Dual Geometric Origins of the Fine‑Structure Constant

This paper explores a geometric interpretation of the fine-structure constant based on a recursive, relational tetrahedral–octahedral lattice referred to as the Honeycomb Unit (HU). The HU is defined as a regular tetrahedron decomposed via midpoint truncation into four half-sized tetrahedra and one implicit central regular octahedron, yielding a self-similar 10-node relational structure (4 tetrahedral nodes and 6 octahedral nodes). Repeated inward and outward recursion of this unit generates a scale-invariant tetrahedral–octahedral honeycomb that tessellates three-dimensional space without gaps or overlaps. Within this framework, two closely related geometric approximations to the fine-structure constant are examined: α ≈ √2 / 120φ and the algebraically equivalent form α ≈ (√10 - √2) / 240 Rather than being treated as numerological coincidences, these expressions are interpreted as encoding distinct but complementary aspects of the same underlying geometry: a local tetrahedral excitation scale and a global combinatorial node measure associated with the HU. The paper emphasizes the role of relational geometry, node counting, and exact volumetric symmetries (including the 4:1 tetrahedron–octahedron volume equivalence) in motivating these expressions. The work does not claim a derivation of the fine-structure constant from established quantum electrodynamics, nor does it propose a replacement for the Standard Model. Instead, it presents the Honeycomb Unit as a mathematically rigid geometric primitive that may serve as a useful toy model for investigating how dimensionless physical constants could emerge from discrete, self-similar spacetime structures. The results are intended as a conceptual and structural contribution to ongoing discussions in quantum gravity, discrete spacetime models, and the geometric origins of physical constants. v1