Consideration of Hierarchical Self-Similarity in Physical Systems Starting from the Fine-Structure Constant

The fine-structure constant α is conventionally understood as a dimensionless constant representing the strength of electromagnetic interaction. In previous studies, the author examined two different geometrical representations of the fine-structure constant. The first is α = rω / c where r is the characteristic length of the internal structure, ω is the angular frequency of the internal phase, and c is the maximum propagation velocity in external spacetime. In this representation, α is interpreted as a coupling ratio between internal phase motion and the external spacetime structure. The second is obtained from an internal rotational structure and the expansion of relativistic energy: α_model = (3/2 − √2)² This value reproduces the observed fine-structure constant with a deviation of approximately 0.8%. The residual may not merely be a numerical error, but may reflect a contribution from a further lower layer within the principal internal structure. In this study, these two representations are not restricted to special properties of the electron, but are examined as a possible basis for generalization to the hierarchical structures of physical systems. Specifically, it is proposed that dimensionless quantities characterizing the coupling between internal degrees of freedom and external structures may also exist between different physical layers—sub-particle internal domains, particles, atoms, molecules, macroscopic matter, astronomical systems, and cosmic structures—and that these quantities may form self-similar structures. The purpose of this study is not to prove the existence of such hierarchical self-similarity. Rather, it is to present this possibility as a generalization derived from the two representations of the fine-structure constant and to clarify physical questions that should be examined in future work.