Surreal Geometry

This manuscript presents a unified mathematical framework that extends Conway's surreal numbers into higher-dimensional geometric spaces to construct a foundation for algebra, calculus, and theoretical physics. By modifying the classic surreal numbers definition to allow bounds to contain nested sets of bounds, the system naturally maps to multidimensional components such as vectors and bivectors. The framework utilizes an angle bracket operator, similar to the Cayley-Dickson construction, and a geometric product, to generate an associative Clifford algebra (Geometric Algebra) of positive metric. The system natively incorporates actual infinitesimals such that, using non-standard analysis, the geometric derivative and a generalized Stokes' theorem are effectively defined without relying on traditional limits. This surreal geometric algebra is then applied to physical space, demonstrating how the fundamental equations of thermodynamics, electromagnetism, and quantum mechanics are naturally expressed within an Algebra of Physical Space (APS) model.