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Discrete-geometric special solution of traceless quadrupole moment — the RFW orthotope

Year: 2026
Type: preprint
Abstract: Starting from the self-similar folding property of the two-dimensional ISO-A paper rectangle, the RFW orthotope, a self-similar hyper-rectangle in N-dimensionalEuclidean space, was constructed in companion work [1]. The present paper focuses on three-dimensional tensor-geometric properties of rectangular corners.We prove that the vertex cone of any non-cubic rectangular block in three-dimensional flat Euclidean space carries a non-vanishing traceless quadrupole tensororiginating purely from volume second-moments, whose components are fully determined by the three edge lengths. This tensor degree of freedom has been largelyoverlooked within Regge-calculus literature, where attention is usually restricted to the scalar 90◦ angular deficit at rectangular corners. We introduce an analogy between a soccer-ball-style polyhedral shell and rectangular-corner elementary building blocks, highlighting how anisotropic volume moments reside locally at vertices independently of scalar angular deficit. It is shown that the three-dimensional RFW orthotope furnishes the unique non-trivial special solution such that this vertex-centred traceless quadrupole tensor retains its principal axes and is only rescaled by a global scalar factor under recursive bisection cutting. This makes it a discrete-scale-invariant eigen-mode for discrete quadrupole sources.Formally isomorphic to the mass quadrupole tensor in linear general relativity,this local geometric tensor motivates a heuristic geometric lower bound for continuous gravitational-wave emission from deformed crust lattices inside rotating neutron stars. All results are derived within flat Euclidean space using volume measures,without additional physical postulates.
DOI: 10.5281/zenodo.22669734
Author: Ren, Fei
Language: English
Open Access status: green
OpenAlex ID: W7212027161
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