Tessellations and Sweeping Nets: Advancing the Calculus of Geometric Logic

**Abstract**

This paper presents an innovative study on the advancement of geometric logic through tessellations and sweeping nets, addressing the challenge of arranging reflecting points for efficient ray tracing under limited time constraints. We introduce the concept of a sweeping subnet alongside a causal barrier to encapsulate the geometrical limitations posed by time, thereby delineating the boundary of influence for light propagation within a defined space. This work delves into the underpinnings of tessellation dynamics, revealing how the spatial arrangement and temporal evolution of tessellated patterns can be navigated and optimized through a novel algorithmic framework. Through a combination of theoretical exploration and practical implementation, including Python code for simulation and visualization, we provide a platform for approximating optimal tessellations that adapt to the constraints dictated by the causal barrier. The exploration of causal barrier dynamics, lattice optimization, and the computational approach to evolving tessellations contributes to the foundational understanding of geometric logic's calculus, with potential applications ranging from rendering engines in computer graphics to dynamic environmental mapping.

In this paper, we have explained how logic-vectors can be interpreted as a geometrical representation over computational engines and how it can be implemented in code using large language models. We have also proposed the usage of Time Compass to derive logic-vector evolution over a quasi-chaotic system and its direct manipulation on the tessellation Colormap.