THE POLYNOMIALS [ n.. Knots–FIGURES ] EDGE - POINTS VIBRATION & M-GEOMETRY

Abstract

The Interactions :

One of the most important concept in Geometry is distance , which is the Quanta in E-geometry , while in Material-Geometry the composition of Opposite , where the Material - Point is the Quanta in Chemistry and Physics. As in Algebra Zero , 0, is the Master-key number for all Positive and Negative numbers and this because their sum and multiplication becomes zero, and the same on any coordinate-System where ± axes pass from zero. In PNS Space , The Rolling of the Positive ⊕ constituent on the Negative Θ constituent , creates the Neutral Material Point which Equilibrium . Angular-Momentum is identical with Spin and consists the First-Discrete-Energy-Monad which occupies , Discrete Value and Direction , in contradiction to the Point which is Nothing , Dimensionless and without any Direction .

Quaternions [(+)(⊕∪⊖)(-)] ≡ Box B_R =_AB carries the Principal stress σ_A , σ_B between Points A(+), B(-) which σ , as Centripetal-acceleration is the minimum Energy becoming from the in-storage AB acceleration and is equal to the Gravity Force g. [108 – 110] . Because of the Revolving and Periodic acceleration of Gravity g ≡ ± σ exists as the First Energy-Box-B_R , while in the Second Box B_P is followed the Local-Extreme-case where Gravity g ≡ ± σ , and is altered Locally by changing the Principal-stress σ with Local-uniform-Pressure → g_L ≡ g_k = g . [Force/Area] = G ← i.e. the minimum Local - Energy acceleration is the known , Universal Gravitational-constant G = g k = k_EG = k_L σ , such for Macrocosm and for Microcosm , Obeying the Newton`s Laws of motion . G ⊥ σ

This Energy in Hydrogen-Cave as E-M, Conductor ≡ Edge Points Vibration ≡ The Pin of Atom → Plug Into their Sockets, which are the Orbit – Bracket–Hooks ≡ The Hands of Atoms ← i.e. The Atoms Plug with their Pins into the other Atoms-Drains = Holes, and so Bond and carry Informations .

This Resonance frequency of Hydrogen is Common to all Atoms and to all Compounds in this Cosmos. The Energy-Quaternion W, ????, Monad-Magnitudes exist as DUAL- Nature for Any { ⊕ , Θ }, { Position , Motion }; { Universe , Black-Holes }; { Gravity ∧Antigravity}, {Action → ← Reaction}, Edge Points Vibration creating Electric = [⊕] and Magnetic = [Θ] Forces as [⊕↔Θ] , The Light and others. The STPL- Line Conductors on the [STPL]-Mechanism, are the Physical-Rotors for the Origination of the Cosmic –Particles which transfer Informations as the Signals - Spectrum. [110]

From Mechanics-Physics , all Systems Possessing Elasticity ≡ motion and Reaction to the motion , the called mass , are capable of free vibration or vibration ≡ Periodic - motion taking Place in the Absence of External Excitation. This Principle issues for Both Systems Closed {The Atoms Nucleus} or Open Systems {The Orbitals}. For instance , In order that shifting of an u-d-Quarks from an Anti-Proton into a Proton , the Spin-Pair requires extra input of Energy (in MeV) , so that would the Proton Paired with a Neutron be Stable. transfer Informations as the Signals . Vibration in a cave Δ , means the Wave Pattern .

Cave–Spin- Spin-S̅ of PNS Space is S̅= r m ∇ and is the first Monad occupying 4-Spaces, i.e.

• a. The n - Spaces of Monad S̅ are the Polygons Sⁿ with n =1 ≈ ∞ Knots ,
• b. The n Anti-Spaces of Monad - S̅ are the Polygons - Sⁿ with n=1≈ ∞ Knots ,

c.. Sub-Spaces of Monad ≡ ± n = 1 to ∞ √5 are the Polygons with n = 1 ≈ ∞ Knots

All n-Regular Polygons end to equations of n-degree Segment, by finding a suitable value of the Segment x, That is we have in the general case to solve one or two equations of the form:

A·R⁰·xⁿ – B·R²·x^n–2 + C·R^n–6·x³ – D·R^n–4·x² + E·R^n–2·x¹ – F·Rⁿ·x⁰ = 0

for The Even Polygons, and

A·R²·x^n–2 – B·R^n–2·x^n–3 + C·R^2(n–4)·x³ – D·R^2(n–3)·x² + E·R^2(n–2)·x¹ – F·R^2(n–1)·x⁰ = 0

for The Odd Polygons, where A, B, C, D are constants.

The Presented Geometrical method is the solution of the above equation in the general case. Because the nth-degree-equations are → the Vertices (n) and the Sides (a_n = λ_n) of the n-Polygon in circle ← number π, is their common Mould. [62].

The Natural Mechanism Continuously Originates the Elementary Particles and Compounds, Atoms and Molecules with the One Action from Opposite. Article [111] encloses many Paragraphs of [110], in order to Distinguish and Prove the Way of Energy and the Stresses-Paths in the Spaces.

For the Regular Polygons is given such the Geometrical Solution as well the Algebraic.

[106] = Programming the Atoms and Compounds, is an Program which solves the Problem of the n-Knots Figures and gives the Energy Spectrum of any Complex-Forced-Vector.

Article [114] an Way of Deceptioning the Cells is prepared.