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

Markos Georgallides — 2025-08-04

The Interactions :
One of the most important concept in Geometry is , distance , which is the Quanta in Egeometry , 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 ⊝ 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 ???? ???? =???????? carries the Principal stress ???? ???? , ???? ???? 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-???? ???? ,
while in the Second Box ???????? is followed the Local-Extreme-case where Gravity g ≡  σ ,
and is altered Locally by changing the Principal-stress σ with an Local-uniform-Pressure
→ ???????? ≡ g k = g . [ Force/Area ] = G ← i.e. The minimum Local - Energy acceleration is the
known , Universal Gravitational-constant G = g k = ???????? g = ???????? 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 w̅̅̅, B̅, Monad-Magnitudes exist as DUAL- Nature for
Any { ⊕ , ⊝ }, { Position , Motion },{ Universe , Black-Holes } , { Gravity ,Antigravity} ,
{Action → . ← Reaction} , Edge Points Vibration creating Electric = [⊕] and Magnetic =
[⊝] Forces as [⊕↔⊝] ,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 and be Stable.
transfer Informations as the Signals . Vibration in a cave ∆ , means the Wave Pattern .
Cave–Spin-????̅ of PNS Space is ????̅ = r m v and is the first Monad occupying 4-Spaces , i.e.
From Number n , of the Equilibrium number of masses m ???? in a System.
a.. The n - Spaces of Monad ????̅ are the Polygons ???? ???? with n = 1 ≈ ∞ Knots ,
b.. The n Anti-Spaces of Monad - ????̅ are the Polygons - ???? ???? with n =1≈ ∞ Knots ,
c.. Sub-Spaces of Monad ≡ ± ????=????-∞√ ????̅ 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. xn-2 + C .Rn-6. x³ – D .Rn-4. x² + E .Rn-2.x1– F. Rn. x0 = 0
for The Even Polygons , and
A .R 2. xn-2 - B.Rn-2. xn-3 + C.R2(n-4). x³ -D.R2(n-3). x² +E.R2(n-2). x1– F.R2(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 (????????= ????????) 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 .

IJO - International Journal of Mathematics (ISSN: 2992-4421 )

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