 and s2), when the surfaces collide by bulging at an imaginary location (O), they will for a very brief instant (t0) be coplanar to an imaginary plane (p) at O, and thus be indistinguishable from each other.  Then upon departure from this location, instant t+1, they may be exchanged or not exchanged by their once respective motions.
    In this event, when and where no exchange takes place, the surfaces will continue to move through each other, creating an ever enlarging ring of intersection (X).  This is shown below.


NON-INTERACTION OF SURFACES ALONG A LINE OF INTERSECTION
A physical surface, such as the fender of a car, requires material for it to manifest.  It therefor has volume and is porous.
 In this study, surfaces are infinitely thin.  Being infinitely thin, they have no volume.  Having no volume, they have no structure.  Having no structure, they can neither be rigid nor conflict with the presence of other surfaces and are able to pass through each other without impedance or resistance of any kind.
In calculus, a surface has no thickness and its shape may be expressed by equations containing three variables.
    In the event that surfaces at point O undergo exchange, a process called interchange commences simultaneously at t0 and t+1.
    Taking a slice of these surfaces coincident to an imaginary flat plane, such as a k-plane
(purely an imaginary tool), passing through the location occupied by point O at t0, portions of the surfaces will continue to move with their original directions such as at points O' and O" respectively, while they reverse their directions relative to location O, once common to both surfaces at t0.  This illustration shows conditions at t+1.
BE REMINDED THAT BOTH SURFACES (S1 AND S2) ALONG THIS FLAT PLANE ARE DESIGNATED
AS LINES L1 AND L2, A CONVENTION TO BE MAINTAINED THROUGHOUT THIS WORK.

    Surface one had been moving downward from above, and now, inside points O' and O", it is moving upward, maintaining the condition that both surfaces are just touching rather than intersecting, creating a continuous ring of non-definity (O' or O") from the original point of non-definity.

    Simultaneously (at t+n) all along this ring, the prospect of exchange or non-exchange are geometrically manifest, when at a virtually infinite number of discrete locations along this ring, directed motions may or may not exchange these two forms, creating a checkerboard composite of both surfaces along the ring.
    Next, as the ring enlarges to conditions at instant t+2, the same conditions continue to occur, producing two smooth surfaces which are now bulging away from each other in the interior region inside the ring.
    This is shown below.  The surfaces in the exterior regions are yet to converge to coincidence at the ring of non-definity.  The surfaces in the interior have already converged and undergone interchange.  Both surfaces in this region, S1 bulging upward and S2 bulging downward, are composites of each other.
    This process continues as the ring of non-definity migrates outward in a sequence of closed loops from it epicenter at O at t0.
    Though we may analyze this sequence as discrete steps, each step is irresolvable from the other.

    As the ring moves outward away from O, the exterior regions converge by motion caused by the change of curvature of either one or both surfaces.  If the process of interchange fails to occur at O, the surfaces will continue to pass through each other with this same motion, generating a ring of intersection (X' and X") concentric to O.  If however interchange commences at O, the surfaces will continue on with this same motion, but with a random exchange of their form, generating a ring of non-definity (O' and O")concentric to O.  A visual representation of either of the two rings is not shown.  Rather, each ring is represented by X' or O' where they pass through the k-plane to the left of O and by X" and O" where they pass through the k-plane to the right of O.

     Inspection of the the pair of blue lines representing surfaces S1 and S2 where they are coincident to the k-plane and undergoing intersection reveals that they are quite different than the pair of red lines representing the same two surfaces undergoing interchange.
    This difference is the effect of both surfaces remaining coplanar during interchange.
    This is a non-mathematical visual assessment of how these surfaces might behave under geometric constraint, mainly being that two surfaces undergoing intersection will remain as such, whereas two surfaces caught in the process of interchange initiated by an exchange of each at O at t0, will not be in a state of intersection, but instead coplanar to each other.
    It is not crucial as to how one goes about drawing these relationships, and it is highly recommended that the reader try it using Adobe Photoshop, Paths menu.
    The process itself commences with each surface being mutually exchanged relative to their respective motion.  Neither time nor space are required for this motion, though it must be relative to something.
    Though both surfaces are contiguous uninterrupted and homogenous forms, they may undergo discrete change of curvature at different locations upon them.  Remember, they are not rigid.  Such changes in curvature at locations adjacent to each other are motion.  If such motion occurs in an unvarying manner or in a consistently variable manner, it is uniform motion.  One such example of uniform motion would be a bell-shaped distribution as shown.

THE INDUCED DISPLACEMENT DUE TO INTERCHANGE (IDDI)

    If two surface are nearly parallel and close together, a bell shaped motion, could readily drive them together, causing them to bump into each other at a single location where they are coplanar.  We of course recognize this location to be O, typical of all such occurrences.
    In this first example, motion is relative to one object, as absolute contiguous relative motion, or change in position, or change in curvature, however one wishes to conceive it.
    Another motion would be relative to two objects, typically two surfaces or two sections of one surface which collide at a point of non-definity, while being concurrently coincident at this common location, despite being an irresolvable event.  Again, time and space are not necessary.
    Thus upon collision at O and surfaces being exchanged, surfaces in the immediate vicinity of O and inside the presence of any ring surrounding point O where the surfaces are still touching, having neither drawn away in the interior nor passed through each other at the exterior regions, must also choose the appropriate motion, again, all being a matter of chance.
    Thus, at each and every location on these surfaces residing on the ring, where they still remain coplanar, they may or may not undergo exchange.
    Of course, in the identifiable interior regions, this has already happened, yielding a random composite of both surfaces;  the one dome bulging upward and the other downward, each a composite of both surfaces S1 and S2.  In the exterior regions, yet to come together, no such exchange as this has happened;  they remain pristinely either S1 or S2 as they originally were.
    This is intriguing, for it suggests that a new form, say S3, generically speaking, has been created or added to this model, as a patchwork of the other two.  But remember, form has no type, such as class A form and class B form, so a patchwork of both would be the same and no different than either.
    It is in this way, that once started at O at t0, the process of interchange may continue indefinitely.

    Basically what we have then is a process commencing at O at t0 and continuing through an indefinite sequence of irresolvable steps, such as t+1 along the ring (O).
    In this above example, we see two such surfaces, the bottom surface S2 moving upward through a series of steps, and the upper surface S1 perfectly flat and stationary.
    The bottom surface is shown moving through a series of steps t-2, t-1 and t0.  Its motion is bell shaped, maximum upward at the central axis, and zero at its end points;  in essence an upward moving dome which collides with the S1 at t0.
     As soon as surfaces are traded between their intrinsic motions, point O evolves into a ring.  All locations along this ring, coplanar to both surfaces, immediately separate as the surfaces continue on with their original motion or in reverse direction.
    These events, which are purely random, allow the ring to move outward, meeting the incoming exterior regions that might still be converging, in which case, the process repeats itself.
    Surfaces which have already undergone this process, now in the inside of the ring (the interior region) are a composite of the original surfaces, and are shown in green.
    This surface position at t+1 closely matches the position of the original surface S1 before being bumped into by S2, thus obscuring the blue line representative of S1 before t+1.
    In order that the surfaces conform to these changes, an additional motion must be introduced, that allows the extremities of both surfaces to be displaced towards each other.  This is permitted, since we have no concept nor construct of distance in this model.
    This change of position, not directly associated with the intrinsic motion of these surfaces, is termed the "Induced Displacement Due to Interchange" (IDDI).  It is noted on on the far right-hand side of the previous illustration, where a descending arrow shows the downward displacement induced on S1, and an upward arrow shows the upward displacement of S2.

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