Natural numbers as 1, 2, 3, - - - to infinity form the basis of mathematics, physics, and technology.  This work scientifically describes how electromagnetic-type sinusoids harbor within the sequence of those ordinal-integer-numbers.  Such waves emerge when enumerating discrete monadic populations, for example action-units representing (energy)x(time), contrasting divisible entities.  Then fractional values are prohibited between integers.  Transitions between n and (n+1) become “orthogonal to” (independent of) the zero-to-n direction.  Sinusoidal-type waves with absolute periodicity of four symmetrical radians-per-cycle result.  The waves start at the origin of the precipitating disturbance, contrasting exponential harmonics that violate causality by initiating at minus infinity in time.  When enumerating continuously divisible objects rather than monads, the newly derived waves can encompass exponentials.  Energy in these new waves proportions to frequency rather than amplitude squared as in photon-energy E = hn.  Based on simple natural numbers, these waves possess many more degrees of freedom than exponentials.  Solution modes allegedly form the foundation for wave equations and quantum mechanics.  Analysis brings out unique properties of the new waves and various associated consequences of physics are speculated.  The derived harmonics should prove extremely useful in numerous fields of science, engineering, mathematics, and biology.

Irwin Wunderman’s technical pursuits focused on developing electronic instruments.  He holds a Ph.D. in Electrical Engineering from Stanford University and Chaired the Northern California IEEE Professional Group on Instrumentation.  His early professional career included responsibility for all solid-state research and development at Lockheed Aircraft Corp., and at Hewlett-Packard Co.  He introduced transistorized circuitry at HP Corporate Labs and managed the transistorized digital instrument and optical instrument programs.  As co-founder of Hewlett-Packard Associates he was Principal Investigator on U.S. Air Force contracts to find uses for junction luminescence.  Through the early '60s his work included development of the first opto-couplers and fiber-optic communication links.  Upon leaving HP in 1967 he founded Cintra Physics International as its President and CEO.  Cintra developed a line of optical auto-ranging digital instruments and a compatible computer/calculator.  He received the 1970 Industrial Research 100 Award for creating the first scientific computer/calculator to employ algebraic notation and having a data bus permitting real-time integration between digital instruments, keyboards, computers and network systems.  Cintra was sold to Tektronix in 1971 who licensed the consumer-product version of the calculator to Texas Instruments.  The Cintra computer-data-bus became prototype for the IEEE-488 bus standard and the calculator the basis of existing Texas Instruments scientific calculators.  Dr. Wunderman received the 1968 commendation leadership letter from President Elect Richard Nixon.  He holds 16 patents and authored 25 papers, some attaining international awards.  This work culminates four decades of independent research attempting to establish how and why ordinary natural numbers and mathematical relationships could express fundamental physical laws through waves.

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 foundation for, and a bridge between, classical mechanics, quantum mechanics, and possibly string theory. Many dilemmas of physics may elucidate and simplify through the newly-derived waves. The theory requires no arbitrary constants. Many conventional mathematical constants like e, p, i, ¥, become unnecessary to represent physical reality using this development for waves. Derivation of the traditional harmonic exponential for sinusoids [exp (iwt)], is shown to violate causality.  It evokes a solution (a frequency) that begins at minus infinity in time, which response precedes the precipitating event that caused the solution.  By contrast the newly developed sinusoidal harmonic waves begin at the precipitating event.