As students of the Reciprocal System of Theory we have become used to a somewhat different set of paradigms than those held by all other students of the physical sciences, and by each of us prior to our acceptance of the concepts embodied within the Reciprocal System. The rest of the scientific community accepts without question the primacy of the observed characteristics of this physical plane of existence. To the community at large, mass is a fundamental characteristic of anything to be called matter and matter is the fundamental building block of the universe. To the establishment it is totally unthinkable to even conjecture that motion could exist unless matter is moving. That is the biggest stumbling block or hurdle to be overcome. Our thought patterns are still inhibited by our previous habitual use of that paradigm, resulting in such extreme difficulty in taking "an old set of data" for interpretation "from a new perspective" that we do not recognize our use of those old habits. By perspective I mean totally new set of concepts as outlined by D.B. Larson in his presentations of the fundamental concepts for a "Universe of Motion." Even Larson had difficulty turning loose of many of the undeclared assumptions hidden in our observations of the physical universe from this region of Time/Space.

Larson has outlined for us an order of complexation of units of displacement motion and given us some of the new representational modes required for many of the phenomena observable in a physical universe of motion; such as two dimensional rotations. Invoking the rules of ordinary mathematics in all regions, including the representational requirements of Euclidean geometry and the concepts of probability relations for any representation of the concept of motion, requires us to really understand exactly what the rules of ordinary mathematics are and also what they imply. It is this requirement for knowing, not only how ordinary mathematics is used, but what its rules imply, that has led to the requirement for six possible modes of representation at three dimensional reference points, not just the familiar four of the Time/Space region.

In a multiple reference point Universe of Motion only the point coordinate axes for any specific reference point combination of representations of the concept of motion is important for that combination representation, regardless of how complex the final representation may become. Critical examination of the idea of a multiple reference point universe reveals that only the individual set of coordinate axes of each and every reference point need ever be considered with respect to any individual reference point. Every photon, every sub-atomic particle, every atom, cosmic or material, is its own reference point.

For the existence of any reference point phenomena, no other reference point is of any importance, so far as the representation of motions or effects of motions at that reference point are concerned. The only possible subsequent importance any reference point may have to another comes about when, and only when, they share the same unit of primary motion and, thereby, become a new reference point for a different reference point phenomena. There are at least two possibilities for this situation: atoms in chemical orientation and photon interactional phenomena. The two interacting components become not two phenomena at the same reference point; they become a new phenomena at a new reference point because the new reference point phenomena is a different combination of motions.

The resulting mathematical expression for this concept must reflect this reality even though the new reference point effect may be measurable in terms of each of the previous reference point phenomena. Conceptual and mathematical consensus for any expression of the effective reality of a reference point phenomena causes the requirement for the concepts embodied in the algebraic expression relating magnitude and direction to be coherent with the magnitudes of the arithmetic. A numerical sequence is required for any expression of quantity, whether that quantity is of substance or direction. One is followed by two and then three.

Let us now consider, "What is it that makes a unit of displacement be a displacement? Is it its opposition to primary motion in whatever required representation primary motion must have, or is it something else?" Since primary motion is the very first possible motion that can be represented, primary motion must be given the very first possible mode of representation in the three dimensions available for its representation. That representation is one Dimensional and one directional in any one of the three dimensions. What must be next? Is it two Dimensional and one directional, or is it one Dimensional and two directional? Can primary motion be directly represented in more than one way? If it could, would there be any consistency among subsequent combinational representation? I have played with as many possibilities as I could think of and have always come back to one and only one possible way of directly representing primary motion: one Dimensional, one directional linear. Any other possibility led to so many possible second steps that it became almost impossible to calculate a required sequence for a third step.

In answer to these questions about displacements and primary motion, it seems clear that since primary motion can be represented in a direct manner only as one Dimensional one directional linear, a displacement must first be able to oppose that kind of representation before a generalization can be considered. With primary motion directly representable only as one Dimensional one directional linear, and that one direction being in either of the two directions of one dimension in any conceivable three dimensional coordinate system, an opposition to primary motion in that dimension has to be represented as one Dimensional two directional linear. It can not be in just one of those directions, because primary motion would be left free to be expressed in the other direction and nothing would have been accomplished and nothing could be represented! That is **why** the first representable displacement motion must be one Dimensional two directional linear in one of the three dimensions, which thereby leaves both directions of both remaining dimensions open in which to represent primary motion. Once the direction of primary motion is selected, it is done, and that's that, so far as that reference point is concerned. Any effects of displacements remaining with that unit of primary motion will seem to have straight line movement relative to any reference system of coordinates. The first possible reference point phenomena must have a structure represented as a combination of a one Dimensional two directional linear displacement and a unit of primary motion in a perpendicular dimension. We call these reference point structures photons!

The question for these photon reference points concerning now the effect of their structure is to be expressed relative to a whole bunch of other reference points of whatever kind must have an answer related to, or given in terms of, the mathematics used for their representation. This requires consideration of the meaning of directionality as it applies to the idea of dimensional systems.

Random orientation of reference point coordinate systems with respect to all other reference point systems requires the use of probabilities for sameness among all such coordinate system. Use of probabilities is limited by the arithmetical system and, thus, the question of which must come first: substantive quantity or direction. An obvious question is: "Is it obvious that the quantity being represented must exist before it can be given direction?" This question is, for us, similar to the question for most physicists of whether motion can exist without the presence of matter; specifically, can there be direction without something (even a concept of something) to have a direction? If so, we have a universe of motion, not direction. This conclusion seems to be the same as that derived by present day physicists; matter, not motion. If a quantity (e.g., the concept of a unit of motion) must be available before directionality can be specified, then the effect of the quantity being analyzed must be one directional, two directional, or multi-dimensional. Since one directional can be in either of two directions, the effect of a **two** directional linear displacement is equally probable in either of the two directions possible. To maintain equality of probability, two such units of displacement must be sequentially related in order to complete the probability function for the representation of either one of the units of displacement.

Considering all the mathematical functions capable of fulfilling the conceptual requirements for representation of the one Dimensional one directional linear primary motion and the one Dimensional two directional linear displacement perpendicular to the primary motion, it is found that only the sine and cosine functions can satisfy those conceptual requirements in an unambiguous manner; i.e., an effect that is linearly positioned and continuously variable and has two directions of possible effect.

By this convoluted path it has been shown that photons must be conceptually represented as a combination of 1D2d_{L} displacements with perpendicularly primary motion and mathematically as sine wave functions. It has been implied that the next step of complexation must be similarly related thru appropriate application of probabilities for the ideas of dimensionality and directionality.

The idea of rotational representation of directionality around an axis causes all linear directions to become partially accessed and thereby related in the resultant effect. Possibilities for subsequent representations require primary motion to be represented with rotational directionality. Direct representational probability of this possibility is so small as to be non-existent. However, it is the augmentation of the concept of directionality for the representation of primary motion that allows for a displacement motion to be represented rotationally and development of the generalization for displacement to be an opposition to primary motion as previously questioned.

Chart 1 indicates what the six modes of representation for units of displacement motion must be at sufficiently compound motion reference points. That which is observable in a generalized three dimensional system is only the effect of Notational Reference Point representations of displacement motions other than 1D1d_{L}. Primary motion is the one dimensional velocity observed for photons and some subatomic particles. Equivalent primary motion is the maximum resultant one dimensional velocity for all atomic and the remaining subatomic structural representation. The order of complexation among the six modes of representation at individual reference points is as shown in Chart 2, increasing complexity from the bottom up. The final Chart shows the resultant physical universe composed of seven principle regions in Three Sectors.

### Chart 1

Equivalent Euclidean mode; (in symbols #of Dimensions #of directions _{type})

Motion Symbol # Dimensions #directions directionalityLinear translation; 1D1d_{L} One Dimension one direction linear

Linear oscillation; 1D2d_{L} One Dimension two directions linear

Unidirectional rotation; 1D1d_{R} One Dimension one direction rotational

2D1d_{R} |
Two Dimensions one direction rotational
Rotational Oscillation 1D2d 2D2d |
---|