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Tessellations and Sweeping Nets: Advancing the Calculus of Geometric Logic

Authors: Emmerson, Parker;

Tessellations and Sweeping Nets: Advancing the Calculus of Geometric Logic

Abstract

**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. This paper focuses on the optimal arrangement of reflecting points for efficient ray tracing given limited sweep time. We examine spatial configurations, employing our core concept of a sweeping subnet and defining a causal barrier to capture constraints imposed by time. We will also discuss the influence of these constructions on the design of an algorithm for approximating optimal tessellations. I have provided code for each of the graphs, as the mathematics is demonstrated unequivocally by their implementation. The reader can test out the reality of this system by visualizing the graphs themselves using Python in a suitable environment like Google Colaboratory.

Keywords

Computational Optics, spatial constraints, Ray Tracing, Geometric Structures, Crystal Lattices, Logic, Path Planning, mathematical linguistics, Biomedical Engineering, Visual Mathematics, Computational Geometry, Tessellation, logic vectors, Solitons, Topology, Viral Capsids, Time-compass, Symmetry, Artificial Intelligence, Three-dimensional Spaces, logic solitons, Computer Graphics, Mamba, topological intelligence, Polar Coordinates, quasi-quanta, Quantum Logic Gates, Generative Art, robotics, Spatial Analysis, runnels, Computational Modeling, Directed Graphs, geometric efficiency, spatial symmetries, Computational Simulations, Spatio-Temporal Constraints, Cartesian Coordinates, Fractals, chaos theory, Double Helix, symbolic modeling

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  • citations
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    0
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
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