This work is a simple straightforward investigation of the realm of numbers in conjunction with the physical world, from a perspective capable of encompassing all major findings already known in this field and rediscovered in this paper as for example Schrodinger’s Equation (Sec3) and Lorentz Transforms (both having as origin the main matrix), or for the first time discovered in this paper as for example the “non primes” primes conjectures, Sec12. In this context, some old findings that were not understood (as for example the Vacuum Catastrophe Sec9, or some aspects regarding the zeta function Secs17 to 22 and 28, or Goldbach’s Conjecture and Poncelet’s Drawings Secs26,27,28) receive a meaning, and a few relatively recent findings regarding the Universe are reinterpreted. All these findings, old or new, are inter connecting components of a unique reality that includes (and has as its driving and unifying force) the observer.
The basic constituents of this paper are a few interrelated naturally repeatedly occurring numerical patterns. One of these patterns (that plays a central skeletal role in this paper) is a remarkable infinite double sequence called the main sequence. The components of these numerical patterns in particular the components of the main sequence, which are numbers (in a generalized sense, with undefined ambiguous significance, i.e., capable of taking various concrete meanings in various concrete situations, or remaining just numbers), have not only algorithmic (necessary) number like properties but also non algorithmic (accidental) object like interlocking pattern like properties (see for example Obs1 of Sec4), with the object like interlocking pattern like properties interlocking themselves with the algebraic properties.
Table of Contents
1 Morley's Theorem Reexamined
2 Morley's Theorem Reformulated (and Proven) for the non Equiangular Case. Coloured Unit.
3 Morley's Theorem (for the non Equiangular Case), Continuation. Colours Assigned to the Colour Unit. Their Geometrical and Physical Meaning. Schrodinger's Equation
4 The Main Sequence and Some of its Outcomes
5 Morley's Theorem "for the Equiangular Triangle Case", Singularities, Angle Renormalization, Perpendicularity, Elliptical Planetary Motion
6 Elementary Particles
7 Lorentz Like Addition Formulas for Velocities, Continued Fractions (CF), Collatz Conjecture, Crossover and Prime Numbers
8 Random Numbers, Continued Fractions, Crossover, and Prime Numbers
9 4.9 % Ordinary Matter, 26.8 Dark Matter, 68.3 Dark Energy Present in the Universe
10 Coupling Constants for Elementary Particles
11 Plank's Constant h as Related to the Gravitational Constant G
12 Some Prime Numbers Conjectures and the Main Sequence. The Two "Non Primes" Primes Conjectures
13 The Number Pi Obtained From Fundamental Non Tautological Considerations
14 Some Particular Logarithms and their Interconnectedness
15 Human DNA
16 The Neutrino, the Main Sequence, and the Prime Numbers Conjectures
17 The Main Sequence and the Riemann Hypothesis, Dichotomy: the First Approach
18 The Main Sequence and the Riemann Hypothesis, the Second Approach
19 The Main Sequence and the Riemann Hypothesis: the Third Approach
20 The Main Sequence, Schrodinger Equation, and the Riemann Hypothesis: the Fourth Approach
21 The Zeta Function, its Crossover Continuity Dual, and Riemann Hypothesis: the Fifth Approach
22 Riemann Hypothesis, the Main Sequence, and Continued Fractions with Crossover: the Sixth Approach
23 More About the Interrelation Between the Set of Prime Numbers, the Arrow of Time, "Actual Infinity", Coupling Constants for Elementary Particles, and the Zeta Function. Some Remarkable Numbers and the "Age of the Universe"
24 Universe/"Standard Observer" Time Scale Discrepancy and the Origin of Mass. The Balanced Universe
25 Goldbach's Conjecture
26 Goldbach's Conjecture, Poncelet's Drawings, and the Main Sequence: The First Approach
27 Goldbach's Conjecture, Poncelet's Drawings, the Double Circle, Prime Numbers: the Second Approach
28 Closing Remarks
Research Objectives and Core Themes
This work presents an investigation into the fundamental relationships between numerical sequences and physical phenomena. By identifying recurring algorithmic and non-algorithmic patterns, the research aims to establish a unified perspective connecting diverse fields such as number theory, quantum mechanics, thermodynamics, and cosmology. The central research question explores whether a specific "main sequence" of numbers can act as a bridge, reconciling abstract mathematical structures with the observed physical reality.
- The exploration of "main sequence" numerical patterns and their recurrence in both algebraic and physical systems.
- Reformulation and extension of Morley's Theorem and its implications for angular and geometric configurations.
- Investigation of the interconnections between prime numbers, the Riemann Hypothesis, and the fundamental coupling constants of nature.
- Theoretical grounding of concepts like dark matter and dark energy through numerical analysis of the main sequence.
- Examination of the "observer" role in physical reality and its relation to entropy and time direction.
Excerpt from the Book
Morley's Theorem Reexamined
Morley's Theorem claims that given triangle ABC with angles trisected as below, triangle DEF is an equiangular triangle.
Since Morley's Theorem does not depend on the size of the triangle, it is natural to expect that its proof should be possible to be formulated and carried out in terms of angles only. Thus it is natural to require that no line segment lengths are used directly or indirectly during the proof, not even in principle.
With relations between angles, assumed to be linear equations, as for example (1) m + n + p = 180^0 (2) y + m + z = 120^0 + b, it is found that any attempt at writing a system of equations that can be derived directly from the drawing, consisting of equations like (1) and (2) and their eventual circular permutations, that would solve the triangle, i.e., would give the values of m, n, p, x, y, z, v, w, t and with these, all the angles related to the figure, in terms of a, b, and c that are assumed to be known, ends up in failure. Such a system of equations can be solved, in a unique way, if we add the following extra equation, "an external equation", i.e., an equation that does not come from Fig1 (3) x = y, and circular permutations.
On the one hand, such a solution for the triangle would be illegitimate because, as we said, (3) is externally imposed, on the other hand, we know that (3) is correct, but not directly from formulas like (1) and (2) nor from algebraic calculations with linear formulas like (1) or (2) and their circular permutations, obtained from the figure above. Therefore this situation leads to a conundrum. Below is presented a solution, based on introducing a "coloured unit", that averts this impasse, shows among many other things as to why formulas (3) are not present in ("visible from") Fig1, and as to why geometrically the use of a circle (or circles) directly or indirectly (by use of circular functions, for example) when proving Morley's Theorem is equivalent to (is a substitute of) formulas (3) used algebraically.
Summary of Chapters
1 Morley's Theorem Reexamined: Discusses the limitations of traditional proofs of Morley's Theorem and introduces the need for an external equation to solve the system of angle relations.
2 Morley's Theorem Reformulated (and Proven) for the non Equiangular Case. Coloured Unit.: Presents an alternative proof method for Morley's Theorem using the concept of an "abstract algebraic triangle" and a coloured unit.
3 Morley's Theorem (for the non Equiangular Case), Continuation. Colours Assigned to the Colour Unit. Their Geometrical and Physical Meaning. Schrodinger's Equation: Explores the geometrical and physical significance of the coloured unit, linking it to patterns found in Schrodinger's equation.
4 The Main Sequence and Some of its Outcomes: Defines the "main sequence" and demonstrates its generation through unfolding and doubling numerical terms.
5 Morley's Theorem "for the Equiangular Triangle Case", Singularities, Angle Renormalization, Perpendicularity, Elliptical Planetary Motion: Applies the developed framework to specific geometric cases and physical motion scenarios.
Keywords
Main sequence, Morley's Theorem, Riemann Hypothesis, Prime numbers, Schrodinger's Equation, Non-primes conjecture, Dark matter, Dark energy, Actual infinity, Coupling constants, Crossover, Dichotomy, Physical reality, Observer process, Numerical patterns.
Frequently Asked Questions
What is the fundamental objective of this work?
The book aims to demonstrate a hidden, underlying unity in numerical patterns that links abstract mathematical conjectures with observable phenomena in the physical universe.
What are the central thematic areas?
The research traverses geometry (Morley's Theorem), number theory (prime numbers, Goldbach's Conjecture, Riemann Hypothesis), and theoretical physics (quantum mechanics, general relativity, cosmology).
What is the primary methodology?
The author uses a method of constructing "main sequence" matrices and identifying crossover patterns between abstract number sets and physical constants.
What does the "main sequence" represent?
It acts as a generating set of numerical values that the author claims appear repeatedly across various physical and mathematical contexts, serving as a unifying mathematical language.
How does the author connect the Riemann Hypothesis to this work?
The Riemann Hypothesis is interpreted through the lens of "dichotomies" within the main sequence, linking the distribution of prime numbers to physical space-time interpretations.
What is the role of the "observer" in this framework?
The observer is central to collapsing wave functions and defining physical states; the work suggests that observation itself is an entropic process linked to these numerical patterns.
How are dark matter and dark energy addressed?
The author provides numerical estimates for ordinary matter, dark matter, and dark energy derived from specific segments of the main sequence.
What is the significance of the "coloured unit" concept?
It provides a tool to resolve analytical impasses in geometric proofs (like Morley's Theorem) by assigning specific values to angular components, which then map to physical variables.
- Quote paper
- PhD Ovidiu Mitran (Author), 2019, The Intricate Realm of Numbers in Conjunction with the Physical World as Revealed by an Exceptional Numerical Sequence and by Some Outstanding Numerical Conjectures, Munich, GRIN Verlag, https://www.grin.com/document/489553
