Planck's Constant and Pi

PLANCK’S CONSTANT AND p
SECTION TITLE & page #

1) ABSTRACT
2) INTRODUCTION AND BASIC POSTULATE
3) MODIFYING EUCLIDEAN AXES
4) SPIRAL MODES SURROUND TRADITIONAL AXES
5) PROPERTIES OF WAVES IN FOUR PLUS EIGHT NEW “DIMENSIONS”
6) MODES OF THE NEW HARMONICS AND THEIR IMPLICATIONS
7) MODES FROM PARTIALLY MONADIC ENTITIES
8) UNIFIED WAVES AFFILIATE WITH POPULATIONS AND UNCERTAINTY
9) CURVATURE OF SPACE INTERPRETED AS A MINIMAL EXCESS ANGLE
10) DIMENSIONAL FACTORS, THE EXPANDING UNIVERSE, AND p
11) PRIME NUMBERS IN RELATION TO QUADRANT AXES OF THE WAVE
12) AN OVERVIEW OF UNIFIED WAVES
13) INTERRELATED RADIAL AND TANGENTIAL VARIABLES
14) GRAPHIC PORTRAYAL OF INTERCONNECTED VARIABLES
15) FURTHER Exponential and Unified Wave Comparisons
16) REFLECTIONS ABOUT DIMENSIONAL UNITS
17) POSSIBLE IMPLICATIONS IN BIOLOGY AND CHAOS THEORY
18) SUBSEQUENT VIEWPOINTS REGARDING MONADIC VARIABLES
19) CORRELATION EFFECTS AMONGST POPULATION MEMBERS
20) DISCUSSING TRANSITIONS BETWEEN NATURAL NUMBERS
21) A GENERAL DESCRIPTION FOR INTER-INTEGER TRANSITIONS
22) ROOT MEAN SQUARE SUMMATION OF INTER-INTEGER TRANSITIONS
23) SPECULATIONS, IMPLICATIONS & ATTRIBUTES OF UNIFIED WAVES
24) SUMMARY
25) APPENDIX A.  ADDITIONAL UNIFIED WAVE CHARACTERISTICS
26) APPENDIX B.  CHORDS WITHIN THE UNIT-CIRCLE ON A CONE
27) REFERENCES
28) PAGE INDEX FOR THE FIGURES

Two and a half millennia ago followers of pythagoras believed numbers and mathematics could explain mysteries of the universe. These pages pay tribute to their insight. Scientific generalizations sometimes necessitate concepts not previously conceived or utilized. They could not know the dominion of their mathematics was inadequate owing to the physical reality of Planck’s constant.

1) ABSTRACT

This work introduces a new interpretation for the space between integers of the   natural number counting system. The material is not easily distilled into a simple, cogent Abstract and Introduction. A mathematical theory is developed built upon the thesis that variables representing a collection of n smallest non-divisible entities called monads or quanta need to be treated as a population of objects. Such variables only occupy integer values and in the variable’s progression occupancy of the space between integers is disallowed.  For something comprised of non-divisible entities, variables can make headway only by orthogonal transition between integers, which transitions are intangible. Such right-angle advancements between integers avoid increase through fractional values relative to the origin.  The consequence of this “orthogonality axiom” will be shown to yield vector-transitions between Integers n of magnitude s_n= √(2n+1) and a spiral progression that embodies harmonic waves. The periodic harmonics emerge because a generic synchronicity exists within the sequence of ordinal counting numbers whenever two consecutive integers n + (n+1) = (2n+1) sum to a prefect square. This occurs whenever n = 0, 4, 12, 24, 40, 60, 84, etc., having respective paired sums, 1, 9, 25, 49, 81, 121, 169, etc. whose square roots are the consecutive odd integers, 1, 3, 5, 7, 9, 11, 13, etc. That circumstance conveys a harmonic periodicity that exists throughout all numbers. It makes accrual of non-divisible monads (as a population variable that increases discretely by integers), behave as a generalized harmonic wave. This unified wave based on summing smallest monads under the orthogonality axiom, is conjectured as the phenomena affiliated with how forces act and particles interact. Exactly what the smallest entities being summed are need not be specified so long as those entities are not divisible. The harmonics occur as a corollary of the natural numbers when orthogonal transitions between integers are accounted for. Smallest units of energy-time for example, can only take on integer values along traditional scales signified by Cartesian-Coordinate Euclidean-type axes. Resultant orthogonal transitions between axis integers are shown to create new cyclic processes surrounding each traditional axis. When counting small objects that are partially divisible, inter-integer orthogonal transitions occur at the fixed fraction 0 ≤ ξ < 1 of the interval between consecutive integers and transition magnitudes become s_n = √(2nξ+1). Harmonic properties sustain for all ξ and encompass conventional exponential harmonics (sinusoids).

Virtually all processes in nature can interpret through the periodicity of waves.  This treatise attempts to show that such periodicities derive from enumerating non-divisible entities, “monads”, and entities that are partially divisible where      0 £ ξ < 1.  Using only natural numbers to illustrate these smallest repeated constituents that accrue to a population, four-quadrant-symmetrical waves emerge that exhibit 4-radian rectilinearity, (in contrast to quadrants associated with 2p radians).  The key postulate that brings this periodic symmetry about is called the orthogonality axiom.  It stipulates that for counting monads, transitions between integer numbers of them must be orthogonal to (independent of) the prevailing population. The inquiry purportedly provides a