# 06 Quantitative Relations

## Fundamental Units

When all quantities are expressed using the natural units of space and time, of which the universe of motion is constructed, the values of all the “fundamental constants” reduce to unity and effectively vanish in the various equations in which they are used. Since our conventional units are in essence quite arbitrary due to the environment of our local planet, the conversion factors between conventional units and natural units are also arbitrary and can be calculated only from specific measurements made of simple basic phenomena. The speed of light is definitely identified as unit velocity and as the natural rate of the progression of space in the spatial aspect of motion. To the extent that theoretical factors have been adequately considered and the experimental determinations accurately made, other fundamental quantities and constants have been determined.

To provide the conditions from which consequences for the postulates of the Reciprocal System could be derived, a progression of one unit of primary motion in one representable dimension must be exactly equivalent to a unit of motion representable in any manner in any direction. Once the quantity of spatial progression involved in one unit of primary motion in a representable dimension is determined, that is the quantity of space represented in a dimension of motion whether that unit of motion is directly representable as primary motion in a specific spatial dimension or as a distributed effect in all directions of dimensional space, including rotational representation. This quantity of space in the arbitrarily selected units of this planet for the amount of spatial progression per natural unit of time is equivalent to any and every unit of scalar space. A similar argument is used for units of time.

The value of unit frequency obtained from an atom of unit magnitude is more consistent with the Reciprocal System theoretical viewpoint and with the general pattern of measured values than that calculated from previously accepted theoretical considerations and infinite mass. The Rydberg constant is a primary multiplier by which energies and velocities of matter in the spatial aspect interact and is, thereby, the value of motion equivalent to one unit of oscillation.17

The fundamental frequency of one transverse vibration is by definition represented as the effect of one unit in one direction and one unit in the opposite direction in a dimension other than that represented as the dimension of progression in generalized dimensional space. The previously discussed theoretical description of radiation requires a photon of frequency “one” to be one cycle of oscillation and involve a minimum of two units of primary motion in the dimension of normal progression because two units are required to give equal probability of representation for the effect of the 1D2dL displacement/s of any photon regardless of its frequency, but especially for the photons having only one unit of displacement from unity.

RH = 109677.6 cm-1

$\frac{1}{R_H} = \frac{1 cm}{109677.6} = 9.117632 \times 10^{-6} \frac{cm}{cycle}$

$s_n = \frac{1}{R_H} \frac{\AA }{cycle} \times \frac{1}{2} = 455.8816$ Ångstroms / half-cycle

Frequency is the rate of presentation of an oscillatory effect at a location in dimensional space. Since primary motion is outward from any spatial location, both the change of natural location of the oscillation being measured and the frequency of the oscillation effect are motion, a speed, a quantity of space progressed and have the units: distance / time; s/t.

$$v_n (natural~velocity) = \frac{s_n natural~unit~of~space}{t_n natural~unit~of~time}$$

$$t_n = \frac{s_n}{v_n} = \frac{4.558816\times10^{-6}}{2.99793 \times 10^{10} cm/sec} = 1.520655 \times 10^{-16} sec$$

One Dimensional motion is s/t whether vectorial or scalar, thus a two Dimensional speed would be s2/t2 and a three dimensionally distributed speed would be s3/t3. Since rotational representation distributes the directionality of a displacement motion in all directions of space, it is a three dimensional speed, a motion that is capable of offering effective resistance to any change of translational motion in any direction. Otherwise, translational motion at light speed could and would be taking place in any vacant dimension; for example, photons and light speed sub-atoms. The concept of resistance to change of motion in all directions of a three dimensional system is referred to as inertia. The magnitude of the effect of resistance to any change of vectorial motion in any direction in space is called inertial mass. The required mathematical representation for the inertial mass effect caused by the rotationally represented displacement of each atom is formulated in general as the reciprocal of three dimensional motion, t3/s3. If this formulation were not correct, inconsistencies would appear very quickly.18

The Avogadro number is commonly given the units of atoms or molecules per gram-mol. When considered as the number of complex motion structural units equivalent to one natural unit of mass per gram of molecular or atomic mass, the reciprocal of that unit gives the value of mass effect (in grams per unit of structure) equivalent to one natural mass unit. The value of the Avogadro number has been determined by several methods, some of which have been adjusted according to a theoretical interpretation which is questionable in the light of the basic assumptions upon which the Reciprocal System of theory is built. The value 6.024861023 units per g-mol is more consistent with the subsequent derivation of various mass dependent constants. Pending further study, 1.6597910-24 grams per atomic mass unit is taken in this work as the unit of inertial mass.

$$M = mv = \frac{t^2}{s^3} \times \frac{s}{t} = \frac{t^2}{s^2}$$

[momentum is the two dimensional analog of mass]

In the definition for kinetic energy K.E. = ½mv2 ; Energy has the units of mv2 = t3/s3(s/t)2 = t/s [energy is the one dimensional analog of mass]

Since v = s/t ; and a = v/t = s/t /t = s/t2:

$$F = ma = \frac{t^3}{s^3} \times \frac{s}{t^2} = \frac{t}{s^2}$$

Thus, force and acceleration are seen to be the dimensional inverses of each other. From the basic units of mass, length of space, and quantity of time the other units of mechanics are derived. All physically meaningful relations are shown to be motion or a relation of motions in which the relative factor dimensions of the numerator are always the same or less than the factor dimensions of the denominator. For example, pressure is defined as the force per unit area;

$$P = \frac{F}{A} = \frac{t}{s^2} \times \frac{1}{s^2} = \frac{t}{s^4}$$

which can also be shown to be energy per unit volume; t/s / s3 . Using the conventional units of centimeters and seconds, the values in Table 5 reflect the natural units and their equivalent conventionally named basic units in mechanics.

Table 5: Space-Time Units

 s space 4.55881610-6 cm 4.55881610-6 cm t time 1.52065510-16 sec 1.52065510-16 sec s/t speed 2.9979301010 cm/sec 2.9979301010 cm/sec s/t2 acceleration 1.9714731026 cm/sec2 1.9714731026 cm/sec2 t/s energy 3.33563510-11 sec/cm 1.4917510-3 ergs t/s2 force 7.31688910-6 sec/cm2 3.27223102 dynes t/s4 pressure 3.520646105 sec/cm4 1.574491013 dynes/cm2 t2/s2 momentum 1.11264610-21 sec2/cm2 4.9759310-14 g-cm/sec t3/s3 inertial mass 3.71138110-32 sec3/cm3 1.6597910-24 g

## Relations Unique to the Universe of Motion

The limitation in three dimensional space to unidirectional motion results in the inability to exceed the speed of light, unit velocity, as effective movement in any direction of three dimensional space. A type of physical relation peculiar to the universe of motion which is not encountered in conventional physics, besides the ability to reduce all quantities to motion terms, is found in a lack of limitation to unidimensional and unidirectional motion.

Translational movement of massive particles at light speed in one dimension is the result of unit effective displacement in one dimension and has been referred to as equivalent primary speed. The postulates of the Reciprocal System of theory require the existence of effective speeds up to and including unit velocity in all three dimensions whether as primary motion, photons and light speed sub-atoms, or as equivalent primary motion. Therefore, the representation of some kind of effect relative to the one dimension of space that is usable for direct representation must be available on which to build toward the effects of equivalent primary motions that are not directly representable in the two perpendicular dimensions.

We have considered one method of representing an effect of displacement motions in more than one dimension by the directional characteristic of rotational representation. One of the results of that representation, quantified by an inter-regional ratio, causes atoms to be effectively much smaller than one natural unit of linear spatial progression. A similar effect is observed for translational velocities that cause equivalent primary motion to exist in one or two of the perpendicular dimensions whose effects are otherwise not directly representable. This is observed also as an apparent reduction in size for the stars classified as white dwarfs.

A factor that has been given only passing attention is that of boundaries between different regions in the universe of motion. The boundaries of the regions are all locations in either three dimensional aspect at which the value of either or both of the aspects of motion are at unity in any one dimension. If only one aspect remains at unity all variability of motion beyond that boundary results from changes in the effective quantity of the other aspect.

The ordinary region of everyday experience involves quantities of motion having net value less than one, but the effective values of both aspects are greater than one in the mathematical relation that is the effective measurable motion between reference points. This is the region of linear movement in extension space at speeds less than that of light. A similar region extends beyond the three dimensional speed of light boundary in which the total quantity of the aspect we call space exceeds the quantity of time involved in the motions; that is to say, at speeds greater than the speed of light in all three dimensions of space.

Since the normal time progression is a constant which determines the relative magnitudes of spatial and temporal effects, increased total time resulting from positive displacements is reflected as less equivalent space. While effective displacement speeds greater than unit speed in one dimension can not be directly represented in generalized three dimensional space, additional motion in a second dimension causes the appearance of an equivalence effect; i.e., equivalent space; in the mathematical expressions for and appearance of appropriate phenomena. The present interpretation of the effects of total displacement motion in excess of unity (i.e., unit motion in a second or third dimension in addition to the dimension of line of sight) is that of being a reduction of size rather than an extension into other dimensions of motion. Thereby, measurements which involve quantities of distributed displacement motion in excess of unity at a reference point have been interpreted as anomalies of size. Equivalent space effects and equivalent time effects are observed in the regions called the time region and the space region, respectively. Similar effects on a very large scale are observed in the regions immediately adjacent to the photon interface region; white dwarfs, pulsars and quasars.

The key factor in the relations between motion in dimensional space and motions in the time region is that in the context of the three dimensional spatial reference system, all motion in the time region is scalar with respect to generalized space even though dimensional in time. Inside unit space the dimensions of all Notational Reference Points are dimensions of time, therefore, motions in dimensional space and motions in dimensional time meet essentially in point contact at the regional boundaries.

Table 6 depicts the physical universe in five principle regions in three sectors: the Material Sector, the Cosmic Sector, and the Photon Interface Sector. The question of the number of regions has only to do with recognition of what constitutes a boundary, not with whether boundaries actually exist. Phenomena of the Photon Interface Sector have displacement in only one dimension, the photons, or no effective displacement in at least one NRP geometric dimension, the result of which is translational velocity at the speed of light; these are light speed sub-atoms of both the Cosmic and Material Sectors. For values of motion for which the quantity of representable space remains constant at one unit in the time region of the material sector, variability of speed is obtained from changes in the effective quantity of positively directed time incorporated from the net quantity of rotationally represented positive displacement motion. Similarly, in the space region of the cosmic sector, the quantity of representable time remains constant at one unit and all variability of speed results from changes in the net quantity of positively directed space incorporated from the required units of rotationally represented negative displacement.

Two additional sub-regions which are being tentatively referred to as Secondary Astronomical regions may be thought of as parts of the Material and Cosmic Sectors or, possibly, as parts of the Photon Interface Sector. In these regions the phenomena involve displacement quantities of equivalent primary motion in one or two dimensions which are not directly representable in the line of sight dimension. Thus, the anomaly of measurement and the error of interpretation of effective sizes of the objects and distances to the objects; to say nothing about theoretical interpretations concerning composition and location of the objects. They are normal matter with more than the normal progression of the temporal aspect of motion between atoms and aggregates, thereby causing them to appear somewhat smaller in three dimensional space than normal.

Cosmic matter aggregates would appear in the Cosmic Sector in the same forms exactly the same as Material matter appears in the Material Sector, as stellar systems organized in galactic systems. It is the ejection of atoms of Cosmic matter into the Material Sector that causes their mass effects in this Sector to be less than unit value, and thereby, apparently sub-atomic from a Material Sector mass standpoint. They are not sub-atomic material structures; cosmic ray particles are Cosmic atoms in the Material Sector.

## The Inter-Regional Ratio

Of all the directions time can take in the time region, only one can transmit its effect across the regional boundary to any given dimension in space. In the absence of factors which might establish a preference, the ratio of the effect transmitted across the boundary to the total magnitude of the displacements in the time region is numerically equal to one out of the total number of possible directions. The transmission ratio thus depends on the mode of representation for the specific displacement motion and particularly on the number of geometric dimensions involved.

The factors involved in the inter-regional ratio are the number of three dimensionally distributed unidirectional rotationally represented displacement units equivalent to one unit of effective linear displacement in one dimension: 23 = 8 units. This is the # of linear displacements between linear datum points raised to the power of the # of dimensions over which the rotational representation extends or is distributed. Due to the limitations by which motion can be directly represented in a three dimensional reference system, the directional vectors for three dimensionally distributed motions can best be visualized by considering a two-unit cube as being an assemblage of eight one-unit cubes. The larger cube being two units by two units by two units to provide the two units of linear displacement in each of the dimensions of space. The eight directions in three dimensional space corresponding to the directions of the diagonals from the center of the assemblage of the eight one-unit cubes represent the most probable directional relationships for the eight displacement units in the temporal dimensions. See Appendix Figure VII.

These eight possible orientations for the one Dimensional rotations along with four possible orientations for each of the two Dimensional rotations make up the atomic orientational system inside unit space. Each of the rotating systems (one for the simple sub-atomic particles and two for the atoms of matter) has an initial unit of vibrational displacement (one for each photon) with three possible orientations, one in each dimension. For the basic two Dimensional rotation of atoms this means 32 = 9 possible positions for the two vibrational units, of which two positions are occupied. This is the number of dimensions that are to be used raised to the power of the number of dimensions that must be taken together. Thus, there is an additional 2/9 direction, due to the vibrational positioning, that can be taken by each rotational displacement unit of an atom.21

The number of possible dimensional directions, which can be taken by primary translational motion, or any displacement in that dimension, or the effect of any one one-Dimensional displacement in each Notational Reference Point compound motion construct, is

Equation 6: Inter-regional Ratio

$$\frac{2^3}{2} \times \frac{2^3}{2} \times \frac{2^3}{1} \left( 1 + \frac{2}{9} \right ) = 4 \times 4 \times 8 \left( 1 + \frac{2}{9} \right ) = 156.444\bar{4}$$

Transmission of effects depend upon, the continuity of motion; as well as the contiguity of position. Therefore, it is the randomizing effect of displacement positioning due to point contact at regional boundaries that causes the measurement of effects of motions in the time region to be reduced in what we have previously considered ordinary space. The fact that we actually measure motions and effects of motions, and interpret the measurements in terms of either space or time, means that our measurements of the motion that is the interatomic distances has been reduced by the inter-regional ratio. Thus, interatomic distances are measured to be in the neighborhood of 10-8 cm even though there is no actual measureable spatial distance less than one natural unit of space.

The concepts of scalar motion, randomly oriented dimensional systems at multiple reference points, and their effects due to representation in a generalized three dimensional coordinate system should eliminate philosophical problems. The idea of more than one unit of displacement motion exhibiting an effect within one unit of extension space, whether thought of as linear or as volume, becomes commonplace.

## The Inter-Atomic Force

An important consequence of motion in time, which takes place inside one space unit, is its equivalence to motion in space. A change in the time aspect of this motion from 1 to t is equivalent to a decrease in the space aspect from 1 to 1/t. Speeds in the time region, when considered from outside of that region, are formulated as equivalent space, 1/t, divided by total time, t, which is equal to 1/t2 in outside region terms. Since space remains constant at one unit for values of motion in the inside region and the velocity in that region is 1/t2 in outside region terms, quantities derived from such velocities exhibit the second power expression for the velocity in the corresponding relations of the outside region.13

This relationship must be taken into account in any relation involving both regions, such as our taking measurements in outside region terms of inside region phenomena. The joint activity of the displacement rotations that constitute an atom is caused by the magnitude of representable space being held at unit value. The joint activity of the rotational representation of motion in an individual Notational Reference Point system exhibits inertial and gravitational mass effects in the outside region. The three dimensions of rotational representation are separate within the region inside unit space, therefore, the one dimensional analog of mass must be used to obtain the equivalent expressions for time region effects in each of the three dimensions of time.

Scalar effects in the generalized dimensional system often appear to be dimensional because of the system in which they are being represented. The interactional force is a function of displacements in all dimensions rather than being limited to the one that happens to be oriented in the association between atomic systems because the individual Notational Reference Point systems are randomly oriented in space. The displacement in the dimension of orientation controls the valence. The effective contribution from displacements in each dimension must be determined separately and then combined.

In evaluating the individual contribution, the outside region expression F = ma becomes:

Equation 7: Force Inside Time Region

$$F = E \alpha = \frac{t}{s} \times \frac{v_i}{t} = \frac{t}{\frac{1}{t}} \times \frac{\frac{1}{t^2}}{t} = t^2 \times \frac{1}{t^3} = \frac{1}{t}$$

in each temporal dimension of the structure under consideration. To obtain the total force that corresponds to t displacement units, it becomes necessary to integrate the quantity 1/t from 1 to t rather than doing a simple summation.22 This is caused by the continuity of motion between and within the units of motion, rather than their simply existing as separate units. The force effect of a one dimensional rotationally represented motion exerted against the unit force of the natural progression in one dimension of an atom is

Equation 8: Force of 1-dimensional rotation

$$F+1 = \int_1^t \frac{1}{t} dt = ln(t)$$

If the force were strictly one dimensional, the one dimensional force that two apparently interacting atoms exert toward each other would be

Equation 9: Force of Attraction (1d)

$$F_1 = ln(t_A)~ ln(t_B)$$

Since every atom exhibits two dimensional displacement, the expression for the one dimensional force must be squared to obtain the two dimensional force exerted by the magnetic displacements.

Equation 10: Force of Attraction (2d)

$$F_m = ln^2(t_A)~ ln^2(t_B)$$

Even though the scalar motion represented as an electric displacement is a distinctly different unit of scalar motion, the fact of its representation as rotational around the common axis of the double photon system causes the effect of the electric displacements to modify the effects of the magnetic displacements in a manner different from that of a direct product of the one dimensional force effect factor. The force modification is caused by the electric rotation being a displacement of a displacement and is obtained by the relation:

Equation 11: Electric Force

$$F_E = \frac{1}{ln(t'_A) ~ ln(t'_B)}$$

In this analytical situation we are not concerned with the translational motions that various atomic structures may have as a result of thermal motions or 1D1dL displacements. We are evaluating scalar motion structures which are displaced from the normal progression of the natural reference system that have a contiguous nature when in apparent interaction. The rotational representation of displacement is the source of the gravitational force effect which brought them into contiguity in space. The net force effect of the rotationally represented motion is inversely proportional to the square of the apparent distance of separation of the apparently interacting bodies and is in opposition to the natural progression (which in the region inside unit space is inward toward the zero of the generalized three dimensional reference system of space). The distance of separation in outside region terms is an equivalent space distance between the two apparently interacting atoms and thus a distance s measured in outside region terms becomes s2 for the inside region equivalent distance. The square of the equivalent distance of separation becomes (s2)2 = s4 in outside region evaluations.

The total force equation is the product of all of the factors: FM, FE, the inter-regional ratio and the inverse proportionality of the square of the distance of separation.

$$F_T = \frac{(1)^4}{(156.4\bar{4})} \frac{1}{s^4}\frac{ln^2(t_A~ln^2(t_B))}{ln(t'_A)~ln(t'_B)}$$

Equation 12

F_T = \frac{1}{(156.4\bar{4})} \frac{1}{s^4}\frac{ln^2(t_A~ln^2(t_B))}{ln(t'_A)~ln(t'_B)}

Since this force is acting against the force of the natural reference system which is unity, substituting this value for F:

$$1 = \frac{(1)^4}{(156.4\bar{4})} \frac{1}{s^4_o}\frac{ln^2(t_A~ln^2(t_B))}{ln(t'_A)~ln(t'_B)}$$

and solving for so yields the equilibrium distance:

Equation 13: Equilibrium Distance

$$s_o = \frac{1}{(156.4\bar{4})} \frac{ln^{1/2}(t_A~ln^{1/2}(t_B))}{ln^{1/4}(t'_A)~ln^{1/4}(t'_B)}$$

Sub-notations A and B refer to different kinds of atoms or to different orientations of the same kind of atom. For application to elements in which A is the same as B, the expression simplifies to

Equation 14: Equilibrium Distance (simplified)

$$s_o = \frac{1}{(156.4\bar{4})} \frac{ln(t)}{ln^{1/2}(t')}$$

Multiplying by the conventional reference system equivalent of one natural space unit gives

Equation 15: Inter-atomic Distance

$$s_o = \frac{455.8816 \AA \AA }{156.444\bar{4}} \frac{ln(t)}{ln^{1/2}(t')}$$

$$= 2.914 \frac{ln(t)}{ln^{1/2}(t')} Angstrom~units$$

The numerical value of t must reflect the fact that from the natural standpoint zero net rotational displacement is unit rotation from the fixed spatial viewpoint even though such rotation cannot be directly represented. The t value obtained is called the specific rotation. The necessity of using specific rotation values also reevaluates all t values to positive rotational displacement equivalencies.23

The initial unit of normal progression from which the displacements must extend, must be added to the displacement values of the notation for atoms in vibration one status. Vibration two status changes the effect of one or more rotational units to half unit values for each displacement unit of the compound motion structure that is rotating on vibration two status. This is primarily due to separate system action in the basic two dimensional rotational structure of atoms.24

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