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

I.    DESCRIPTION AND PURPOSE OF THIS STUDY

    Surface Mechanics studies surfaces in motion.   The purpose of the study is to provide the student understanding as to how purely abstract objects can provide a tangible basis for natural law pertinent to the physical sciences.
    This study is an extension of Dynamic Geometry.
 

II    DEFINITION OF A SURFACE

     In understanding what a surface is, one should know what a surface is not.
    In this study, a surface is not a membrane, a thin plastic foil, a transition boundary, the surface of a material object, the boundary between fluids or anything else material.  It is not a Euclidean surface as a collection or locus of points.
    Rather, a surface is an intrinsic form unto itself;  consisting of nothing else.  It is therefor an object rather than a collection of objects.
    It resides in three-dimensional Being, as the superimposition of pure form within the substance of Being.  A surface is an object limited in presence and behavior by its defined nature;  comprising only contiguous relative adjacent position.
    A surface may be expressed as form extending infinitely great (a) along two dimensions and infinitely small [infinitesimal or zero (0)] along the other.  If a point is an object represented by (a, b and c), where a = b= c = 0, a surface may be represented by (a, b and c) where a = 0 and b = c = a.
    By convention, if  b and c are equated to any imaginary lines or geodesics coplanar to a surface, such lines, which may also be denoted as b and c, and set mutually orthogonal to each other, and to any line as an extension of dimension a.  It follows, that dimension a and its corresponding line, represent the thickness of the surface.
    If Being may be associated to any three-dimensional rigid frame of reference, such as a Cartesian frame of reference (X, Y and  Z) and if a, b and c of a surface do not necessarily correspond to X, Y and Z of the frame of reference, they may be oriented in any direction relative to frame X, Y and Z, providing the junction of any set of the orthogonal vectors of the family a, b and c, falls upon and is coincident to the surface at any given location on the surface.

III    ORIGIN OF A SURFACE

A.    Because existence and non-existence may be equated as being one-in-the-same; each having identical characteristics and both mutually exclusive because (1) both are perceived as universally permeable throughout Reality and (2) because categorically each is conceived as being diametrically opposed, neither empty space nor non-existence can uniquely exist.
    Accordingly, it can be directly surmised that only existence exists throughout the unbounded Cosmos as homogeneous substance, or what might call stuff.  In both Dynamic Geometry and Surface Mechanics, such substance is referred to as the Infinite Volume.
    Though the Infinite Volume was initially discovered as substance, it serves in a dual role as form, being represented as (a, a, a) along any three orthogonal axes within it.
    As form, the Infinite Volume belongs to a class of four forms, uniquely common in the measure of their simplicity, and hence probability of occurrence (P), where the probability of existence is inversely proportional to their complexity (C) as depicted in the following table:

    What this means is that all four forms each have the highest chance of existing;  more than anything else.  It also means that any number of each may exist throughout the Cosmos.
    Just as it would be inconceivable that "a perfect vacuum" could be created, the same is true of the Infinite Volume, along with the other members of its family in the class of form.  Each exists, because no other option is probable, principally because their degree of complexity is zero.
    In respect to the destruction of any, this would seem implausible, because there is nothing to destroy.  Thus as both form and substance, the Infinite Volume (Being) is temporally permanent.  Likewise, as form superimposed within the Infinite Volume, points, lines and surfaces are also permanent.
    Despite this theorized permanency of the surface, there is the observational suggestion, that though a surface cannot be annihilated, it can be drastically alter to the extent that its theorized definition is violated.  What I believe to be a pragmatic example of this, is the Great Wall in astronomy.

IV    DESCRIPTION AND BEHAVIOR OF A SURFACE
    A.    STATIC'S
       1.     CONDITION OF STATE - In geometry, the instantaneous curvature of a surface is determined through the approximation of limits and represented as a value called the radius of curvature.  Because this radius of curvature applies to a singular location on the surface itself, and not the coordinate system against which the surface might be referenced, the value for the radius of curvature is an absolute value, rather than a relative value.  No matter how you move or turn the surface relative to the frame of reference, this value remains constant in magnitude and in direction relative to the surface, always remaining orthogonal to it.
    In physics, though it is theorized that the condition of state should have some semblance of being absolute, it is hardly possible under normal circumstances.  For example, though it would be nice to know the absolute energy potential, usually the measure of some accelerative impulse between objects separated by some distance, such as gravitation, there is no absolute reference level.
    To the physicist, energy potentials for various systems, is important, since work can only be done through the change in these energies, such as gravitational potential undergoing conversion to kinetic energy potential.  Thus work, is equivalent to change in the condition of state.
    A surface relative to a frame of reference, such as a three-dimensional Cartesian frame of reference, possesses three qualities of state which can undergo change.  One of these is its position (x,y,z) at any exact location on it, its degree of curvature (R) at the same location and its orientation (m) at this location, where m is usually the slope corresponding to a straight line segment or a flat plane passing through this location.  Under these conditions, only its freedom of degrees of curvature are absolute, the other variables pertaining to slope or position being relative to the frame of reference, rendering them as being other than absolute.
    However, if other than a frame of reference, we compare the conditions of state of the same location on a surface at two different conditions of state, these same variables inclusive of position, orientation and curvature, become intrinsically absolute to the surface itself, and nothing else, inclusive of some independent, imagine or real, frame of reference.
    In the previous illustration, we see the surface at some original position inside the Infinite Volume.  In the adjacent illustration to the right, we see the same surface having gone some separation without any change in the other two variables.
    In the illustration to the left and below, the surface has undergone some change in orientation only but not in position at some given location, designated by the white dot in each illustration.
    In each of these first two examples, the radius of curvature (R) has remained constant.
     In the last illustration below and right, only the radius of curvature has changed.
    From these brief excursions, one can realize that there are three variable of state which can change, each being intrinsically absolute to changes of a singular surface, without the need or introduction of a frame of reference, real or imagined.  Under Dynamics, we shall learn how the system energy directly relates to these changes of state, and hopefully the phenomenon of work, though first we shall determine their more basic dynamic and collective behaviors.

2.     CHARACTERISTICS
              a.    Because a surface is infinitely thin (a = 0), it has no volume, and is therefor without mass and structure.
              b.    Because a surface is infinitely thin, without mass volume, structure and conventional material properties, it is transparent, colorless and invisible.
              c.    Because a surface has no structure, it cannot be destroyed.
              d.    Because it has no volume, any change in the condition of state of its curvature, neither thins nor draws upon its extremities to maintain its form along the a dimension, at any given location on it.

3.     AREA
            a.    Given a finite region of a surface, as denoted by the yellow lines, what would the area be?  In this example, one may consider the white arrows as unit arrows and the finite bounded region, marked by the yellow lines, as closely approximating a square shape.
    The question to be asked, what are we actually measuring inside the yellow lines?  If the surface was a single sided membrane, that would be one thing;  yielding one half the total area of a double sided membrane.  But what membrane has only one side;   one whose thickness along the a-dimension approaches zero, or one whose a-dimension equals zero?  Clearly, a paradox has arisen regarding how one measures surface area, reflecting a perceptual paradox concerning if a surface has any sides or not, and if so, how many?
    Since form is nothing, or in the most contiguous relative position, in the case of the surface, how could one measure the area of nothing?
    There is the tendency among most students to revert back to traditional views of the matter, such as the typical perceptions of "space", such as a stable Newtonian space, perhaps thought to be strung out between the yellow lines, or a more flexible and fluid like Cartesian space, neither of which is correct.  Clearly, the surface is not defined to be in space, but superimposed within the substance of Being.
    In this regard, because Being and Oblivion (nothing) are synonymous, both are immeasurable, hence, the area of a surface is indeterminate.
    What about a surface having undergone change from one condition of state to another by undergoing a change in curvature (R)?  Given a surface initially flat, or nearly so, if its curvature changes in some uniform way as to avoid rifts and other discontinuities, thus remaining uniform and somewhat smooth in general resolution, its second condition of state might be a bulge as shown.  The uniformly spaced and once nearly parallel black lines represent geodesics;  lines whose characteristics are wholly dependent upon the surface as frame of reference.  Assuming the outlying body of this surface barely changes, thus maintaining an overall change in state between these two conditions approaching zero, the geodesic within the bulge are therefor consider the most variable, reflecting the most probable adjustment to our image as to what happens in this region of the surface.
    In our minds we clearly imagine that the surface seems to stretch in this region, perhaps somehow drawing upon its infinitely thin extremities.   But such is not the case, because what we imagine, is change in area, and area really has been logically factored out.  Further, our concerns is not the surfaces, it just does it, as though adhering to our abstract definition.  The two blue areas between geodesics were the same size, typically, at the original condition of state;  the larger area being the size of the smaller.
            b.    Since area is indeterminate, perceived changes in area is meaningless and changes in area are not changes in state.

    B.    DYNAMICS

        1.    Within an unbounded spatial realm, motion is not impossible, and therefor motion is considered to be a Fundamental Virtue.
            a.    If the realm is of one dimension, bounded by and upon a line, anything which might move within this realm, must travel upon the line.
            b.    If the realm is of two dimensions, bounded by and upon a surface, anything which might move within this realm, must travel upon the surface.
            c.    If the realm is of three dimensions, bounded by and within a volume, anything which might move within this realm, must travel within it.
            d.    If the realm is of three dimensions, bounded by and within a volume, and if that volume is unbounded and not finite, then anything which might move within this realm, can move anywhere.
        2.    Though things may move because of the Virtue of Motion (Dynamics), the condition of rest is not disallowed.
        3.    Because a surface is without structure, it cannot be rigid, and is therefor flexible.  Such flexibility though, cannot in any way be related to material malleability, or inertial and elastic qualities of ordinary substance.
        4.    Because a surface is not rigid, it cannot be stationary, but is instead free to move, bend and undulate in Being.  The motion of a surface (S) at any given point (location) on the surface may be resolved into orthogonal vectors, one directed coplanar to the surface at that point, the other orthogonal to the surface at that point.  All motion of such a point can be described by the resultant vector (R) between these two vectors,  However, since a surface is homogeneous, motion directed coplanar to the surface has no meaning since such motion does not alter the surface's position nor intrinsic condition, such as its condition of state.  In other words, the motion of the surface at point a, where such motion is directed coplanar to the surface at a, produces no apparent change of the surface, relative to the next contiguous position of the surface at a'.  On the other hand, motion directed orthogonally to the surface, say at point b, has meaning and produces a definite lateral displacement of the surface.  Orthogonal motion is motion directed at a right angle, which is denoted by the symbol q (meaning at ninety degrees).  Also the symbol for perpendicularly, ^, is used.  In this illustration, the surface S is shown as simply a straight line because we are only showing the surface where it intersects the plane of this paper.  The following illustrates this in three dimensions, where the plane of the paper (K) is shown.  It should be understood that most depiction of surface behavior will be restricted to the behavior of a surface at its intersection with a flat plane, called the plane of intersection, which is usually denoted as K, and where the plane K is purely imaginary (Euclidean) and not construed as being real, as is the surface, and instead merely serves as a device to simplify conceptual complexities regarding surface behavior.  The same surface, without plane K, is illustrated in full three dimensions in Figure 4.
    Since only a finite section of an infinite surface can be shown, dashed lines at the corners indicates the infinite continuation of the surface.  In actuality, surfaces are completely invisible, since they are merely form as position imbedded within Being, and are unable to reflect nor impede the progress of electromagnetic quanta, such as light.  Shading and geodesic lines are used to enhance surface features, such as contour, as well as to delineate between different surfaces in one illustration.
    These three techniques, which illustrate surfaces either as lines, in full three dimensions, or in three dimensions showing the plane of intersection, will be used interchangeably.  All that needs to be remembered, when the text describes a surface, yet the illustration depicts a line, that the line represents the surface where it is intersected by a flat plane.  This is usually does as a means of simplification.  Occasionally, reference will be made to line behavior, and as the drawing will also show a line, it will also be assigned the symbol l (lower case L), which indicates a line, whereas surfaces are labeled with an S, even though they may appear as lines.  The plane of intersection, also called a k-plane, is labeled with a K.
    Though we can say that at any location on a surface, the surface might be moving in any direction, since any motion directed coplanar to the surface is discounted, the sum total of all this motion could only be perceived as a lateral displacement of the surface to itself.  Following, we see this behavior, where hypothetical motions of many locations on the original position of the surface S at time zero are shown as faint vectors which join the surface's new position at t=1.

    Since by definition a surface is contiguous position, all vectors emanating from St=0 must coincide with St=1, in order that the surface remains contiguous and unbroken.  The bold vector in the middle represents the average motion of all vectors.  Following, we see both surface positions shown, initial and final, without seeing causative vectors.  Can you tell which way the surface has moved from its position at t=0, in order for it to arrive at its new position at t=1?   Again, because there are no discrete landmarks or what might be called identifiable locations of the surface, only lateral displacement carries any meaning.

    Consider the very special conditions of a flat surface moving through itself, where at a given location, such as at point a at t=0, the surface moves to a new position at a' at t=1.  Such motion is unobservable to the observer and meaningless to the surface.
    It is through this line of reasoning that only motion orthogonal to the surface is considered.  In the case of our first example, where the surface moves laterally between positions at t=0 and t=1 (Figure 5), this motion is correctly illustrated in the following illustration.

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.

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.
 

V  RELATIONSHIPS BETWEEN TWO SURFACES

In the same way that lines move, surfaces can also bulge together,  meeting at a singular position.

Just like two lines in these conditions share the same orientation, position and curvature, so do surfaces.

In their case, they are said not to be collinear, but coplanar.

Again, point O, where the surface first meet, is called a point of non-definity; marking the prospects of an exchange  or non-exchange of surfaces  to their respective motions prior to convergence.

Just as the distribution of motion of a line may be represented by a series of uniformly spaced orthogonal vectors along the line, the distribution of motion of a surface may as well be represented by a set of uniformly space orthogonal vectors over the surface. Following is a transcendental distribution of motion of a surface.

VI  THE UNIVERSAL FIELD

    As we have discovered, all three forms, the point, line and surface are of least complexity; a complexity equivalent to zero.  Since they are without causal mechanism or governing means; each one being impossible to construct or destroy, it is axiomatic that their presence is a matter of extreme possibility, as it is therefor axiomatic that their probability of existence is unity (1.0).  If indeed any one of these can exist, so can many of the same. This means that Being is filled with an infinite number of Forms. This is called the Random Field.
    Simply put, the field is of infinite expanse and eternal and consists of myriad surfaces, lines and points; all bending, convoluting and moving, as the case may be. In order to visualize this, consider the basic backdrop of Being (the Infinite Volume) as being clear and dark, where within it we can see transparent silver surfaces floating like the finest gossamer, but not all aligned the same but with different and varying orientations and movement, all of them cutting through one another as though the others were not even there, and thin spider web like lines of infinite length, floating and bending and passing through the surfaces, and a myriad silver points rushing to and fro, penetrating and passing through one surface after another, and on rarest occasion, striking a line. It is a great grinding, but silent, primordial playground, invisible to direct detection.
    In this illustration of the field, though the surfaces shown are seen as finite, they are infinite, without detectable edges.
   One of my first questions concerning the field, if given a finite cubic region of the field, say one centimeter square, how many surfaces might be passing through this unit volume?  My current best is estimate, based on the decay of a radial configuration is 1.51 x 10³⁷ surfaces per cm³, along with probably the same number of points and lines.  This would represent the fine structure of our physical universe;  the ultimate resolution of the size, weight and movement of even the smallest particles.
 
 
 

   In the most exotic and esoteric studies concerning natural behavior, the field density is deemed to be inversely proportional to what is called space-gauge;  the mean separation between surfaces.  The space-gauge, as it should be, is remarkably similar in magnitude to Planck's constant lying in the range of 10^-34, which represents the least measurable magnitude of photon energy; essentially, the smallest conceivable quanta.
    Taking the value for field density and inverting it gives us the maximum average distance between field surfaces, a value of 6.6 x 10^-36 centimeters, a value representing the smallest resolvable magnitude of the field's fine structure.
    The correspondence between these two extremely small and rationally associated numbers demonstrates a profound correspondence between the hypothetical and the real.
   Now as observers sitting on our small earth racing around our bigger sun racing around our yet much larger Milky Way galaxy, we cannot presume this great field to be moving with us, but rather that we are moving through it. Perhaps to a certain extent, our spinning galaxy does in part move the field with it, drawing it along at roughly 200-300 kilometers per second at its edges. This, we cannot be sure, since one might think that some sort of detectable trace, or some variation in the sun's bow break might be noticeable as we both race across this field; having the physical material within us replaced by 3.7 x 10⁴⁴ surfaces per second! It is a staggering possibility to think that this could be happening, but then again, surfaces are not mass, so the usual material relations do not hold up, notwithstanding inertial as well as relativistic concerns.
    This is the field, which, because of the purely random and rather disorganized order of surfaces which makes it up, it is termed, the Random Field, though I generally refer to it as the field.  All well and good, except that what we have is an inert Supreme Ultimate Being, the Virtue of Motion, and the existence of Form, namely being surfaces.  If this is Nature's bottom line, where's interaction?  After all, the universe you and I know is filled with interaction.
    In Dynamic Geometry we discovered the mechanism of interchange. Of course, we were dealing with imaginary surfaces. Is it possible though, that interchange might work for real surfaces, if in fact there is a field of such surfaces?  If so, we might have the embryonic semblance of interaction at the physical level.
    One requirement is that our surfaces be infinitely thin, which they are. Being infinitely thin, surfaces have no volume. Not having any volume, they cannot have material structure, or structure of any sort. Not having structure, they cannot be rigid, nor contain internal stress or resultant forces. Not being rigid, they are incapable of not moving; hence are willing and able to bend, flex, move and undulate. Of course, not having internal structure, force or stress they can pass through one another without the slightest impediment to each other's motion. All things considered, since our real surfaces and imaginary surfaces share in the same qualities, there is every reason to think that interchange can serve as a potentially real process.
    Significantly, at this level, there is no concept of energy, only Motion, Motion being more elementary than even Form, though it is through Form we see its manifestation. We know that Motion cannot be exchanged and cannot be destroyed anymore than the Forms upon which it manifests cannot be destroyed; in essence all Motion being permanent. Again, Motion is a virtue. Another way to put it. If there are no Forms, we cannot see Motion, even though it is there. Further, it is there despite the presence or non presence of the Being, since such Being is inert and wholly unrelated to Motion, where Motion is a natural option.
    From this we know that though waves themselves might decay, their underlying motion is still there, causing the formation of newer and presumably even smaller waves than before, which then begin their own propagation. Clearly though, if these smaller waves are spawned by the residual motion of larger waves before them, how could they even begin to move through the field? Essentially then, if a wave does decay, the field in this region becomes vibrantly active with the assertion of motion, unable to escape because of a field density which is too low. The process again repeats itself, producing more convolution and smaller and smaller waves; a microcosm within itself.
    This process is of course not confined to only one place, but is happening everywhere through the field; essentially pockets of trapped motion trying to reach out, but unable to because of the low density of the surrounding ambient field.
Eventually though, as each micro-domain grows larger and larger, and new ones are being generated between the old, all eventually encroach upon each other, enriching the overall Random Field with greater density than it had before. This process is considered to be in direct correspondence to the variance of space-gauge, inversely so, which in turn relates to the size of manifest objects in the field.
    When I speak of manifest object, I am referring to physical objects whose presence results from the field and whose size is determined by the field.
    Consider then a wholly artificial and contrived object, such as a cube, consisting of twenty-seven smaller cubic regions as part of a very uniform field of orthogonal surfaces. This if of course very unique and unreal, but despite being so, it clearly demonstrates what I am talking about.
    In a past field of a given density (Frame A), this object will be of a certain size, but later, as the field increases in density because of all these decaying waves (Frame B), the same object of twenty-seven cubic regions will be smaller. These dimensions, as they should be, are relative to the observer's standards of measure, in this case frames A and B, which we may consider as being unit dimensions.

    What this tells us is that as the field gets older, objects get smaller, by virtue that the size of objects is inversely proportional to the field density. This of course is one of the foundations for the Contracting-Universe Hypothesis, since simple waves, thought to be analogous photons, are not manifest field objects, and are therefor unaffected by this changing field density (variable space-gauge). Thus after traveling vast time and distance, these simple waves remain very much the same as they did at their time of emission, despite the rest of the real, material world of matter, getting smaller.
 

VII   FIELD ORIGINATION

Given a surface undergoing motion in free space, what will it do?  Following, this is illustrated, starting with a surface  perfectly flat at t₀ (a unique condition) and moving with a motion as shown (dashed), through a sequence of uniform intervals after t₀.  At each given interval, the surface is shown to be solid, except at t₁, where it is shown to be dashed.  Here, at t₁, both the surface's position and distribution of motion are visually coincident to the observer.  As a matter of convention, only its motion is shown, its position understood to be the same, otherwise had its position be drawn solid, its distribution of motion (dashed) would be masked by this solid line.

Here we see the surface as it moves through a position approximately equivalent to t₂.

If the surface is unable to undergo interchange with other surfaces, it will eventually move to position t₈ where it undergo auto-convolution. When this happens, the surface appears as a button whose diameter is about four times larger than its original simple wave diameter.

Along the line of non-definity (a closed loop not necessarily round or oblong), where the bottom side of the bottom touches the exterior region of the original surface  the surface may or may not undergo interchange with itself.  If interchange does not occur, the surface will pass through itself, its motion now in the opposite direction than the original simple wave.  If interchange occurs, the surface will literally bounce off itself, causing its exterior portions to move in the opposite direction than the original simple wave.  The surface at the location of the ring of non-definity, will come to rest.

When this happens, the original surface will progressively become more convoluted and complex within this finite three-dimensional region.

This will continue within a fixed finite volumetric region, causing surfaces within this region, to become more and more complex.  It is this complexity which is inversely proportional to the space gauge, a measure of the distance between apparent surfaces within a finite region.

The rate of auto-convolution within this region is of course directly dependent upon the number of simple waves passing through.  It is also inversely dependent upon wave size;  very large waves being least likely to undergo auto-convolution.

This process is occurring everywhere throughout the field at a consistent universal rate,  viz., all finite regions increase in field density (or decrease in space gauge) at the same rate.  This is because all finite regions may be seen as inter linked by the common simple waves passing through them;  the more waves, the faster the process.

Since these waves can be generated within finite regions, such as photon emission, the greater the field density within any region, the greater the prospect for more waves and smaller waves.   If any region achieves a higher density than surrounding regions, proportionately fewer of these waves will undergo auto-convolution, being that they can readily move between surfaces which are closer together, without auto-convoution, this relationship serving as a natural governor to the process.  On the other hand, if any region is considerably less dense than its surrounding regions, a greater number of small waves, generated within the outlying regions, will pass through, thus causing its density to increase more so than if only waves internal to this region occurred, thus causing lower density regions to "catch-up", so-to-speak.

Overall, the process is irreversible; the entire field progressively becoming more dense.  At this level there is no such thing as time;  the process being time independent.  Since time manifest at the phenomenal level, within the Fundamental Domain, and since there are several other processes, such as interchange and radial configuration decay, one might say that rather than entropy, changing space-gauge is "time's arrow".

Since the changing curvature of a surface is absolute, rather than relative, and since it can be directly equated to the dynamics of energy and condition of state, as surfaces collide and undergo interchange, "new" energy becomes available.  Thus, unlike entropic considerations related to the Inertial Domain, the reverse is true in the Fundamental Domain, in that there is always an abundance of energy, namely change in curvature, which can be directly translated as contiguous relative surface change of position, which, if correlated to time, is contiguous relative motion.

Such motion, as changing curvature, continues unabated, being that there are no resistive forces since a surface has no structure, being infinitely thin.

Within any finite region, no matter how complex it may appear, can be represented by one surface, or by many.

FIELD OBJECTS

    Within the random field, there are a number of theoretically possible structures, some permanent, some fleeting, some very small and others consisting of huge collections.  Among them are simple waves (l) moving through the field in all directions, some traversing vast distances in terms of light years, and others decaying almost immediately.  There are also arrays, configurations, dions and trions, inclusive too of ISSs and OSSs, all of which are classed under one category as field objects.
    At this fundamentally advanced level, where the student might recognize the interchangeability between elementary physics and geometry, a simple wave would have a spin equivalence of one.  Also, each simple wave would have the potential of acquiring one of six of the quantum chromodynamic colors, though at its onset, this would not be defined, since prior to the presence of any field objects, it would not have color, thus any simple wave following its typical microscopically zig-zagged* yet macroscopically rectilinear path through the field would be colorless.
     But in a field of many objects, many of which might have color, the simple wave may now be associated with one of these colors, providing its motion ends in an orbital motion whose color corresponds to the color of many field objects within the same finite region of the field.
    As I have mentioned, there are two types of simple waves relative to the field density: one type able to propagate indefinitely if its wave diameter is greater than 3h;*  the other decaying if its wave diameter is less than this.  The value 3h is dependent on the field density, where h is the reciprocal of the field density.*
    Just as a simple wave is nothing more than a logically discernible order of the field, so is a radial order.  A simple wave moving into a radial order:  the conjunction of both, is defined as a radial array.  An array most likely will persist longer than an order, which without the presence of a simple wave, would quickly disappear.
    Though all are both geometrically and functionally different, all three are defined as field objects because of their logically discernible.
    As these three principal objects evolve, radial arrays may become spiral arrays, which in turn may evolve into spiral configurations, sometimes becoming inside-shelled spiral configurations (ISSs) or outside-shelled spiral configurations (OSSs).
    Along with these possibilities are a potential seventy-five combinations of dions and two mutually exclusive families of ninety trions each.
    All these mentioned are defined as field objects.