Structural Properties of Pascal Pyramids and Pascal Simplexes: Classical Results and Some Extensions

Pascal’s Triangle, renowned for its geometric elegance and profound applications across combinatorics, algebra, and probability, has fascinated mathematicians for centuries. While its origins can be traced to Chinese, Persian, and European mathematical traditions, the study of its higher-dimensional analogues remains notably underexplored. This paper offers a systematic and self-contained study of Pascal Pyramids and Pascal Simplexes with their proofs. It encompasses both classical results (such as multinomial identities) and novel contributions (including boundary and scaling properties), as well as fresh perspectives (such as graph-theoretic interpretations) that are rarely documented in the existing literature.