This paper introduces a novel method for generating Pythagorean quadruples using a single-variable algebraic approach derived from the classical Platonic sequence. While traditional methods for generating Pythagorean triples are well known and deeply rooted in number theory, their extension to quadruples has remained less accessible. By building on the established formulas for triples and applying recursive logic, this work presents a set of formulas that yield valid Pythagorean quadruples with only one input variable. The approach distinguishes between even and odd values of the input variable and constructs each quadruple component step-by-step using simple algebraic expressions. Although the method does not account for all primitive quadruples, it offers a highly efficient and elegant means to generate a wide range of valid solutions. Additionally, the paper provides full algebraic proofs for correctness and includes a Python implementation that can be used to automate the generation and verification process. The simplicity of the method makes it particularly useful for educational, computational, and exploratory purposes. This work also forms a basis for potential future developments in higher-dimensional Diophantine analysis and applications in cryptography, where integer-based geometric structures play an important role.