RFW-Orthotope: Self‑Similar Geometry and Vertex‑Tensor Structure

This paper systematically studies a class of high‑dimensional orthotopes possessing strict self‑similarity, termed the RFW orthotope. This construction generalizes the self‑bisecting property of the two‑dimensional $\sqrt{2}:1$ rectangle (ISO‑A paper size) to $N$‑dimensional Euclidean space, with edge‑length ratios given by the geometric progression $2^{(N-1)/N}:2^{(N-2)/N}:\cdots:1$. For the three‑dimensional special case the edge‑length proportions read $\sqrt[3]{4}:\sqrt[3]{2}:1$. We prove that within the family of orthotopes, the bisective self‑similarity condition uniquely fixes this geometric‑progression form of edge lengths; the RFW orthotope is the unique non‑trivial solution up to translation, orthogonal transformation and global scaling. After bisecting along the midpoint of the longest edge, the resulting sub‑block can be made congruent to the original body after edge‑order permutation and global rescaling. Within the $N$‑dimensional framework we derive the closed‑form volume $V_{N}=2^{(N-1)/2}$ and the volume ratio $R_{N}=\pi^{N/2}\big/\big[2^{N}\Gamma(N/2+1)\big]$ of the largest axis‑aligned inscribed ellipsoid relative to the RFW orthotope. It is demonstrated that $R_{N}$ decays super‑exponentially for $N\ge 5$: the inscribed ellipsoid occupies only a negligible fraction of the orthotope volume, and the overwhelming portion of volume resides in boundary regions outside the inscribed ellipsoid. We further investigate the combinatorial skeleton, vertex‑and‑edge counting, and sector decomposition at vertex corners. It is proven that sector volumes increase strictly monotonically with their corresponding edge lengths, exhibiting strong anisotropy. In three dimensions, a traceless second‑moment tensor $Q_{ij}$ can be defined at each vertex corner, fully determined by the three edge lengths. This tensor characterizes shape anisotropy of the vertex corner and provides complementary information to the scalar angle defect in Regge calculus. All derivations are carried out in flat Euclidean space, obtained via volume measure, moment integration and combinatorial counting, without introducing any physical assumptions or dynamical postulates.