The Unified Sierpiński Gasket Model of Philology: Crossroads Between English, Arabic, and Azerbaijani Languages

Old ways of studying how languages change, like family trees or wave patterns, work okay for showing how words and grammar shift over time within language groups. But they have trouble explaining things that aren't simple, like when languages borrow from each other in complicated ways, how similar structures pop up at different levels within a language, or how languages from totally different families line up structurally. This paper introduces a new approach called the Unified Sierpiński Gasket Model of Philology (USGMP). It uses math to describe how languages evolve, thinking of it like a repeating system that works in a kind of fractal space. We used English, Arabic, and Azerbaijani as examples. By placing them at the corners of a Sierpiński Gasket shape, we showed how this repeating, self-similar pattern helps explain how languages interact, how words and meanings form fractal patterns, and how sentence structures break down into smaller, similar parts. We calculated specific fractal measurements, like dimensions and scale-invariant matrices, to describe how languages have influenced each other historically, how dominant languages affect others, and how new borrowed words fit in. The paper also includes visual diagrams of language structures, observations from real language data, unsolved questions, and ideas for future research in areas like digital humanities, quantum linguistics, and automatic translation.