Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach

TL;DR

This paper explores both commutative and non-commutative parallelogram geometries through an accessible, software-based experimental approach, integrating geometry, algebra, and axioms for educational and research purposes.

Contribution

It introduces a novel educational method combining software experiments with algebraic and axiomatic approaches to parallelogram geometry.

Findings

Demonstrates the transition from commutative to non-commutative geometry.

Provides exercises accessible to students and researchers.

Bridges geometric intuition with algebraic formalism.

Abstract

By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via exercises using dynamical software (such as geogebra), hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebra and elementary group theory), and (3) axiomatic geometry.