RFW Orthotope: Measure-Geometric Properties and Formal Physical Analogies —A Self-Similar Hyper-Rectangle

This paper systematically investigates a family of high-dimensional hyper-rectangles with strict self-similarity, termed the RFW orthotope. The construction generalizes the self-bisection property of the two-dimensional √2 : 1 rectangle (ISO-A paper format) to N-dimensional Euclidean space, with side lengths proportional to2(N−1)/N : 2(N−2)/N : ··· : 1. After cutting through the mid-point of the longest edge, each resulting sub-block becomes congruent to the original geometry uponsuitable rotation and rescaling. Within the N-dimensional setting, we rigorously derive the volume formula VN = 2(N−1)/2 and the volume ratio RN = πN/2/2NΓ(N/2 + 1) between themaximal axis-aligned inscribed ellipsoid and the RFW orthotope. We prove that RN decays super-exponentially for N ≥ 9, yielding an ellipsoidal shell-like geometry where the overwhelming fraction of volume concentrates near the boundary. We further study its combinatorial skeleton: vertex and edge counting, combinatorialexplosion induced by recursive cutting, and decomposition into vertex-associated pyramids and sectors. It is proved that sector volumes increase strictly monotonically with corresponding edge lengths, giving rise to strongly anisotropic geometric behaviour. On top of the flat Euclidean background, we introduce an auxiliaryeffective metric. Under this effective geometric picture, we observe gradient-like effective curvature, conical singular behaviour near vertices, and chiral distinctionof barbed spike structures.Based purely on these geometric results, we report several formal similarities and numerical coincidences between this toy model and general relativity as wellas gauge-field theories. We do not claim that this geometry corresponds to our physical universe, nor do we establish any rigorous mathematical equivalence. Asan analytically tractable toy-geometric object, its complete mathematical structure and potential physical connections provide concrete starting points for future investigations. The main rigorous contributions lie in high-dimensional convex geometry; physical-like analogies are heuristic observations only.