[Paper Review] The Transform of a line of Desargues Affine Plane in an additive Group of its Points

Orgest Zaka, Kristaq Filipi|arXiv (Cornell University)|Aug 20, 2016

Mathematics and Applications15 citations

TL;DR

This paper establishes a group structure on the points of a line in a Desargues affine plane by leveraging the little Pappus theorem to define an additive operation. It proves that the points on any line in such a plane form an abelian group under this operation, providing a geometric realization of additive group structure in incidence geometry.

ABSTRACT

In this paper we present a set transformation of points in a line of the Desargues affine plane in a additive group. For this, the first stop on the meaning of the Desargues affine plane, formulating first axiom of his that show proposition D1. Afterwards we show that little Pappus theorem, which we use in the construction of group proofs in additions of points on a line on desargues plane, also applies in the Desargues affine plane.

Motivation & Objective

• To establish a geometric construction of an additive group on the points of a line in a Desargues affine plane.
• To demonstrate that the little Pappus theorem holds in the Desargues affine plane, enabling the definition of a group operation.
• To show that the group operation on points of a line is associative, commutative, and has an identity and inverses.
• To provide a foundation for understanding additive structures in incidence geometries via synthetic geometry methods.
• To bridge incidence geometry and group theory by showing that Desarguesian planes support natural abelian group operations on lines.

Proposed method

• Utilizes the first axiom of the Desargues affine plane to define the geometric framework.
• Applies the little Pappus theorem as a key tool to define and verify the group operation on points of a line.
• Defines the addition of two points on a line via geometric constructions involving parallel lines and intersection points.
• Verifies the group axioms (associativity, commutativity, identity, inverses) using incidence and collinearity properties.
• Relies on synthetic geometry techniques rather than coordinate-based algebra to derive the group structure.
• Establishes that the resulting operation is closed, commutative, and forms an abelian group under the defined rules.

Experimental results

Research questions

• RQ1 Can the points on a line in a Desargues affine plane be endowed with a natural additive group structure?
• RQ2 Does the little Pappus theorem hold in the Desargues affine plane, and can it be used to define a consistent group operation?
• RQ3 Is the group operation on points of a line in the Desargues plane associative and commutative?
• RQ4 What are the geometric properties that ensure the existence of identity and inverse elements under the defined addition?
• RQ5 How does the group structure on a line relate to the underlying incidence axioms of the Desargues affine plane?

Key findings

• The points on any line in a Desargues affine plane form an abelian group under the geometrically defined addition operation.
• The little Pappus theorem is valid in the Desargues affine plane and is essential for proving the associativity of the group operation.
• The group operation is commutative, with the identity element derived from geometric symmetry and intersection properties.
• Each point on the line has a unique inverse point such that their sum yields the identity element.
• The construction is purely synthetic, relying only on incidence and parallelism axioms without coordinate systems.
• The resulting group is isomorphic to a standard abelian group, confirming the consistency and algebraic richness of the geometric structure.