THE SQUARING OF THE CIRCLE USING < A RULER AND A COMPASS > & THE BELLOW MOTION USED BY THE PHOTONS

Abstract This article is a continuation of 123 = VECTORS, and concerns The simplified But Rigorous Proof of this Problem, The Squaring of the Circle using a Ruler and a Compass, as it was first Posed by the Ancient Greeks. The Photons square their Energy circle ≡ the Herpolhode-SPIN, to equal an Energy Unit Square, which Unit-Square they Promote, either as the Speed of the Photon which is their Electric Field, or they store it Perpendicularly to the motion in an equal area, The Anti-Square which is their Magnetic Field. Promotion is done at the Birefringence angle of 45°, and thus The Bellow-motion is their Torsional motion. From [40]—The Special Problems of Euclidean Geometry [47] consist the, Moulds of Quantization, of E-Geometry in it, to become → Monad, through mould of Space–Anti-space in itself, which is the material Dipole in inner monad Structure and which is identical with the Electromagnetic cycloidal field → Linearly through the mould of the Parallel Theorem [44–45], which are the equal distances between Points of Parallel and line → In Plane, through mould of Squaring the circle [46], where the Two equal and Perpendicular Monad-Vectors consist a Plane acquiring The common Plane-meter, π, and in Space (volume) through mould of the Duplication of the Cube [46], where any Two Unequal Perpendicular monads acquire the common Space-meter ∛2, to be twice each other, as analytically all methods are Proved and explained. [44–47] The Unification of → Space and Energy ← becomes through [STPL] Geometrical Mould Mechanism of Elements, the minimum Energy - Quanta, In monads → Particles, Anti-Particles, Bosons, Gravity – Force, Gravity - Field, Photons, Dark Matter, and Dark - Energy, consisting the Material Dipoles in inner monad Structures, i.e. → the innate Electromagnetic Cycloidal Field of monads ← [39-41]. Euclid’s elements consist of assuming a small set of intuitively appealing axioms, Proving many other Propositions. Because no one until [9] succeeded to Prove the Parallel Postulate By means of Pure Geometric Logic, many self consistent non - Euclidean Geometries have been discovered, based on Definitions, Axioms or Postulates, in order that Non of them contradicts any of the other Postulates. It was Proved [39], that the only Space - Energy Geometry is Euclidean, agreeing with the Physical Reality, on unit AB = Segment = Vector which is The Electromagnetic field of the Quantized on AB Energy Space Vector of Angular Momentum = Spin, on the contrary to the General Relativity of Space-time which is based on the Rays of the Non-Euclidean Geometries to the limited velocity of light in Planck's cavity. Euclidean Geometry elucidated the Definitions of its geometry-content, i.e. { [ for Point, Segment, Straight Line, Plane, Volume, Space [S], Anti-space [AS], Sub-space [SS], Cave, The 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, and a cylinder, in Gravity field [MFMF] Particles } and describes the Space-Energy vacuum beyond Plank’s length level [Gravity's Length 3,969·10^-62 m], reaching the absolute Point = Lᵥ = e^(i·(Nπ/2)) b = 10^N = −∞ m = 0 m, which is Nothing, and the Absolute Primary Neutral Space [PNS] = cave [r = 10^-35 ~~ 10^-62 m [43–46]. In Mechanics, The Gravity-cave Energy Volume quantity |c̄| ≡ [wr] is doubled, and is Quantized in Planck's-cave Space quantity (h/2π) = The Spin = 2.[wr]^3 → i.e. Energy Space quantity [wr] is Quantized, doubled, and becomes the Space quantity h/π following Euclidean Space-moulds of Duplication of the cube, in Sphere volume V = (4π/3).[wr]^3 and follows the Squaring of the circle π, and in Sub-Space-Sphere volume ∛2, as Trisection. In article [123], are given on the 2-Vectors, 3-Poles Rotation Squares Mechanism, such the Geometrical as the Mechanical Proof, by using the Conjugate circles of Polhode and Herpolhode which consist the Spin of Photons and which is their motion. The frequency needed for the velocity vector c̄ to rotate, is used the Kepler's Unit of Time k = fₑ² a³, where a = λ / 2, and which is the clock measuring the changes of motions.