An Observer-Based View of Euclidean Geometry

Newshaw Bahreyni (), Carlo Cafaro and Leonardo Rossetti
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Newshaw Bahreyni: Physics and Astronomy Department, Pomona College, Claremont, CA 91711, USA
Carlo Cafaro: Department of Nanoscale Science and Engineering, University at Albany-SUNY, Albany, NY 12222, USA
Leonardo Rossetti: Department of Nanoscale Science and Engineering, University at Albany-SUNY, Albany, NY 12222, USA

Mathematics, 2024, vol. 12, issue 20, 1-24

Abstract: An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2 + 1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

Keywords: Euclidean geometry; lattice; partially-ordered set; causal set; geometric algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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