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Template for Preparing an Article The Quantization of E-geometry to Material Geometry and the Origin of Energy Particles . The Origin of Energy Particles , from Material Geometry , and Relativity . Markos Georgallides 1Larnaca (Expelled from Famagusta town occupied by the Barbaric Turks Aug-1974) , Cyprus Civil-Structural Engineer (NATUA) , Athens * Markos Georgallides : Email address : [email protected] : To cite this article: Markos Georgallides : E-geometry , The Unification of Space and Energy through Geometry-moulds . Abstract : In article is shown the How Positive (+) and Negative (-) are joint to create material point which is Zero and by adding it to monads , to create all Primary Particles , following Extrema Geometry-Spaces-Moulds , and which are the In-Periodic Table Elements in Planck`s Space-Level . Also the How Rest- Gravity constituent MFMF field is joint and creates the Electromagnetic Waves E,P and Gravity-Spin , which External Spin , is continually converted into the Inner Electromagnetic Waves E,P and which continue this eternal cycle. Euclidean-Geometry elucidated the definitions of geometry-content {Point, Segment, Straight Line, Plane, Volume Space [S] , Anti-space [AS] , Sub-space [SS] , Cave, Space-Anti-Space Mechanism of the Six-Triple- Points-Line , that produces and transfers Points of Spaces , Anti-Spaces and Sub-Spaces in a Common Inertial Sub-Space , the Field [MFMF] and Particles} and the same for Material-Geometry which describes the Space and Energy beyond Nπ i.( )b=10 ͞ N= − ∞ Plank´s length-level [ Gravity-Length 3,969.10 ̄ 62] , reaching Quantized Point L v = e 2 m → 0 . Energy monads presuppose Energy-Space Base which is the cause of Spaces existence and the motion of particles. This Energy-Space Base , MFMF = Medium-Field of Material-Fragments is the Material-Geometry in the way used in Mechanics and Physics .Property for Doubling Energy in the same Space is encountered in Stationary wave where energy is proportional to angular velocity ,w . Now is shown the How Material-Fragments are built-in E - Geometry and the Why follow its moulds . The Special Problems of E-geometry consist the , Mould Quantization of Euclidean Geometry in it to become → Monad , through mould of Space –Anti-space in itself , which is the material dipole in inner monad Structure as the Electromagnetic cycloidal field → Linearly , through mould of Parallel Theorem ,which are the equal distances between points of parallel and line → In Plane , through mould of Squaring the circle , where the two equal and perpendicular monads consist a Plane acquiring the common Plane-meter ,π, → and in Space (volume) ,through mould of the Duplication of the Cube ,where any two Equal or Unequal perpendicular monads , producing zero Work , acquire a common Space-meter tobe twice each other. The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of Elements , the minimum Energy-Quanta to , Monads → Particles , Anti-particles , Bosons , Gravity-Force , Gravity-Field , Photons , Dark Matter , and Dark-Energy , which consist the Material Dipoles in inner monad Structures , i.e. The parallel motion of DM and DE Heap Mixture , by rolling on Gravity-field MFMF ,and which travels with light velocity and continually formulating the , Zero → Discrete → Infinite Geometrical Universe . Keywords : Material Geometry , Energy Particles , The Periodic Table . 1 The Quantization of E-geometry to Material Geometry and the Origin of Energy Particles . 1. General homogenous differential equation mẍ+cẋ+kx = 0. Periodic motion . For a flexible string of mass ,ρ, per unit dx is Harmonic Periodic motion x(t) is when motion is stretched under Tension T and analyzing Newton repeated itself regularly , in equal intervals of time Laws for tiny length ,dx, then Net Force ,T ẍ = ρ a̅ 1 T (the period of oscillation) and is designated by and the equation of motion is ẍ = [ ] a̅ or the time function , v² x(t) = x(t+T) = x = A.sin(2π.t/T) = A.sin.wt = ∂²y 1 ∂²y T T = [ ] . ….(1) , where v = √ = √ = A.cos(kq-wt) because is sinusoidal , where ∂x² v² ∂t² ρ m A is the amplitude of oscillation measured from The general solution of (1) is y =F1(ct-x)+F2(ct-x) equilibrium position and for repeated motion t =T. where F1,F2 are arbitrary functions and Quantity (2π/T) = w = 2.π.f is circular frequency, regardless of the type of function F , the argument or f = 1/T = w/2π , is the frequency and ,k, is the (ct ±x) upon differentiation leads to the equation , wave number k = 2π/λ and the speed of a wave is ∂²F 1 ∂²F = [ ] . ….(2) , where F = The tension , v = λ.f or w = v.k and because of relation of ∂x² v² ∂t² angular velocity v̅ = w.r = w(1/k) → w = v.k then v = the velocity of wave propagation r . k =1 . Another general solution of (1) is that of Velocity v̅ = ẋ = wA. coswt = wA.sin(wt+π/ 2) separation of variables as y(x,t) = Y(x).G(t) …(3) 1 ∂²Y 1 1 ∂²G a̅ = ẍ = w²A.(- sinwt) = w²A.sin(wt+π) where then (1) becomes → = [ ] . Y ∂x² v² G ∂t² i.e. Velocity , ẋ , and Acceleration ẍ are also and because of independent variables x ,t are both harmonic with the same frequency of oscillation , constant the general solutions are , w w and when evaluated lead to the displacement ,x , Y = A sin x + B cos x by π/2 and π radians respectively and the whole v v G = A sin wt + D cos wt , where arbitrary system reveals at ẍ = - w²A , so that In harmonic motion acceleration to be proportional to the constants A,B,C,D depend on boundary conditions displacement and directed toward the origin , and and for y(0,t) = 0 will require B=0 and for y(l,t) because also Newton`s second law of motion = 0 the solution lead to equations , w states that the acceleration is proportional to the y = [ C sin wt + D cos wt ] sin x and v force , then harmonic motion can be expected with wl Wn.l 2π. l sin = 0 or = = n.π where force varying as kx. (which is Hook`s law F = kx v v λ v and k , the stiffness coefficient , directed in n = 1 , 2, 3,..n λ = the wavelength , f centrifugal velocity vector v̅r, on radius r ) . n n T x fn = v = √ is frequency and Y = sin nπ In Free vibration of monads AB = q =[s+v̅i] and 2l 2l ρ l because velocity vector is composed of the In case of vibration initiated in any manner y(x,t) = ∞ centrifugal velocity v̅r, and the rotational velocity v̅q, perpendicular to displacement ,x, and because ∑[Cn. sinWn. t + Dn. cosWn. t]. sin (nπx / l) viscous damping represented by a dashpot , is n=1 described by a force proportional to the velocity as Example 1 : holds F = cẋ where ,c, is the damping coefficient , To determine the Cn and Dn of above equation . it is a constant of transverse proportionality and solution : this because ẋ dx , then it is directional to At t=0 the displacement and velocity are , ∞ transverse velocity v̅y = ẋ/dt and is holding the y(x,0) = ∑n=1 Dn. sin (nπx/ l ) and velocity 2 The Quantization of E-geometry to Material Geometry and the Origin of Energy Particles ∞ ẏ (x,0) = ∑n=1 Wn. Cn. sin (nπx/ l) = 0 and both area , and the shear modulus ,G, of elasticity , and multiplying by sin(kπx/l) and integrating from ∂T ∂²θ which net Torque is → dx = IpG dx . x=0 to x=l ,all terms become zero except n= k and ∂x ∂x² 2 l kπx The differential equation of motion becomes by D = ∫ y(x, 0). sin ( ) dx and C = 0 k =1,2,3. k 푙 0 l k equating torque to the product of the mass moment For standing waves equation of motion is the sum of inertia ρIp dx of the element and the angular of a right moving and a left moving wave , or acceleration ∂²θ in it and is , ∂t² Ysta = A.sin.( kn x + wn t ) + A.sin.( kn x + wn t ) = ∂²θ ∂²θ ∂²θ G ∂²θ ρIpdx = IpG dx and = . …(2) [2A.sin(kn x).cos(wn t) where k n = 2π/λn and , ∂t² ∂x² ∂t² ρ ∂x² G E 퐬퐢퐧(퐤퐱) → is the Spatial dependence at locations where θ≡ replace u , ≡ of longitudinal vibration λn ρ ρ x = 0 , λ/2 , λ , 3λ/2 , n called the nodes . 2 with the general solution given for θ is …(3) ρ ρ At nodes amplitude ,A, is always zero and at θ =[A sinw√ x+B cosw√ x].[C sinwt+D coswt] locations x = λ/4 , 3λ/4 , 5λ/4 called the Anti- G G nodes where the amplitude becomes maximum . Example 2 : To find the equation of the natural frequencies of a 퐜퐨퐬(퐰퐭) → is the Time Oscillation dependence uniform rod in torsional oscillation with one end where corresponding frequency f1 = v /2x = Fix and the other end Free . The solution : v /n.λn and fn = n.f1 , The boundary conditions are , i.e. n , more times Energy is stored in fn frequency , showing the way of Energy a.. when x = 0 then θ = 0 resulting to B = 0 quantization in constant caves [49] . b.. when x = l then Torque = 0 or ∂θ/∂x =0 and Since energy E = h.f then frequency f = E/h . resulting to cos w√ρ/Gl = 0 and which is Angular velocity w = 2π/ T = 2π.f = 2π. (E/h) = satisfied with the following angles , E / [ h/2π] , or w = E/(h/2π) , E= w.[ h/2π] i.e 훒 퐰 √ l = π/2 , 3π/2 , 5π/2 ….
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