The Geometrical Construction of the Regular Polygons Using a Ruler and a Compass & the How Gravitational - Force Keeps Them Still.

2026-11-05T11:31:54+00:00

The Method

a). Document Article 132-REGPOIY, is The NEW Method of Constructing All Polygons and Based on Three Triangles & Three Circles {1=Apothems, 2=Constant-Differences-Squares 3=Constant-Sum-Squares} where this Mechanism of these Triangles-Circles is Drawn Step by Step with < A Ruler and a Compass only >. In Mechanism and onto Any Even-Sequence -(n, n+2)-- of Polygons, The Inbetween Odd -(n+1)- is Detected on the Two Circles and Defined by the Third one on the main circle. The Resolution of Constructibility with Ruler and a Compass for All the Regular Polygons was achieved with, the Clear Step by Step Geometrical Analysis and Proofs, while is Highlighted The Erroneous Solution at the time.

b). In the System of a circle [O, OK], of Diameters KK′ ⊥ l 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 Collectively and Constantly Related to each other;....It is Proved.

b1). On Triangle KAK′ any Chord KA is related to its Angle φₐ as KA = 2R. sin φₐ, and Squared |KA|² = 2R. hₐ, While on the Triangles KADₐ, KAEₐ, sin φₐ = ADₐ / KA, and Differences |KA|² - |KEₐ|² = |KA|.|OₑₐDₐ| = k², where Point Oₑₐ is Midpoint of KEₐ = KA.

b2). On Any 2-Even Polygon Set-[n,n+2]--of Sides KA,KB is Defined the Constant Apothems on circle & Archimedes 2n-Sided-Vertices Polygon from n+1. Midpoints Mₑₐᵦ, Mᵦᵦ from Distance |Eₐ-Eᵦ| And |BB′|, and The Golden-Ratio-Point A₂.C on Apothems (rₐ – rᵦ) Sector |A₁.A, A₁.B| which Denotes Apothe m A꜀, and KA꜀ Cuts [O,OK] Circle at Point C such that Side KC IS The ODD-Polygon`s, & The Gravitational-Force as Stress σ = 2Gπrf/Φ ≡ The Holding Energy [Fig-7-3].

Skew-Cyclic and Constacyclic Codes over Finite Chain Rings for DNA-Based Data Storage and Error Correction

2026-11-09T10:30:34+00:00

DNA has considerable potential for long-term data storage, but the process of synthesising, preserving, and sequencing DNA strands can introduce substitutions, insertions, and deletions. In addition, biochemical requirements restrict the sequences that can be used reliably. This paper develops a structured framework for constructing error-correcting DNA codes from skew-cyclic and skew-constacyclic codes over finite chain rings. We first select a chain ring whose additive structure can represent the four DNA nucleotides and define a nucleotide map under which Watson–Crick complementation corresponds to addition by a fixed ring element. We then introduce the skew polynomial ring and describe skew -constacyclic codes as left ideals of . Their generator representations and cardinalities are presented alongside sufficient conditions for reversibility, reverse-complement closure, and balanced GC-content. Error performance is considered through Lee weight and a distance-preserving Gray map. Finally, we outline an end-to-end DNA storage workflow and discuss illustrative constructions: a length-four code over and a genuinely skew construction over that demonstrates non-unique factorisation. The examples clarify the algebraic framework; practical biochemical performance remains a matter for further testing.

GPH-International Journal of Mathematics

The Geometrical Construction of the Regular Polygons Using a Ruler and a Compass & the How Gravitational - Force Keeps Them Still.

Skew-Cyclic and Constacyclic Codes over Finite Chain Rings for DNA-Based Data Storage and Error Correction