Title: Non-Standard Geometry – Computational Geometry of Complex Systems

Abstract

This paper explores the application of non-standard geometry to computational geometry, focusing on the development of a novel method for defining and computing geometric properties for complex systems, particularly fluid dynamics and protein folding. Traditional geometric computation often struggles with the inherent complexity and self-organization of these systems, necessitating the creation of intricate geometric constructs. We propose a 'geometric language' – a system of rules and symbolic representations – that enables efficient manipulation and analysis of these complex shapes. The core mechanism involves establishing a hierarchical structure within this language, allowing for the generation of novel geometric configurations through a combination of geometric transformations and parametric modeling. This approach aims to overcome limitations in current computational geometry by offering a framework for tackling problems that are currently computationally intractable. The paper will detail the conceptualization of this language, its implementation through a set of rules and algorithms, and initial explorations into its potential for solving specific problems within fluid dynamics and protein folding. Finally, we present preliminary results demonstrating the feasibility of this approach, highlighting its potential for advancing the field of computational geometry.