THE GEOMETRICAL CONSTRUCTION OF THE REGULAR POLYGONS USING A RULER AND A COMPASS & THE OVERTHROW OF THE THEORY OF THE NON - CONSTRUCTIBILITY

Markos Georgallides — 2026-10-02

The Method

a). In the System of a circle [O, OK], of Diameters KK′ ⊥ aa′ = 2R, KK′ ⊥ K(E), and a Chord KA = KEₐ on K(E), Represents The Fixed Twin - Couple of the Triangles KAK′, KAEA. Simultaneously, The Midpoint Oₐ of side KEₐ, The Isosceles triangle KAEA, The Base Point Dₐ of Perpendicular ADₐ, and The Perpendicular triangles KADₐ, KAK′ Are Constantly Related to each other;......It is Proved

1... On Triangle KAK′ any Chord KA is related to its Angle φₐ as KA = 2R. sin φₐ, and |KA|² = 2R. hₐ, And on the Triangles KADₐ, KAEA, sin φₐ = ADₐ / KA and Difference of |KA|² = |KEₐ|² = |KA|.|OₑₐDₐ| = Constant, where Point Oₑₐ is Midpoint of KEₐ = KA.

2... The locus of Any Point B, which Summation of squares [BK² + BEᵦ²] = Constant from the Distances to Fixed Points K, Eᵦ, with KB Fixed and equal to k², IS the Circle [Oᵦₑ, OᵦₑB], at Point Oᵦₑ, and to Be Distant from the Midpoint Oₑᵦ of the Side KEᵦ → The distance is equal to ⇒ [BK]² + [BEᵦ]² = k² = 2.|OₑᵦB|² + [KEᵦ]² ←

The Prior :

a...The Unsolvability of the Three famous Ancient Greek Problems—Trisecting the Angle Doubling the Cube, Squaring the Circle and The Construction of the Regular Polygons With a Ruler and a Compass, Stands on the Base justification of the Algebraic Field Theory (specifically Galois Theory) and the Theory of Constructible Numbers, which were developed in the 19 th century.

b... The Greeks often used other Techniques (like Conic sections or Mechanical tools) to Solve these Problems, But their self-imposed "Euclidean" constraint with a Ruler and a Compass, (straightedge and compass) made these Specific Problems impossible to solve.

c...In the Published Articles [123],[124],[126], it is clearly evident that the Solution of the Ancient, Unsolved Greek Problems, using a Ruler and a Compass (as the constraint has been set by Euclid), Has Become Possible, and is in the Critique of Both, Human Logic Thinking and, The Artificial Intelligence when it uses The Path of Knowledge to the Truths of Nature, and which is the Dialectic Logic of the Euclidean Geometry.

In article [123], are given on the 2-Vectors, 3-Poles Rotation Squares Mechanism, where Ē = M̄, 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 from Kepler`s Unit meter of Time k = f²e a³, where, a = λ / 2, and which is the clock measuring the changes of motions,

In article [124], It was Noted that When in Photons, Ē = M̄, these acquire the common Plane-meter the Square CMNH = [CM]² = π.[EC]², where EC is the circle`s Radius, Using the Bellow-Motion Method by creating the Square CMNH = [CM]² from E, M, and from [E/√2]² = π.[M/√2]², π ≡ [E/√2]² / [M/√2]² and Thus Photon is Squared to an UNID SQUARE.

In article [126], It is important to Note that When in Photons, Ē ≠ M̄, these acquire the common Space-meter ³√2, Using Quantum-Cloning Method for creating a Perfect-Copy of E, M, as are when E = 2.M and, [Ē ≠ M̄]³ = 2.[Ē = 2.M̄]³. With this Way Photon is Duplicated to a New-One. The simplified But a Rigorous Proof of the Problem, The Squaring of the Circle using a Ruler and a Compass, as it was first Posed by the Ancient Greeks, is followed by The Photons which Square their, Energy circle ≡ The Herpolhode ≡ SPIN, to equal an Energy Unit Square, which UNIT-Square they Promote, common Plane - meter, 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. At Phase Angle where Ē = 2.M̄, Energy Cube [M / 2]³ is Duplicated and enters INTO . [E]³ = [2.M]³ = 8.M³ ≡ Tetrahedron-Cube-Sphere-Mechanism. [102]

d...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, E = M, consist a Plane acquiring The common Plane -meter, π, and in Space (volume) through mould of Duplication of the Cube [46], where

Any Two Unequal Perpendicular monads Ē≠M̄, acquire the common Space-meter ³√2, Using Quantum-Cloning Method to be Twice each other as analytically Proved-explained. The Unification of → Space and Energy ← Becomes through [STPL] Geometrical Mould Mechanism of Elements, the minimum Energy - Quanta, In modads → 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, Axiom 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 [MF/MF] 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 = eⁱ.(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 Physics, there is No single Process called "Duplication of Energy" for Photons Because Energy must always be conserved, or when Two Photons are "merged" into a single Photon.

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, Sphere volume V = (4π/3).[wr]³ and follows the Squaring of the circle π, and in Sub-Space-Sphere volume ³√2, as Trisection.

GPH-International Journal of Mathematics

THE GEOMETRICAL CONSTRUCTION OF THE REGULAR POLYGONS USING A RULER AND A COMPASS & THE OVERTHROW OF THE THEORY OF THE NON - CONSTRUCTIBILITY