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Science & Mathematics Publications
The Fractal Root of Numbers
Formally, the principal roots of numbers, i.e. square or cube roots, etc., are believed to exclusively consist of numerically identical numbers. In this paper, we expand this formal definition by exploring the existence of fractal roots; roots that are identical in their numerical structure but not in their relative magnitude. This new type of root has its own mathematical and geometrical peculiarities with implications intersecting many fields of science.
It is known that prime numbers occupy specific geometrical patterns or moduli when numbers from one to infinity are distributed around polygons having sides that are integer multiple of number 6. In this paper, we will show that not only prime numbers occupy these moduli, but non-prime numbers sharing these same moduli have unique prime-ness properties. When utilizing digital root methodologies, these non-prime numbers provide a novel method to accurately identify prime numbers and prime factors without trial division or probabilistic-based methods.
Recently, a 24-based distribution of numbers was used in a novel method for an infinite and accurate prime prediction and factorization. Therefore, further investigation of this particular configuration of numbers, labeled here the icositetragon wheel, is essential if we to expand our understanding of this method and to further improve it. We will show that using the icositetragon wheel is not an arbitrary choice by elaborating on the unique properties this configuration has, not only in regard to prime numbers distribution, but also for the many symmetries and complementary properties that numbers, prime and non-prime, observe in it.
The concept of waves is very fundamental to classical and modern physics alike, being essential in describing light, sound, and elementary particles, among many other phenomena. In this paper, we show that the wave/particle duality is a phenomenon manifested not only in the physical world and the mathematics that describes it, but also in the simple numbers that form the basic matrix upon which most of our sciences rest. We will also show how this wave-based approach to numbers could be essential to our understanding of the mathematical and physical constants that govern the physical laws as well as the natural elements emerging from them.
In this paper, we show how the 12 notes of the octave have inherent decimal references that correspond precisely to the internal angles of regular polygonal shapes that exhibit symmetries found abundantly in nature. This exact correspondence manifests itself only when the standard pitch tuning is set to 432Hz instead of the modern 440Hz, which is an indicator on the importance of this tuning, from a mathematical perspective at least.
Prime Number Pattern and Discovery of Quasi-Prime
When integers are continuously plotted around each side of an icositetragon (24-side polygon), patterns of primes and a new classification of prime numbers (Quasi-primes) emerge.
When integers are continuously plotted around each side of an icositetragon (24-sided polygon), patterns of primes and a new classification of prime numbers (Quasi-primes) emerge. This information presents a method to predict prime number incidence.
Quaternion Symmetry of the Icositetragon
When integers are continuously plotted around each side of an icositetragon (24-sided polygon), quaternion sets are demonstrated, allowing the identification of charge-associated quadripolarity.
Unified mathematical models must establish numerical connections between all types of physical phenomenon. Inherent relationships between geometry and music are shown through the inscription of regular polygons within a unit circle.