Mathematical Modeling of Pascal Triangular Fractal Patterns and Its Practical Application

摘要

This article focuses on the mathematical modeling of triangular fractal patterns and carpet design in textiles, demonstrating the classic and new arithmetic, combinatorial features of the generalized arithmetic triangles of Pascal's triangle. Binomial, three-term, and other combinatorial numbers constructed based on recurrence relations have been investigated. The software "Pascal's Triangle" was developed in the C# programming language. A convenient user interface of the program has been developed. Based on the software implementation, arithmetic triangles are constructed from residues in the form of Pascal's triangle and combinatorial numbers. The developed software can be used in industry, when drawing patterns, for their subsequent stamping on carpets, fabrics, ceramic tiles, etc.