The Geometrical solution, of the Regular n-Polygons and the Unsolved Ancient Greek Special Problems

The Special Problems of E-geometry [47] 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 monad Structure → Linearly , through mould of 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 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 [46] , where any two Unequal perpendicular monads acquire the common Space-meter ³√2 , to be twice each other . [44-47] The Unification of Space and Energy becomes through [STPL] Geometrical Mould Mechanism , 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 [39-41] . Euclid's elements consist of assuming a small set of intuitively appealing axioms , proving many other propositions . Because nobody 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 in [39] that the only Space-Energy geometry is Euclidean , agreeing with the Physical reality, on AB Segment which is Electromagnetic field of the Quantized on (AB) ̅ Energy Space Vector , on the contrary to the General relativity of Space-time which is based on the rays of the non-Euclidean geometries . Euclidean geometry elucidated the definitions of 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 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 ≡ L_( v) = e^(i.(Nπ/2)b=10 ͞ N= - ∞) = 0 m , which is nothing and the Absolute Primary Neutral space PNS .[43-46] . In Mechanics , the Gravity-cave Energy Volume quantity [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-mould of Duplication of the cube, in Sphere volume V=(4π/3).[wr]³ following the Squaring of the circle , π , and in Sub-Space-Sphere volume ³√2 , and the Trisecting of the angle .