Generated Space and the Ω-Cycle : Relational Geometry, Scale Transition, and Processual Memory

Abstract

This article develops a relational account of space within the Four Chamber Model. It begins from the classical contrast between absolute, relational, and transcendental conceptions of space and revisits the historical notion of imaginary spaces. The central proposal is that neither real nor imaginary space should be treated as a pre-existing container. Spatial order is instead generated through relational organization, so that different chambers correspond to different relational regimes of a common generative field. The proposal is explored through three-dimensional space-filling structures: the cuboctahedral–octahedral honeycomb, a non-convex stella-octangula–octahedral honeycomb, and the tetrahedral–octahedral honeycomb. Their relations provide a geometrical model for four functionally distinct transformations. Ω1 reweights already active relations, Ω2 activates latent relational positions, Ω3 differentiates a higher-order composite structure, and Ω4 introduces a bifurcation: in its positive form, Ω4+, completed lower-level spatial organizations are coarse-grained into the effective relational units of a new scale; in its negative form, Ω4-, the terminal tetrahedral–octahedral organization is modified within the existing scale. Under Ω4+, the resulting process is not merely cyclic but spiral: relational type recurs while scale changes. This provides a basis for fractal organization and processual memory, since higher-order vertices retain the relational history of the lower-level spaces from which they emerge. Under Ω4-, local modifications of the terminal regime may further generate relational non-uniformity, possible curvature, and biased trajectories.