[Paper Review] Different conceptions of Euclidean geometry

Yuri A. Rylov|arXiv (Cornell University)|Sep 18, 2007

Advanced Mathematical Theories and Applications7 citations

TL;DR

This paper proposes three distinct representations of Euclidean geometry—E-representation (points, segments, angles), V-representation (points, vectors with linear space), and σ-representation (single element with world function σ = ρ²/2). The σ-representation, where distance is encoded as a fundamental structure via σ, is shown to be optimal for modifying Euclidean geometry, as any such modification leads to multivariant geometry with multiple non-equal vectors of the same magnitude and direction.

ABSTRACT

Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In σ-representation there is only one basic element and additional structure: world function σ = ρ 2 /2, where ρ is the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the σ-representation. The σ-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q0Q1,Q0Q ′ 1,..., which are equal to the vector P0P1, but they are not equal between themselves, in general.

Motivation & Objective

• To analyze and compare different foundational representations of Euclidean geometry based on their basic elements and structures.
• To identify the most suitable representation for modifying Euclidean geometry while preserving geometric consistency.
• To investigate how the introduction of a world function σ = ρ²/2 as a fundamental structure enables systematic geometric generalizations.
• To demonstrate that modifications of Euclidean geometry naturally lead to multivariant geometries where equal vectors are not necessarily identical.
• To establish the σ-representation as the most appropriate framework for exploring non-Euclidean or generalized geometries.

Proposed method

• Defining E-representation using three primitive elements: point, segment, and angle, without additional structures.
• Introducing V-representation with points and vectors, augmented by a linear vector space structure.
• Formulating σ-representation using only points as basic elements, with the world function σ = ρ²/2 as the sole geometric structure.
• Using the world function σ to encode distance and geometric relations, replacing traditional metric concepts.
• Analyzing the implications of modifying σ for geometric consistency and vector equality.
• Demonstrating that in modified geometries, multiple vectors can be equal to a given vector without being equal to each other, leading to multivariance.

Experimental results

Research questions

• RQ1 How do different foundational representations of Euclidean geometry differ in their basic elements and structural requirements?
• RQ2 What advantages does the σ-representation offer over E- and V-representations for geometric generalization?
• RQ3 In what way does encoding distance as a world function σ = ρ²/2 fundamentally alter the structure of geometry?
• RQ4 Why does modifying Euclidean geometry via σ-representation inevitably lead to multivariant geometry?
• RQ5 How does the σ-representation enable a more systematic and consistent approach to geometric modifications?

Key findings

• The σ-representation, based on a single basic element (point) and the world function σ = ρ²/2, provides the most suitable foundation for modifying Euclidean geometry.
• In the σ-representation, the concept of distance is not a derived quantity but a fundamental structure, encoded directly in σ.
• Any modification of the proper Euclidean geometry within the σ-representation results in a multivariant geometry where multiple vectors can be equal to a given vector without being equal to each other.
• The E-representation lacks a formal structure for distance, while the V-representation relies on a linear vector space, both being less suitable for generalization than σ-representation.
• The σ-representation allows for a consistent and systematic exploration of non-Euclidean geometries by treating σ as the primary geometric invariant.
• The transition from univariant to multivariant geometry upon modification is a direct consequence of altering the world function σ, highlighting its central role in geometric structure.