A Quantum Constants' Relation to Natural integers

A Quantum constant�s relation to Natural integers
A simple, Natural, geometric construction of the Planck constant

This thread is a reply to questions and responses posed at an outside forum
at topic: Physics, Math, & Philosophy Homework;
Thread: Proofs; in posts #3 through #5.

The Planck constant (PC), and its derivatives, directly link Pulsoid Theory (PT) to Standard Model (SM) quantizing . . . the quantum theories.

PT�s proof of the value of �One� is important, beyond its refutation of Kurt G�del�s �Incompleteness Theorem,� because its proof is dependent upon the Elliptical Constant (EC), the common denominator of all integers, which is a heuristic description of the Conceptual Unit (CU), the common denominator of the radii of the smallest manifestation of Nature (Resoloids) and the unit of fundamental, intrinsic time (FIT).

The �smallest manifestation of Nature� is an apt description of the Planck constant (PC) that is at the foundation of the most respected insight of the many intellectual giants that is contained within academia�s evolved Standard Models (SM) of physics.

Linking the concepts of the Planck constant (PC) and Natural integers is profound, subtle, beautiful . . . and, refreshingly, simple. Possibly, comparable to Einstein�s concept of relativity. The linking of the Planck constant to the common denominator of the Natural integers, �One,� has the added benefit of resolving G�del�s �Incompleteness Theorem� (GIT) as well as directly tying GIT resolution to Nature; and, thus, linking all mathematics to Nature.

The resolution of GIT strengthens the foundation for both the disciplines of physics and philosophy (philosophical logic).

The link is easily seen through the Elliptical Constant (EC) that is the crux of the symbolic, geometric (mathematical) unification of Reality; whereas, the Unified Concept (UC) is the philosophically logical unification of Reality.

There are many derivations of the Planck constant; so, first, it is necessary to define what Planck meant by his original quantum constant. Said constant was a construction to mathematically describe what Planck considered as the minimal quantum of Nature, at that time known as: the Wien �energy element.� Wien's �energy element� must have come from somewhere; therefore, it will be considered as an emerging, �energy element� or �eee,� which will be appropriately designated as a backward three, "3," or �ε,� εpsilon, which designates the constant of the minimal energy quantum.

From observation, this minimal energy quantum, εpsilon, �ε,� must expand radiantly while oscillating in some manner; and, being fundamental, it must be without any distinctive variations and will be, thus, assumed as oscillating seminal motion, which has no dimensions, originates as a sphere (without dimensions) with its radius/vector being designated �One�; of course, if there was anything smaller that exists, it would be designated �One.� Thus,

εpsilon = One.

As there is no quantum that is smaller, by definition, than εpsilon, �ε,� it can be said that nothing exists that is smaller; thus, a sphere with a radius of One, �1� is the limit of Reality; or simply, a sphere of radius one is the infinite (which is dimensionless) and a point (which is dimensionless) within the sphere (there is only a center) is the infinitesimal.

As there are no dimensions, the sphere and the point are congruent . . . as, in a sense, are the Natural integers Zero, "0," and One, "1," as demonstrated by the Natural function, x� � x, (NF) that is the major radial complement of the Pulse/time for any ellipse, which when rotated about its major diameter, �D,� is an ellipsoid, which is the necessary, and required, geometry of the emerging �energy element, �ε.�

Motion is the hallmark of Reality. That something does move is the saliency of the Unified Concept (UC). Said motion results in the dynamic, emergent separation of the dimensionless point and the dimensionless sphere that envelops said point.

At the instant of separation there is
an ellipsoidal quantum, εpsilon, �ε,�
with vectors that are minimally
larger than One.

Thus, it can be said that "One" is as representative of non-existence as "Zero"; or, that One, Zero, the infinite, and the infinitesimal are all singularities. As there can be only one singularity, they must all be equivalent to one another, or symbolic of . . . Infinity.

It is of the utmost significance, for the structural stability and internal harmony (the basis of resonance, which is the essence of fermions and bosons) of the Triquametric motion, of Natural quanta, that the internal, salient, structural parts (SSP) of the Emergent Ellipsoid, and its Pulse, "P," are always multiples of the Elliptical Constant (EC). Such stability and harmony occurs when the vectors, "v," are the square of the Pulse, "P," for any Emergent Ellipsoid.

The square of any value, from the infinitesimal to the infinite, approaches "One" as said value approaches "One."

Thus, approaching the minimal limit, the increment is moot and the vector/radius is conceptually �One.� While, interestingly, because of the dynamics of the separation, the vectors are the square of the radius; and, the spherical quantum has become a spheroidal Emergent Ellipsoid. All ellipses are geometric constructions with limits between a circle and a straight line with vectors that are the square of the Pulse/time; and, most importantly, as such, all ellipses have an Elliptical Constant (EC) that equals . . . One, which is the same "One" as the minimal energy quantum: εpsilon, "ε."

All Ellipses are defined by the Brunardot Theorem: c� = 2v� � s�;
and, by its corollaries, the Pulsoid Theorem: v = εP�;
and, the diameter chord = c = P(square root(v + r)).

The integer two, "2," for the Pulse, P,� is most significant, because it results in the geometry that creates first evidence of existence and the evolution of the Natural integers; this is because, most significantly, the smallest (first) resonances initiate the first "tick" of fundamental, intrinsic time (FIT). The energy transfers between the motion of the vectors and Resoloids accounts for the pulse/�tick� of the quantum.

The first resonances that evolve from harmonic Triquametric motion have radii of One; and, each subsequent hypotenuse Resoloid has a radius that increases by one Conceptual Unit (CU) with each "tick" of FIT . . . as does the "gap" between each pulsing Emergent Ellipsoid's major diameter. The value of the Conceptual Unit (CU), One, "1," remains constant for an ellipse of any size when the vector, "v," is set as the square of the Pulse, "P,"/time. It is this constant that is described heuristically by the Elliptical Constant (EC).

(If no image appears below, "Click" your browser "Refresh" icon.)

Legend for an Equilateral Emergent Ellipse (vector = wave)
Acute and Obtuse Ellipses are congruent

ε = εpsilon = 1
P = Lines BE, BC, CD, DG = Pulse, �P� = 2
s = Lines BC, CD = soliton, �s� = Natural Function (NF) = P� � P = 2
r = Line AD = radius = 2P � ε = v � ε = 3
w = Line BD = wave = 2s = 4
v = Lines BF, CE, CG, DF = vector = w = P� = 4
h = Line AB = hypotenuse = w + ε = v + ε = 5
Hr = Hypotenuse radius of circle
..................inscribed in right triangle ABD = ε = 1
ε = P � Hr = h � w h � v = P� � r = v � r
c = P(Sq rt(r + v)) = 2√7

It is at the second "tick," that is symbolized by the Pulse, "P," equals 2, that forms an equilateral Emergent Ellipsoid wherein the Natural integers evolve as lengths of the internal, salient, structural parts of said ellipsoid with the EC as their common denominator. It is the integer values, of said structural parts, that create the harmony within ellipsoidal quantum that creates the resonances, with integer value radii, that are heuristically symbolized by the circle inscribed in the above right triangle ABD. As the Pulse/time increases by Conceptual Units (CU) said resonances separate with some groups of resonances (orbital Resoloids) moving toward the ends of the major diameter, �M,� and inside the radii. The remaining groups of resonances (nuclear) converging at the intersection of the major and minor diameters. Thus, the basic Standard Model of fermion distribution.

And, thus, the EC that is always One, and is the radius of the minimal resonance, that is the minimal essence of mass, is equivalent to the Planck constant that symbolizes the smallest, emergent, ellipsoidal, energy element:

εpsilon, "ε."

And, it is the EC, as the Natural value of One that establishes the value of all subsequent integers. And, the dynamics of the ellipsoidal quantum, particularly, its Triquametric motion (TM), is the source of all fundamental arithmetic manipulations/operations.

Therefore, the Planck constant; that is related to the EC, that is equal to the Natural integer One, "1"; establishes the fundamental quantum constants' relationship to Natural integers. And, in a very real sense, the Planck constant's requisite geometry can be considered the source of all mathematics; as well as, the resolution of G�del's Incompleteness Theorem (GIT).

�Copyright 2006-2008 by Brunardot. All rights reserved.
Terms: Dialogue21.com, Brunardot, and Pulsoid Theory must be cited.

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