[It is shown, elsewhere, that the integer values of the Soliton (s) and the Force (F) can also be half integer values.  This is because their constructs in Nature derive from the major and minor diameters of ellipsoids, rather than the partial, half constructs that they represent in Conceptual ellipses (CE).]

Conceptual ellipses (CE) are an heuristic illustration of the internal structure of Light; whereas, BHE's are a similar illustration for the structure of seminal matter, which is, commonly, referred to as "exotic" dark-matter structures (edams).

When Force (F) reaches a critical point twelve linked solitons are simultaneously formed along the doubled, mirrored axes of the radius (r), the vector (v), and the diagonal (d).  The resultant individual structure (a proto-atom) of linked solitons is referred to as an ultron, which manifests the unique structural properties of mass.  It is, thus, obvious that mass is a phenomenon composed entirely of Light that is existing in one of its three states, which is referred to as oscillescents.  The other two states of Light are:  ight are:  coalescents and dissipents.

There are many amazing aspects concerning the integer components of the Brunardot Harmonic Ellipses (BHE) as reflected by the data in all the BHE tables.  The data in all these tables is arranged in the same cell format.  The formulas that are associated with each cell are the same for all BHE tables.

Any single integer, Alpha (A), with any value from +2 on up and -1 on down, relatively, without limit, with the same calculations, creates a unique BHE, and a unique BHE series of ellipses, with all the same integer attributes of any other BHE.

With each BHE Table, only the Alpha value differs; all of the other integer values are derived using the same formulas; much like each value for the radius of a circle creates a unique circle.  That the same formulas apply is not very surprising; but, that the same integer, prime, and square attributes are maintained and generated by such simple formulas, for every integer value of Alpha (except "0" and +1), is quite surprising!  One conclusion to be drawn is that the Natural Prime numbers are uniform in distribution; each one maps to an integer; and, the integer structure itself, practically, defines uniformity and unit value; something considered the Grail of arithmetic proof, that has always been lacking in the philosophy of numbers. 

Incidentally, a circle is an ellipse with congruent foci . . . just as a straight line is an ellipse with the foci extended to near the limit of Infinity.   It can, thus,  be assumed that the internal "geometry" of the "strings" of String Theory, in so far as the theory is applicable to Reality, are also ellipses.  String Theory touches upon some fundamental "truths"; however, String Theory has been developed to fit rigidly into the confines of conventional Big Bang-General Relativity-Quantum Mechanics' physics.  A Paradigm Shift is required . . . - i.e.  The Equilibrium Theory of Reality (ETR) is missing from String Theory.  

All data is the result of very simple calculations; beginning with ANY, single integer, as defined; as demonstrated by the formulas that are shown as corollaries to Brunardot's Theorem.  It should be kept in mind that any Theory of Reality must, by definition, be simple !

The salient formulas can be located by "clicking" on numbers that are generally along the left side of the Tables; all formulas remain the same for all cells to the right unless indicated otherwise.  Definitions of terms can be found by "clicking" on the appropriate text.

Once observed, the calculations of BHE's seem quite simple and ordinary.  Remember, complexity is simplicity . . . and vice versa.  Also, don't forget, these endless series of related ellipses are generated with many profound relationships that reflect the Natural order . . . all from the most minimal of calculations . . . numbers of 20, and more digits arise from single digits.  The nature of Nature's scale is exponential.

The BHE Tables are intended for two-dimensional, heuristic illustration; there are many Minor Harmonic series, lower number relationships, that become more amazing as the depths of integers are plumbed; and, the dimensions are increased.  The obvious conclusion is that numbers are driven by Nature and NOT man . . . man's numbers are a reflection of Nature; and, the Hindu-Arabic numbering system we are familiar with is excellent for counting sacks of grain; but, a poor contrivance by which to describe Nature; note the difficulties and ambiguities of calculus as an example that is readily at hand and familiar to all physicists.

To demonstrate the complexity of the simple formulas of the Brunardot Theorem and its corollaries: try doing similar calculations with the three sides of a triangle.  Calculate a series of related, right triangles with all the sides as integers using the Pythagorean Theorem:

The significance of: Why integers? is that: Nature evolves in integers.  There are no partial solitons in nature; there are either complete solitons; or, no solitons.  Thus, all fundamental Natural phenomena involves integers and simple manipulations or calculations.  BHE's, as are all Conceptual ellipses, are representations of naturally occurring phenomena.  BHE structure evolves from the coalescence of simpler Conceptual and Brunardot elliptical structures.

All Conceptual ellipses are ellipses that are formed by the oscillations of a state of Light; and manifest as radiant energy.  Light oscillates in three very different ways: "sliding," "pulsing," and "swinging."  These three types of oscillations will be discussed elsewhere in more detail.  For now, it is sufficient to know that all three forms of oscillation are represented by Pulsoids.

Pulsoids represent every element, or phenomena, that exists within Reality.   The Pulsoid's unimetry is analogous to Conceptual, Brunardot, and Brunardot Harmonic ellipses (BHE).

When Force "F," as a consequence of a state of Light's coalescence, increases until it is an integer, Brunardot ellipses and BHE's are formed . . . manifesting as twelve solitons, which manifests as "exotic" dark matter structures (edams), which is the seminal state of mass.