IOSR Journal of Dental and Medical Sciences (IOSR-JDMS) e-ISSN: 2279-0853, p-ISSN: 2279-0861.Volume 15, Issue 8 Ver. XI (August. 2016), PP 128-183 www.iosrjournals.org Material Geometry the Periodic Table and Chemistry Markos Georgallides 1larnaca (Expelled From Famagusta Town Occupied By The Barbaric Turks Aug-1974) , Cyprus Civil-Structural Engineer (Natua), Athens Abstract: Point in E-Geometry , which is nothing and dimensionless , can be derived from the addition of a Positive (+)and a Negative (-) number while Material point has dimension , the Segment ds=[⊕⊝] , and originates in the same way . Adding it to numbers i.e. Monads , creates Primary Particles, the Rest-Gravity constituent and the Atoms of Periodic System in Planck`s Space-Level. Monads are Spinning because of the inner ElectromagneticWaves E,P which create External Spin and again the Inner Electromagnetic Waves E , P continuing this eternal Cycle . Now is shown the , Why and How the Spinning of Particles and the Way and the Place conserved. 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 and Energy beyond Plank´s length -level [ G -Length 3,969.10 ̄ 62] , to Point L v = ei.(Nπ2)b=10 ͞ N= − ∞ 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 is the Electromagnetic cycloidal field . The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism of the minimum Energy-Quanta to Monads → Particles, Anti-particles, Bosons, Gravity-Force, Gravity-Field, Photons, Dark Matter , and Dark-Energy , consisting the Material Dipoles in inner monad Structures , with light velocity and continually formulating the ,Zero → Discrete ,Particles-Atoms-Elements →Infinite Geometrical Universe. Conservation of Energy, as Torsional momentum, is obtained by Dynamical stability of Nucleus-Wheel-Rims. Keywords: Material Geometry, Energy Particles, Periodic Table and Chemistry. MaterialGeometry the Periodic Table and Chemistry. General Periodic motion. Harmonic Periodic motion x(t) is when motion is repeated itself regularly , in equal intervals of time T (the period of oscillation) and is designated by the time function , x(t) = x(t+T) = x = A.sin(2π.t/T) = A.sin.wt = = A.cos(kq-wt) because is sinusoidal , where A is the amplitude of oscillation measured from equilibrium position and for repeated motion t =T. Quantity (2π/T) = w = 2.π.f is circular frequency, or f = 1/T = w/2π , is the frequency and ,k, is the wave number k = 2π/λ and the speed of a wave is v = λ.f or w = v.k and because of relation of angular velocity v̅ = w.r = w(1/k) → w = v.k then r . k =1 . Velocity v̅ = ẋ = wA. coswt = wA.sin(wt+π/ 2) a̅ = ẍ = w²A.(-sinwt) = w²A.sin(wt+π) i.e. Velocity ,ẋ , and Acceleration ẍ are also harmonic with the same fre quency of oscillation , and when evaluated lead to the displacement ,x , by π/2 and π radians respectively and the whole system reveals at ẍ = - w²A , so that In harmonic motion acceleration to be proportional to the displacement and directed toward the origin , and because also Newton`s second law of motion states that the acceleration is proportional to the force , then harmonic motion can be expected with force varying as kx. (which is Hook`s law F = kx and k , the stiffness coefficient , directed in centrifugal velocity vector v̅ r, on radius r ) . DOI: 10.9790/0853-150811128183 www.iosrjournals.org 128 | Page Material Geometry The Periodic Table and Chemistry. In Free vibration of monads AB = q =[s+v̅ i] and because velocity vector is composed of the centrifugal velocity v̅ r , and the rotational velocity v̅ q , perpendicular to displacement ,x, and because viscous damping represented by a dashpot , is described by a force proportional to the velocity as holds F = cẋ where ,c, is the damping coefficient , it is a constant of transverse proportionality and this because ẋ dx , then it is directional to transverse velocity v̅ y = ẋ /dt and is holding the homogenous differential equation mẍ +cẋ +kx = 0. For a flexible string of mass ,ρ, per unit dx is stretched under Tension T and analyzing Newton Laws for tiny length ,dx, then Net Force ,T ẍ = ρ a̅ and the equation of motion is ẍ = [ 1v² ] a̅ or ∂²y∂x² = [ 1v² ] . ∂²y∂t² ….(1) , where v = √Tρ = √Tm The general solution of (1) is y =F1(ct-x)+F2(ct-x) where F1,F2 are arbitrary functions and regardless of the type of function F , the argument (ct ±x) upon differentiation leads to the equation , ∂²F∂x² = [ 1v² ] . ∂²F∂t² ….(2) , where F = The tension , v = the velocity of wave propagation Another general solution of (1) is that of separation of variables as y(x,t) = Y(x).G(t) …(3) where then (1) becomes → 1Y ∂²Y∂x² = [ 1v² ] 1G . ∂²G∂t² and because of independent variables x ,t are both constant the general solutions are , Y = A sinwv x + B coswv x G = A sin wt + D cos wt , where arbitrary constants A,B,C,D depend on boundary conditions and for y(0,t) = 0 will require B=0 and for y(l,t) = 0 the solution lead to equations , y = [ C sin wt + D cos wt ] sin wv x and sin wlv = 0 or Wn.l v = 2π. l λ = n.π where n = 1 , 2, 3,..n λ = vf the wavelength , fn = n2l v = n2l √Tρ is frequency and Y = sin nπ xl In case of vibration initiated in any manner y(x,t) = Σ[Cn.sinWn.t+Dn.cosWn.t].sin (nπx / l)∞n=1 Example 1: To determine the Cn and Dn of above equation . solution : At t=0 the displacement and velocity are , y(x,0) = ΣDn.sin (nπx/ l )∞n=1 and velocity ẏ (x,0) = ΣWn.Cn.sin (nπx/ l)∞n=1 = 0 and both multiplying by sin(kπx/l) and integrating from x=0 to x=l ,all terms become zero except n= k and Dk= 2푙∫y(x,0).sin(kπxl)dxl0 and Ck = 0 k =1,2,3. For standing waves equation of motion is the sum of a right moving and a left moving wave , or Ysta = A.sin.( kn x + wn t ) + A.sin.( kn x + wn t ) = [2A.sin(kn x).cos(wn t) where k n = 2π/λn and , 퐬퐢퐧(퐤퐱) → is the Spatial dependence at locations x = 0 , λ/2 , λ , 3λ/2 , n λn2 called the nodes . Material Geometry the Periodic System and Chemistry . 3 At nodes amplitude ,A, is always zero and at locations x = λ/4 , 3λ/4 , 5λ/4 called the Anti-nodes where the amplitude becomes maximum . 퐜퐨퐬(퐰퐭) → is the Time Oscillation dependence where corresponding frequency f1 = v /2x = v /n.λn and fn = n.f1 , i.e. n , more times Energy is stored in fn frequency , showing the way of Energy quantization in constant caves [49] . Since energy E = h.f then frequency f = E/h . Angular velocity w = 2π/ T = 2π.f = 2π. (E/h) = E / [ h/2π] , or w = E/(h/2π) , E= w.[ h/2π] i.e Proportional to energy E , since h/2π is constant. Energy in a standing wave : Kinetic energy is EK = dKdx (x,t) = mv²/2 = = m2 ∂y²∂t² = m2 A².w² ……..(1) Potential energy is EU = dKdx (x,t) = [ F/2 ] ∂y²∂t² and Power transmitted P(x,t) = -F ∂y∂x ∂y∂t Total Energy ET = EK + EU = [ m2 ] ∂y²∂t² + [ F2 ] ∂y²∂t² In Sinusoidal waves dKdx = dUdx = [m4] ∂y²∂t² = m4 A² w² Torsional Vibrations of Rods is similar to that of of longitudinal vibration of rods and for the angle of twist in any length ,dx, due to Torque T is dθ then dθ = T dx/ Ip.G where , Ip.G is the torsional stiffness as the product of the Polar-moment , Ip , of Inertia of the cross-section area , and the shear modulus ,G, of elasticity , and which net Torque is → ∂T∂x dx = IpG ∂²θ∂x² dx .The differential equation of motion becomes by equating torque to the product of the mass moment of inertia ρIpdx of the element and the angular acceleration ∂²θ∂t² in it and is , ρIpdx∂²θ∂t² = IpG ∂²θ∂x² dx and ∂²θ∂t² = Gρ . ∂²θ∂x² …(2) where θ≡ replace u , Gρ ≡ Eρ of longitudinal vibration with the general solution given for θ is …(3) θ =[A sinw√ρG x+B cosw√ρG x].[C sinwt+D coswt] Example 2: To find the equation of the natural frequencies of a uniform rod in torsional oscillation with one end Fix and the other end Free .