The Interaction of Electromagnetism and Gravitation along Equipotential Lines

The Interaction of Electromagnetism and Gravitation along Equipotential Lines

A Prelude to Advanced Energy and Propulsion Technology

Russell Kramer

When I first encountered the Reciprocal System of Theory it absorbed my attention immediately with its myriad of dynamic concepts. These include motion in time, rotationally distributed scalar motion, the three dimensions of time corresponding to the three dimensions of space and the reciprocal relation between space and time which is motion itself. Larson's ability to integrate all this and many more key insights into an integrated general holistic theory is truly an outstanding achievement. All of these and many more viable insights into universal a priori physical events led me at one to reconcile The Reciprocal System of Theory with various other geometric systems also based on scientifically verifiable phenomenon. This includes 'Synergetic Geometry' and The 'BiRadial matrix'. These will be explained in the proper context later in this paper.

While exploring the vast world of possibilities opened up to us by The Reciprocal System of Theory, I am also synthesizing this with the aforementioned Synergetic Geometry, The BiRadial matrix and related systems. There is a common purpose for this and more to this story; we shall begin.


There are technological applications of the Reciprocal System of Theory, which deal with clean energy and propulsion. This has enormous potential benefit for mankind. In order to develop a specific basis for describing these technologies (both new and prior arts) geometric models will be employed extensively. Larson, in reference to The Reciprocal System of Theory states: " lends itself readily to representation by pictures and models." There are certain descriptions of atomic structure, three dimensionally rotationally distributed scalar motion and the space time progression. These models are very useful and they will be developed, refined and interpreted in this paper.

The geometric modeling used throughout this work represents a parallel, partially overlapping development to the Reciprocal System and was conceived independently. Any conclusions derived from this, which are consistent with the Reciprocal System of Theory, will only further validate both systems. Here is a direct relation between the present work and 'Synergetics', as originally propounded by the late Buckminster Fuller. His nature based coordinate system gave rise to a detailed system he named "Synergetic Geometry". There are profound similarities and equally profound differences between these two systems. Also there are two key aspects of the present work. The BiRadial matrix itself which is a concise geometric model as such consisting of lines, angles and so on, and the subsequent physical interpretation of this model as it relates both to the Reciprocal System of Theory and observed physical phenomenon. There are two basic premises in using the geometric topological models illustrated in the following:

It will be demonstrated that the operational principals of certain unconventional energy and propulsion technology result from the interaction of electromagnetic and gravitational forces along equipotenetial lines. There are various "N" machines that are reported to have attained "over unity" energy output. "N" machines, as I have come to understand the term, are based on the original homopolar generator constructed by Michael Faraday. It consists of a cylindrical conducting disc immersed in an axial magnetic field that can be used as a generator with sliding brushes extracting current caused by the voltage induced between the outer and inner regions of the disc when the rotational energy is from an external driving source. Later scientists, including Bruce DePalma, have constructed N machines based on variations of Faraday's original design with varying degrees of over unity energy output reported.

Many of these machines are presently explained in terms of the "tachyon" theory and other related theories. There are also the "æther" theory (past and present), "Zero point energy" and various extensions of quantum and relativity theory. The Reciprocal System, The BiRadial matrix, along with Synergetic geometry and related systems in concert with each other offer, I believe a more generalized and comprehensive foundation for explaining these machines; including how these phenomenon are related to the innate geometry and topology of the universe.

Moreover there are a number of "electro-gravitic" energy and propulsion systems whose underlying process and performance characteristics can be readily explained by the present systems combined. The sudden bursts of acceleration and stops along with right angle turns and other difficult maneuvers with the presence of radiation are often noted during or just after sightings of "Unconventional Flying Objects" (UFO's). The present work will provide a foundation upon which such a propulsion system and related energy systems with radical performance characteristics can, in principle be built with our present technology on a sound scientific basis.

Larson states: "Gravitational energy is not interchangeable with other forms of energy... It is true a change in location results in the release or absorption of energy, but the gravitational energy at point A cannot be converted to any other type of energy at point A nor can the gravitational energy be transferred unchanged to any other point B [except along equipotential lines.]" (Italics mine) This is important and the very nature of these equipotential lines and a geometrical interpretation of these in light of Larson's statement will be explained in due course.


The technological applications of The Reciprocal System are enormous. From quantum mechanics and Relativity came photo-voltaics and atomic "nuclear" energy in wide spread applications. The Reciprocal System and its co-existing geometrical counterparts are a major development since Relativity and Quantum Theories evolved. Are there technologies that constitute an analogous improvement over existing technology based on the Reciprocal System of Theory? How can The Reciprocal System be translated from theory to practical benevolent technology?

The Reciprocal System, in addition to predicting the existence of phenomenon inaccessible to observation (motion in time for example), also anticipates novel forms of energy and propulsion technology. What new technologies will arise from the Reciprocal System? What light does this shed on the understanding of "Over Unity" energy and "zero point" energy? How can it shed light on the technical challenges we face in:



Requirements for the Technological Applications

Properly establishing the technological applications of the Reciprocal System for radical energy and propulsion systems requires evaluating the three basic laws governing electricity, magnetism, and gravitation with a review of the term "F". In terms of electricity "equipotential" describes points in an electric field, which are at the same electrostatic potential. An equipotential line traverses all those points with the same electrostatic potential and is in 90-degree orientation in relation to the field lines. The electrostatic potential at a point is the work done against the forces of the electric field in bringing unit positive charge from a point at zero potential to the assigned point. There are analogous magnetic and gravitational "equipotentials" to be dealt with presently.

Three fundamental inverse square laws for attraction and repulsion.


Fg = G M1M2/dh²
Eq. (1)

G = Gravitational Constant

F = Force of Gravity

M1 = Mass 1

M2 = Mass 2

dh = Distance between centers of M1 and M2


Fm = A P1P2/d²
Eq. (2)

A = Constant of proportionality

F = Force of magnetism

P1 = Pole 1

P2 = Pole 2

d = distance between centers of P1 and P2


Fe = K Q1Q2/d²
Eq. (3)

K = constant of proportionality

Fe = electrical force

Q1= point charge 1

Q2 = point charge 2

d = Distance between centers Q1 and Q2

Diagram one represents a gravitational mass whose center is at N1. The direction of the field is everywhere radially inward and the spacing of the lines shows the field is stronger at the surface and weaker away from the surface. As many radial lines as desired can be drawn to represent the field but they must be equally spaced. The more lines, the greater the corresponding mass.

These diagrams represent an "ideal mass"; that is one with uniform distribution of mass. Rotation can be imparted in one or more axis (corresponding to rotationally distributed scalar motion). We will momentarily freeze this motion to glimpse at the geometrical and topological structure of gravitation. In diagram 1 there are 36 equi-spaced radial lines. The specific units of measure (e.g. metric tons, micrograms, etc.) will be dealt with later. The model consisting of concentric radial lines is employed extensively by Buckminster Fuller in is epochal work "Synergetics", in which he develops a comprehensive model of natures coordinate system. The extensive use of equi-spaced radial lines through out both volumes (Synergetics 1 and 2) is including detailed explanation as to how these radial inward-outward lines correspond to gravitation and radiation respectively. This same model us employed in more conventional textbooks as well. For now let us allow that the number of equi-spaced radial lines corresponds to "mass" in some way.

Referring again to diagram 1, N1 = number of rays from N1 and N2 = number of rays from N2. The following point is crucial. That is the model of a gravitational mass (a single pole with equi-spaced radial lines) found in textbooks appears to be derived from a thought process not unlike the following: Observation indicates that objects fall towards the Earth's center (of gravity). Also the rate of acceleration of falling objects from the same elevation above sea level any place on Earth is the same; therefore the gravitational field of the Earth is uniform (in theory). This is thus represented by the uniformly distributed equiangular radially inward lines. The closer the lines are the stronger the force of gravity or the "denser" the gravitational field. This is generally consistent with observation. This model is further reinforced by the following facts: the rate of a falling object toward the Earth conforms to an inverse square law. In geometry, as a concentric surface (sphere or a regular polyhedron) contracts toward or expands from a central point or "pole," the area of the surface changes by the square of the distance from the center; thus also conforming to an inverse square law. (See diagrams 4 & 5 and equations 1, 2, & 3.

From all of the aforementioned (and likely other factors as well) it is concluded that gravity is mono-polar (i.e. emanating from or centered about a single source or point.) This is hardly ever stated, yet it is implied in most extant theories and models dealing with gravitation. While it is true that the inverse square relation in diagrams 4 & 5 holds true for most if not all polyhedrons, and while it is also true that Newton's law of gravitation (equation) is also an inverse square law, it does not follow that the former is the basis of the latter. At some point it was concluded that since gravity and the monopolar surface expansion and contraction phenomenon both obey an inverse square law that gravity must be monopolar. I believe this is a basic epistemological error that has greatly hampered our efforts to relate gravity with electromagnetism in a precise and quantifiable manner; as will be demonstrated.

When observing objects "falling" towards the Earth, (Newton's legendary apple, for example) because of the relative vastness of the Earth in relation to the apple (among other possible factors) it is assumed that the Earth's center of Gravity is the sole reference point which the apple is falling towards. Again this model assumes ideal conditions where the density of a given mass is uniform. In any case the Earth is (if only to a slight degree) "falling" towards the apple. In fact an observer on the apple would measure the Earth falling toward the apple at 32 feet per second squared. The point is that any attempt to measure the Earth's, or any other body of mass's gravitational field means introducing another instrument (as simple as a falling apple). This instrument is not measuring the Earth's gravitation; rather it is measuring the (apparent) interaction of the Earth's gravitation with its own gravitation complete with its own center of gravity. With regards to the Reciprocal System, Larson states that gravity is not an interaction or "action at a distance" as such. "Like the gravitational motion, the motion which gives rise to the electric or magnetic forces is a motion of the individual atom or particle with respect to the general structure of space time, not an action of this unit on some other charged or uncharged object as it appears to be." Returning to equations 1, 2 and 3 it becomes clear, or at least entirely plausible, that the two poles of gravity are the two masses M1 and M2. This is a critical distinction. It clearly reveals that gravity has at least a bipolar (BiRadial) structure if it is not entirely bipolar or multi-polar. It appears to suggest that gravity, as expressed in equation 1, requires the presence of (at least) two masses and in this respect can be considered bipolar or "BiRadial". Buckminster Fuller states "Unity is plural and at minimum twofold". In the present context this suggests that to have unity (unification) of a field requires the presence of two masses or poles of some sort. We will in fact arrive at an inverse square equation based on BiRadial geometry.

Once it is established that gravity has bipolar, even multi-polar properties, the geometrical and mathematical unification of gravitation and electromagnetism is closer at hand. In fact, for a single equation to unify the three above equations would require at least two qualities: first it would need a variable in place of the three constants contained in the three separate equations. (The variable would correspond to either one of the three constants.) Also, there would need to be a variable unit of measure, which would correspond to electricity, magnetism or gravity. As we know from the Reciprocal System, the units of measure are rotationally distributed scalar motion, which can be further simplified into discrete space-time relationships.

Diagram 3 represents two identical masses separated by a distance "D". Diagram 6 shows the intersection points of field 1 and 2. What is the significance of these points? How do they relate to gravitation or Electromagnetism? They are reminiscent of the points on the expanding balloon used by Larson and others to represent scalar motion and the outward space time progression. This array has both a macroscopic and microscopic physical analog. For example if we view the distance between the poles as very great and increasing over time this model depicts the outward scalar motion of the universe. If we view the distance between the poles as extremely minute, we can view this array as depicting the highly concentrated "Lines of force" present on the atomic and sub-atomic level. There is a direct relationship of these points to scalar motion to be further elaborated on. There are several symmetrical patterns, which can be derived from this array.

This investigation will examine two variations that relate directly to novel forms of energy and propulsion. Emphasis will be placed on the physical interpretation of the patterns. Certain assumptions are made and rigorously justified before moving on to subsequent analysis and conclusions. Interconnecting the nodes by a certain criterion (which relates to describing the curve mathematically), yields diagram 7. This represents what appears to be a magnetic field (attraction mode), resulting from two opposite poles attracting each other. Interconnecting the nodes by a similar criterion yields diagram 8. This represents what appears to be a magnetic field resulting from two like poles repelling each other. The original rays can be extended in three dimensions, allowing for a more comprehensive model. There is a very distinct relationship between these attraction and repulsion lines and actual magnetic lines of force. This demonstrates the utility of The BiRadial Matrix in revealing basic physical properties of "fields". The criterion for connecting the "lines of force" is crucial and involves deriving actual BiRadial equations describing the quantized "lines of force," discovered by Faraday and easily revealed with iron filings in close proximity to a magnet.

We have developed this model to illustrate to masses and the gravitational interaction surrounding them, after having rigorously justifying the use of this model as a viable analogy to gravitation. Could it be that gravitation has a geometrical and topological structure similar to magnetism and also a quantized structural basis for its description within the BiRadial Matrix? The concept of a quantized description of gravitation is a major goal in modern physics. Any complete physical system, be it a theory or model, must account for the specific geometrical and topological structure of primary "motions" of nature. Also it must explain both the similarities and differences between their properties.

As regards to the similarity of the conventional models, such as the electric "dipole", it should be noted that this similarity is not exact. The distance D in the conventional dipole is equal to zero and the related math is divergent from the inherent math in the matrix. Returning to Diagram 3, we can assign specific variables and derive a number of simple yet revealing equations including equation 4. Upon close analysis of this equation it becomes apparent that it has the same form as Equations 1, 2 and 3 and helps reveal the relationship between all these three equations showing the interrelationship between gravitation and electromagnetism. Derivations aside for now we have:

1 = sin b / sin a + sin a / sin b - cos q (a b ) / D²
Eq. (4)

There is a variable in place of the three constants--in place of the force factor on the left side there is a "1". How is this to be interpreted?

Syndyne is conducting research and development in this area with a firm conviction that extensive analysis of the BiRadial Matrix and related math can reveal such interactions and provide a greater subtlety of understanding the basic "forces" of nature. What is happening is that both the inherent geometry of the matrix along with the related math are both simultaneously pointing to the interrelationship between gravity and electromagnetism; there being total internal consistency. As will be further elaborated upon, the BiRadial and PolyRadial Matrix explains the structural aspects of the interaction between gravitation and electromagnetism. While the Reciprocal System addresses this, and thoroughly explains other related issues such as why gravity cannot be shielded, and by integrating the BiRadial Matrix, the Reciprocal System, Synergetic Geometry and related systems, a true comprehensive approach to investigating the Unified Field along with its verifiable and measurable effects (including "cold fusion," "zero point energy" and radical propulsion, for example) is possible.

Returning to equation 4, there is an obvious reciprocal relationship being expressed, which shows a direct relation to the Reciprocal System; the variables a and b are prominent. So far as I can determine, a and b are related both geometrically and mathematically to the equipotential lines referred to by Larson, and equation 4, with the corresponding diagrams, strongly suggest that electromagnetism and gravitation are indeed capable of interaction along equipotential lines. In fact there are numerous relationships, which directly emanate from this basic system of modeling, of which this is but one example.

The matrix is indicating a coupling effect between electromagnetism and gravitation. This is consistent with current explanations of electro-gravitic energy and propulsion devices and related experiments. At its utmost, this matrix can provide a thorough and detailed foundation for describing primary field interactions, clean energy and propulsion technology, both existing and under development.

When the distance "D" between N1 and N2 (see diagrams 3, 6, 7 and 8) decreases, this predicts a collapsing field and enormous inward accelerations of energy. When the distance "D" increases, this predicts an expanding field, and enormous outward accelerations of energy. Also in Diagram 6, by increasing N1 and N2 (the number of rays from each pole) to higher values a series of finer wave interference patterns emerge. The relationship of these to the distance between the poles expresses the basic harmonic sequence. Harmonics are logically integrated with the BiRadial Matrix and related to electromagnetism and gravitation. In fact, there is a key contingency of researchers who are keenly focused on the relation between harmonics and gravitation.

The BiRadial Matrix describes this phenomenon implicitly. Just as electricity and magnetism can be derived from each other, the results of the present investigation predict that (electro)magnetism and gravitation, in various states (forms), can be derived from each other is an analogous way and that they are both related to harmonics. This will be elaborated upon in the next article. To be sure the technological applications of this are enormous. It stands to reason that this is the empirical basis for the conversion of gravitation to electromagnetism (i.e. the interaction of gravitation and electromagnetism along equipotential lines.) When we regard the distance "D" as very small on the atomic scale, corresponding to the atomic mass, the repulsive "lines of force" depict a concentrated magnetic potential. The "forces" on the atomic scale are highly concentrated and exist over short distances. By decreasing the number of rays from each pole (corresponding to lower masses such as electrons) and decreasing D, this yields a highly concentrated pattern, which exists over a very short distance. Further illustrations continue to reveal this fact. By following this geometric analogy concisely this model helps explain the unique property and how it differs from magnetism and electricity. Further interpretation accounts for the inter-atomic region often referred to as the "nucleus" and shows how it is a concentrated form of the same basic fields being discussed at present.

During my presentation of this BiRadial matrix at the I.S.U.S. 22nd annual meeting, Frank Meyer commented on what the definitions of the "lines of force" are. Much has been observed of these by notable experimental scientists such as Michael Faraday, Nicola Tesla and others. In the present context I am compelled to reevaluate the precise meaning of "force lines" as they play such a pivotal role in the BiRadial and PolyRadial Matrices.

From what has been revealed thus far, we know that they obey certain mathematical formula which greatly simplify the equations, as compared to the Cartesian mathematical equivalent. We know that these lines of force are quantized. From the Reciprocal System, we can add to this fact that the so-called "force" is really a form of rotationally distributed scalar motion. We know that the precise pattern of these initial "Lines of force" are not exactly similar to the electric "dipole," for instance, and that the BiRadial Matrix takes into account the discrete interaction of forces. As discussed in my previous article, fusion, as such, is not what is occurring in the sun or on the subatomic level. Advanced prototypes of the present model yield a more general set of equations accounting for both the microscopic and macroscopic phenomena including the inter-atomic region. This model has an enormous ability to predict numerous process and physical relationships based on "trans-discrete superstructures". The corresponding mathematics is much simpler using essentially algebra, trigonometry and primary number structural patterns. This is in keeping with the Reciprocal System.

The lines of force in diagram 8 clearly illustrate a repulsive phenomenon, which is the apparent basis for "anti-gravity" propulsion. There will be more on this in the next article. Diagram 9 has a direct relation to atomic structure and provides a detailed description or analogy of the inter-atomic region. It is one of a series of related diagrams where the number of poles "N" can vary from N=2 through N=92 and perhaps beyond. It is a logical extension and generalization of the BiRadial Matrix, and shows further relationships concerning the fundamental motions or "forces" of nature. The general concept, in any case, is that the motions prevalent on the atomic scale are the same basic "lines of force" as electromagnetic and gravitational "Lines of force"; the distinctions are in the relative configuration of the lines of force and their density and degree of rotational scalar motion. In all cases, the common denominators are the lines of force consisting of their quantized segments and angular progressions. Both of these directly result from the geometrical constraints of their finite number of rays emanating from (in this instance) the 12 poles. An entirely new picture of the atom emerges where the surrounding poles form the central region as opposed to the center "attracting" the outer region of the atom. At the same time this model is clarifying the relationship between quantum physics with both electromagnetism and gravitation. This is a major step and the precise mathematical equations along with the geometric structure (as in the BiRadial and PolyRadial matrix) along with related systems, and the theoretical construct (as in the Reciprocal System) are now both in place to proceed on course to bring this research to fruition and culminate in a thorough assessment and description of the "Unified Field" and its unlimited applications. Further in depth explorations of the BiRadial and PolyRadial matrices and there their relation to the Reciprocal System is in progress and shows great promise.



Enormous possibilities lie ahead. We are on the thresh hold of a great era of understanding in science and technology. Progress on both the theoretical and technological fronts is unfolding at a spectacular pace. It is crucial that theoretical and applied physics is integrated and the direct relation between them fortified. The structural aspects of the primary "forces" (equipotential lines among them) are a vital link between physics theory and application.

Post script

The present author, through his company Syndyne, is seeking mathematicians, scientists, engineers, computer programmers and sponsors to accelerate the research on the BiRadial Matrix and related technologies. Those interested please contact Russell Kramer at: Syndyne.


  1. There is a direct correspondence between the known motions of the Universe (otherwise known as "forces") and the geometric-topological elements including lines, angels, genus, inwardness and outwardness.
  2. Our own perceptual and conceptual apparatus (our means of interacting with our surroundings) is directly interrelated to the three dimensions of space and the three dimensions of time. (This second premise will be dealt with in a later article. This is an important adjunct to one or both of the fundamental postulates of The Reciprocal System of Theory.
    • Permanent Magnet motors?
    • Methods for harnessing "atomic" energy?
    • Designing new propulsion systems?
    • Cold Fusion?
    • Review of the applications of the Reciprocal System.
    • Various geometrical models of the Reciprocal System were cited mainly: atomic structure, scalar motion and the space-time progression.
    • A working definition of equi-potential lines was given
    • A geometric model consisting of equi-spaced radial lines was introduced as an equivalent to mass. The assumptions on which this was based were justified.
    • The three primary inverse square laws were reviewed.
    • Two distinct geometric models for deriving inverse squares laws were shown
    • A case was made for why a BiRadial system should be incorporated in the treatment of gravitation.
    • A quantized attraction vector matrix was derived, as was a quantized repulsion vector matrix, solely from geometrical and topological considerations, in particular a BiRadial matrix.
    • Numerous physical interpretations presented themselves. These models were interpreted in light of the Reciprocal System and other models including the harmonic sequence.
    • A unique inverse square relation was derived from the model, which offers some clue on how electromagnetism and gravitation can interact along equi-potential lines.
    • Once the physical relation to the model was established, possible applications in clean energy and propulsion were discussed.
    1. Arfken, George, Mathematical Methods for Physicists (Academic Press, New York, 1966).
    2. Cramp, Leonard G, UFOS and ANTI-GRAVITY: a piece For A JIG-SAW (Adventures Unlimited Press, Kempton, Illinois, 1996).
    3. Custo, Hans, The Cosmic Octave (Life Rhythm Publication, Mendocino, CA 1988).
    4. Fuller, Buckminster, Cosmography (Macmillan Publishing Company, New York, New York).
    5. Fuller, Buckminster, Synergetics 2 (Macmillan Publishing Company, New York, NY, 1979).
    6. Garbell, Maurice A., Soviet Research on Gravitation (Garbell Research Foundation, San Francisco, CA).
    7. Haken H., Wolf, An Introduction to Atomic and Quantum Physics (Springer-Verlag, New York 1987).
    8. Hill, Paul R., Unconventional Flying Objects, a scientific analysis (Hamton Roads Publishing Company, Charlottesville, VA, 1995).
    9. Kramer, Russell, The Social and Technological Implications of The Reciprocal System of Theories, Reciprocity, Volume 25 No. 3 (North Pacific Publishers, Salt Lake City, UT, 1996).
    10. Larson, Dewey B., New Light on Space and Time (North Pacific Publishers, Portland Oregon, 1965).
    11. Larson, Dewey B., Basic Properties of Matter (North Pacific Publishers, Portland, Oregon, 1988).
    12. Lehrman, Robert L., Physics The Easy Way (Barrons Educational Series Inc., Hauppauge, NY, 1984).
    13. Longman, Dictionary of Scientific Usage (Logman Group Ltd., Longman House, Burnt Mill, Harlow Essex).
    14. Mallove, Eugene F., Fire From Ice (John Wiley and Sons Inc., New York, 1991).
    15. Meyer, Frank, Correcting Discrete Time/Space Measurement Procedures, Reciprocity, Volume 19, Number 3 (ISUS, Inc., Salt Lake City, Utah, 1990).
    16. Ouspenski, P.D., A New Model of the Universe (Vintage Books, New York, 1971).
    17. Pais, Abraham, Subtle is the Lord, The Science and the Life of Albert Einstein (Oxford University Press, New York, 1982).
    18. Price, William, Effects of Phase Relationship on the Interactive Components in a Bipolar Matrix (Unpublished).
    19. White, Harvey G., Introduction to Atomic and Nuclear Physics (D. Van Nostrand Inc., Princeton, New Jersey, 196

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