Generalizing Pascal’s Triangle through Mathematical Modeling: Meta Triangles and Cross-Domain Applications in Energy, Education, and Systems Analysis

Abstract

This paper introduces an innovative mathematical construct known as the Guelph Expansion, which enables the generalization of Pascal’s Triangle into a family of Meta Triangles. Two efficient algorithms, the Embedded Pascal Triangles (EPTs) method and the Staircase Horizontal–Vertical (SHV) method, are developed to systematically generate these structures while enhancing numerical efficiency and scalability. In addition, the paper utilizes the Guelph expansion to formulate a novel Inhour Polynomial and a corresponding Coefficient-Based Model (CBM) for nuclear reactor kinetics, contributing to robust, and reliable reactor kinetics modeling.  Beyond energy applications, the proposed framework demonstrates broad interdisciplinary relevance, including computational approaches for surface generation in computer graphics, genome tagging and DNA sequencing, fault tree analysis, and probabilistic modeling. By unifying diverse applications within a single mathematical tool, the study supports knowledge integration, and decision making across scientific disciplines.