**The Brunardot Series (BS)**

The Brunardot Series (BS) is a simple, additive series of unending sequences, each with simple, additive, unending terms, such that: when the first term is "x" the second term is the Natural function (NF), "x² – x," which NF is referred to as the soliton, “s,’ (half a wave, crest or trough) of a Brunardot Ellipse (BE).

Thus the Brunardot Series is:

It is the Brunardot Series (BS) that unifies all phenomena. BS is born of complex oscillations of vibration, slide, and swing; and, is thus, physically, a Natural function (NF) rather than as conventional mathematics is: mere contrived symbolism evolved from unproven, fundamental axioms.

And, all of mathematics is Natural and fundamentally provable that depends upon the Natural integers (NI) and the manipulations and functions, such as the Brunardot Series (BS), that are derived from seminal motion’s (SM) evolution as an Emergent Ellipsoid that is heuristically described by the Emergent Ellipse (EE); all of which is provable in accordance with Gödel’s Incompleteness theorem (GIT).

Pi, Phi, Circles, Ellipses, Sinusoidal curves, and the revised Fibonacci Sequence (rFS) are directly related. Something that Einstein would love; as, he sought, until his death, to relate the sinusoidal equations of Light (special relativity) with the elliptical equations of Gravity (general relativity).If the first term is One, "1," both a circle and the revised Fibonacci sequence (rFS) are generated.

If the second term is One, "1," the first term is the Golden Ratio.

If the first term, is Two, "2," (which also generates a second term of Two, “2”) an ellipse is generated that yields the Natural integers: 1, 2, 3, 4, 5, 6, 7, 8, for its most salient structural parts (SSP).

From the above considerations, it is not only easy to understand why the Fibonacci sequence and Golden Ratio are ubiquitous throughout Nature; but also, why Stephen Hawking edited with commentary the nearly 1,400 page compendium,

*“*

**God Created the Integers**, The Mathematical Breakthroughs that Changed History.”(If no image appears below,

*"Click"*your browser "Refresh" icon.)Integer

The Brunardot Series demonstrates the

*uniform*distribution of the Natural prime numbers . . . Nature's Units. That is; said primes, that are the value of certain focal radii, can be mapped with a simple algebraic function to the perigee of its respective Brunardot Ellipse that is generated from the sequences of said series.

And for

*any*ellipse so generated, when the perigee, "p," is an integer, so is the hypotenuse . . . and the apogee, and the radius, and the soliton, and the vector, and the wave, and the glyph, and the major diameter. Also, as is often, and predictably, the amplitude, diameter chord, and diagonal radial.

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