Rigid and Deformable Bodies in a Dynamic Medium: Packing Problems for Basic and Decomposable Bodies

The paper introduces a formal language for describing bodies of existents in a dynamic Medium. We define the body of an existent, connected and disconnected bodies, the hardness function and the degree of hardness, the basis and decomposability, accumulations at scale , scattered existents, coherence of accumulations, and the degree of coherence of a cohort. It is shown that classical methods of packing theory (geometric, computational, analytical, computational methods, models of deformable bodies) work with rigid objects (or with fixed area/perimeter) and do not cover bodies with a hardness function in a dynamic Medium. Three problems generalizing classical formulations to bodies with a hardness function are formulated: an analogue of the sausage conjecture, an analogue of packing in a bounded container, and an analogue of Apollonian packings. Open problems are listed, including the maximal packing density of bodies with in 2D, saturation for deformable bodies, a transformer and a saturation reachability length for deformable bodies, as well as a generalization to . The language introduced in the paper is applicable not only to packing problems but also to the description of structures in demography, urban planning, and biology.