Unimetry describes

 the Natural creation of mathematics

that begins with Brunardot Ellipses,

which are two-dimensional, heuristic,

manifestations of a seminal pulse.

Ellipses are defined by the Brunardot Theorem:

c² = 2v² - s².

c = major chord, v = vector, s = soliton.

The Pulser of an ellipse defines every ellipse.

Said Pulser is the perigee, p; and vector, v.

The vector always equals

the sum of the perigee and the soliton, s:

v = p + s.

The cardinal value of any two terms,

or structural parts, of any ellipse,

defines that ellipse.

The Brunardot Series consists of

unending sequences, each with unending terms,

that have any Natural integer

for the first term of each sequence and,

the third term is the square

of the first term; and,

as with successive terms,

the third term is the sum of

the preceding two terms.

The ratios of the terms of any Brunardot Series Sequence converge at Phi, F, the Golden Ratio, as the terms increase, when any term is divided by the preceding term.

Phi, F, the Golden Ratio equals one plus the square root of 5, all divided by 2, or, 1.618...,

Within the Brunardot Series; and the related Tinicir arrays, and their elliptic, circular, and generated three- and four-dimensional manifestations; are found all the fundamental concepts of all the fundamental phenomena of Reality: mathematics and physics . . . science, theology, and philosophy.

Tinicir is an acronym for

Tangent, Infinite, Integer Circles.

Tinicirs have curvatures that are

Natural Integers; and,

Tinicirs are similar to Brunardot Ellipses because

they are generated from

unending sequences with unending terms

that are Natural Integers.

Brunardot Ellipses are defined by

a Pulser, which is:

the first and third terms of any Brunardot Series;

thus, the Brunardot Elliptic Theorem is:

v = p².

The cardinal value of either term of

the Pulser of a Brunardot Ellipse,

or any other single, structural part,

defines a unique Brunardot Ellipse.

The six most salient structural parts

of Brunardot Ellipses are:

 perigee, p; soliton, s; vector, v;

diagonal, d; radius, r; and, force, f.

Other structural parts are:

apogee, a; and, major chord, c.

The perigee can be any Natural Integer;

the soliton is an even integer;

the vector is a square,

as is the radius in the higher ellipses.

The diagonal is unusually salient;

the diagonal is a Natural Prime number,

which demonstrates that said

Natural Prime numbers are

uniformly distributed,

as the diagonal can be mapped to the perigee;

which can be any Natural Integer.

The distribution of

conventional prime numbers

appears random because

they are the result of

three combined sets

of Natural Prime numbers.

Some Important Terms, Factors, and Ratios

of Brunardot Ellipses are:

CU = Conceptual Unit,

HF = Harmonic Factor, and

K = Unimetric Factor.

Primary Corollaries of the

Brunardot Elliptic Theorem are:

2p - r = d - 2s = 2d - r² = r² -  4s = p - K = CU = 1,

2s + 1 = (r² + 1) / 2 = 2v - r = d,

2p - 1 = v - s² / v = SqRt(2d - 1) = r,

SqRt(rv) = pSqRt(r) = HFp = f, 

(r² - 1) / 4 = p² - p = s,

a = 2p² - p,  v = (a + p) / 2.

The only universal constant is the Conceptual Unit, One, 1; the speed of Light is inversely proportional to the local value of Light, L, or 2s.

There are many classes of Brunardot Ellipses.

Four such classes are:

Seminal Ellipses, Natural Ellipses,

Conceptual Ellipses, and Harmonic Ellipses.

Seminal Ellipses are the simplest,

or first order, of Brunardot Ellipses, which have

the soliton, s, and the diagonal, d,

as Natural integers.

Natural Ellipses are Seminal Ellipses

which have the perigee, p; and the vector, v; 

as Natural Integers.

Conceptual Ellipses are Natural Ellipses,

 which have the force, f, as a Natural integer; and,

the radius, r, as a square.

Every Natural Integer generates

a unique Conceptual Ellipse.

Harmonic Ellipses are Conceptual Ellipses

that are generated by

every Conceptual Ellipse

in unending iterations, of four cycles.

Each Harmonic Ellipse is defined by its

cycle within an iteration, i, and its generating integer.

All Brunardot Ellipses,

in accordance with the subsequent terms

of each Brunardot Series Sequence,

converge to a Unimetric Ellipse.

A Unimetric Ellipse is defined by a Pulser that is

any two adjacent terms, after the first term,

of a Brunardot Series Sequence.

The Unimetric Ellipse is an ellipse,

which is similar to an ellipse with a Pulser of:

1/F and F.

F, Phi, the Golden Ratio is:

(The Square Root of Five, 5, plus One, 1)

divided by Two, 2.

The soliton and radius

of a Unimetric Ellipse are:

each equal to a Conceptual Unit, CU, One, 1;

also, for all Unimetric Ellipses:

s / p = v / s = a / v = Phi, F.

Unimetric Ellipsoids,

when compressed from every direction,

evolve to Tinisoids,

which are analogous to Pulsoids.

Tinisoids are heuristically described,

two-dimensionally, by Tinicirs.

There are four forms of Tinicirs.

Tinicirs are described by two terms that establish the largest first two tangent circles.  All following smaller tangent circles are of three types which are determined 

Before time, there was a soundless, infinitesimal pulse.

Its locus is Infinity.

Its shape is spherical . . . a special ellipse.

 Because of its congruity, it is a singularity.

This spherical pulse is referred to as . . . Reality.

And; with limitless time; subsequent, elliptic pulses; and, their sequential and cyclic evolution: converge, coalesce, compress, and dissipate; this simple complexity becomes the sustaining, endless, cyclical Universe.

The Seminal Pulse, Reality, is defined by the first three terms, of the second sequence, of the Brunardot Series.  The second sequence is the "revised" Fibonacci Series.

The second sequence of

 the Brunardot Series is:

1, 0, 1, 1, ... ,

which represents: perigee = 1, soliton = 0, vector = 1, and apogee = 1, which generates  a special ellipse in the form of a circle with a radius of 2p - 1 = 1.

The second sequence of the Brunardot Series, beginning with the third term, is commonly known as the Fibonacci series.

The salient features of all the Brunardot Sequences are:
1.) the first two terms generate an ellipse that describes the Conceptual Unit, 1, as 2p - r; and,
2.) the ratio of all subsequent terms when divided by the preceding term converge at Phi, F, the Golden Ratio.

The most salient feature of the Fibonacci series is that subsequent terms converge at Phi, F, the Golden Ratio; this feature of said convergence is a function of the first two terms of the Brunardot Series.

Therefore, it follows that the Fibonacci series is a part of the Brunardot Series; and, in accordance, the first two terms of the first Brunardot Series should be included as a part of the Fibonacci Series.

Each pulse represents three forms of oscillation which create the manifestation of that pulse; somewhat analogous to Einstein's "fields."

The soliton vibrates, the vector slides, and the diagonal/radius (vector/vector) swings.

All motion is to and from the extreme dualities of Infinity.

When time is minimal, size is minimal, and the speeds of oscillation are maximal.

In Summation:

The significance of all numbers of the Brunardot Series and the Tinicir Series being Natural Integers is that they represent a soliton or a multiple of a soliton.

Partial solitons do not exist anymore than partial waves or partial "charges" exist.

The underlying principles of the most complex, by definition, must be the most simple.

Mathematics originates with Natural phenomena; therefore, the simplest mathematics must reflect the most underlying principles of the most complex Natural phenomena.

All the endless structures which result from the Brunardot Series and the Taicir Series are heuristic representations of three-dimensional energy pulses which are generated from and about axes that are infinitely long and infinitesimally narrow.

As said pulses dissipate with the increase of time, to their originating singularity, their internal oscillations slow, they expand relativistically, and their dissipation is countered with increased pulses from along their locus of Reality, from which they originated.

Thus, the Equilibrium Theory of  Reality with endless cycles of pulsating ellipsoids; their compression to atoms within quasars; and their dissipation as pulsating, elliptical strings.

Conventional dark matter and dark energy are largely manifestations of said pulsating ellipsoids.

Gravitation and Cosmic Inertia are largely a manifestation of said compression.

Conventional Light, dissipents, is a manifestation of said dissipation.