The Brunardot Series (BS)

[Epsilon=One]

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.”

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Integer

Any ellipse generated by any sequence of the Brunardot Series establishes The Inverse Square Law and orthogonal dimensions.

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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[Midgar21]

Re: The Brunardot Series (BS)

I see, now, your reasoning behind the revision of the Fibonacci Sequence. My understanding of this is as follows:

In the Brunardot Series, X is the perigee, "p," of an ellipse.

The first term is the pure perigee; the second is the soliton; the third is the vector; the fourth is the apogee. What about the fifth term, and all of its successors? I admit that this is extremely interesting, especially the linkage between Fibonacci's Sequence and the modifiers... But how does the rest of the series correlate with ellipses, in terms of structural, salient parts?

Also, given what you have stated, could you elaborate on what repercussions/implications this has on the Fibonacci sequence's seemingly inherent pervasiveness in the natural, physical world? Obviously, if this sequence is derived from the elliptical equations, which is in turn a product of seminal motion, this number series represents the fundamental numerical foundation that formulates existence...

[Epsilon=One]

Your keen insight has saved me a bit of explaining.

I see, now, your reasoning behind the revision of the Fibonacci Sequence.

Your keen insight has saved me a bit of explaining. I thought the Brunardot Series (BS) might help.

Basically, the Fibonacci sequence (along with the related Golden Ratio (GR) is noted for being ubiquitous throughout Nature. Because the revised Fibonacci sequence (rFS) is derived directly from Nature (the first sequence of the BS), I believe, the rFS (1,0,1,1,2…) is a more general/logical/complete/appropriate sequence than those conventionally assigned to Fibonacci (0,1,1,2,3… or: 1,1,2,3,5…). And, of course, the rFS opens the door for the other unlimited sequences of the BS, which are not insignificant . . . Also, not to forget, the rFS demonstrates the precise relationship to the GR (which is also Naturally ubiquitous) through the second term, the “soliton”/Natural function (NF).

The rFS has a logical reason for the selection of the first two additive terms; I can find no such reason for the first two terms of either of the conventional Fibonacci sequences. In a simple additive series only the first two terms are of significance; which, in the rFS 1st term is the first integer, the common denominator of all integers, the Elliptical Constant (EC), which is the perigee, “p,” of a Natural ellipsoid (as the first BS sequence, it is a sphere that happens to be symbolic of Infinity,; and, the 2nd term is the Natural function (NF), the soliton, “p² – p.”

My understanding of this is as follows:

In the Brunardot Series, X is the perigee, "p," of an ellipse.

The first term is the pure perigee; the second is the soliton; the third is the vector; the fourth is the apogee.

Yes!

What about the fifth term, and all of its successors? I admit that this is extremely interesting, especially the linkage between Fibonacci's Sequence and the modifiers... But how does the rest of the series correlate with ellipses, in terms of structural, salient parts?

Only the first two terms of a simple, additive series are significant; just as with an ellipse, where only ONE term is required and the knowledge of which, of unlimited, parts of an ellipse it symbolizes. (This, and rapid exponentiation, is why elliptical mathematics is so important to cryptology.)

Also, given what you have stated, could you elaborate on what repercussions/implications this has on the Fibonacci sequence's seemingly inherent pervasiveness in the natural, physical world? Obviously, if this sequence is derived from the elliptical equations, which is in turn a product of seminal motion, this number series represents the fundamental numerical foundation that formulates existence...

You seem to have adequately answered your own (rhetorical, perhaps) question. There is not much more that I can add beyond emphasis.

[Midgar21]

Practical Appeal-ication?

It is the Brunardot Series (BS) that unifies all phenomena.

It is difficult to understand/comprehend the actual significance that lies in this statement/series. It describes a system that establishes the fundamental numerical foundation that forms existence, which is profound in itself, but could we derive a practical, every-day application for this series for use as a prominent example to the public, perhaps, thereby increasing its general appeal?

If the first term, or third term, is 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).

I believe you intended to say "first term, or second term." If the third term is 2, then the vector is 2, which means the perigee and key are ~1.4, in which case, they are not integers.

[Epsilon=One]

Excellent thought.

It is difficult to understand/comprehend the actual significance that lies in this statement/series. It describes a system that establishes the fundamental numerical foundation that forms existence, which is profound in itself...

You seem to have answered your first statement.

...could we derive a practical, every-day application for this series for use as a prominent example to the public, perhaps, thereby increasing its general appeal?

Excellent thought.

I thought I would accomplish such by tying it to the Natural, enigmatic ubiquitousness of both the Fibonacci sequence (which I also revised) and the Golden ratio. Obviously, I, or the rest of the world, do not understand the situation.

And of course, also, only the second term of the BS is of much significance. I named the second term both the Natural function (NF) and the "soliton," which is probably the most enigmatic and ubiquitous Natural phenomenon for well over 100 years. The ellipse, itself, is only unusual, and unigue among all geometric figures, because of the soliton, "s," and its vectors, "v," wherein one is the resultant of the other.

I've done as much as I can imagine for some 50+ years. I'd appreciate some suggestions.

I believe you intended to say "first term, or second term." If the third term is 2, then the vector is 2, which means the perigee and key are ~1.4, in which case, they are not integers.

You're better at this than I am. And, you are correct. Not only have you caught a mistake, but you've corrected it.

I am going to edit the original post with your correction; as many Viewers will probably not read this far.

Nice neoligism. I appreciate such. In the same vein: one of my favorites is "ignore"-ance; another is "Amerigance."