Classical Fractional Dimensional Scale Invariance

Physical systems that behave according to laws that are independent of their size are emergent and often epi-phenomenal. The understanding of these systems is not well discussed in university, albeit the mathematics is attainable for advanced undergraduates and graduate students. The idea is simple. The laws that govern one scale are isomorphic to laws at another scale. This is easily seen in the holographic principle, exemplified by the Stokes' theorem and divergence theorem. These lemmas, extended to a more erudite argument, imply that dimensions, in their integer formation, relate through surfaces and volumes extending into n dimensional theory. As such, string theory has become popular in the extension of dimension. What is interesting is the fractional dimension and the application of the divergence theorem to the fractal shape. Here, the motion, for lack of a better term, between dimensions of fractional order, is made to align through invariants or constants of a gauge condition. By applying the rules of isomorphic analysis of dimensions, in physical systems, the divergence and Stokes' approach generate constants of motion or scale inherently tied to Platonic solids and laws of regularization. In essence, fractional and integer dimensional laws are correlated to the one higher and one lower dimensions through a calculus dependent on gauge conditions. Modification of the fundamental theorem of calculus implies a zeta function time step, in effect, to correlate the spacetime landscape and epi-phenomenal state to the philology of invariant mathematics.

This is all too complicated. What is important to take home is the idea that scale invariance manifests in physical systems and could be the technique to better analyze physical systems that would otherwise require a massive experimental apparatus. By studying the overarching scale and the connection, or isomorphism, one can correlate dimensions and scales to basic mathematics and logic. This will allow for the creation of a saturating physical axiom, or a set of axioms, that depend on one physical system and then correlate to all physical systems.