Arithmetic Expression Geometry

What is the geometry of a simple arithmetic expression? This presentation explores this fundamental question, starting from the observation that basic operations like addition and multiplication do not commute symmetrically in a vector space. We propose a new perspective: "Arithmetic Expression Geometry," where any arithmetic expression can be seen as a path in a specially constructed space. We derive a "flow equation" that acts as the "law of motion" in this space, unifying different arithmetic operations into a continuous flow. Our primary example is a hyperbolic space where the geometry is intrinsically tied to the arithmetic it represents. We demonstrate this link by showing that the evaluation of an expression corresponds to a scalar field which is an eigenfunction of the Laplacian on the space. This work opens up new avenues for exploring deep connections between arithmetic, geometry, analysis, and the theory of computation, suggesting that even the most basic mathematical structures hold undiscovered geometric beauty.