From Space-Filling Particles to a Porous Heap: Formation-Generated Collective Closure under Gravity

This working paper extends the concept of formation-generated structural closure from spherical particles to compact, orientation-dependent particle geometries. It addresses a simple physical question: Why can congruent particles that are geometrically capable of a void-free arrangement nevertheless form a persistent porous heap under gravity? A three-dimensional first-principles model is developed for homogeneous, rigid, convex, dry and noncohesive particles with frictionless unilateral contacts. The analysis distinguishes geometrical admissibility, gravity-driven motion, static support, neutral escape mechanisms, finite energy barriers and boundary-connected collective closure. Particular attention is given to the influence of particle orientation, contact position and contact-generated rotational lever arms. An exact four-sphere configuration provides the reference case. The corresponding regular triangular-prism support motif is shown to provide vertical support while retaining lateral translations and yaw. Suitably displaced lateral contacts can eliminate these residual motions. The analysis demonstrates how a porous terminal structure can persist even when a void-free tessellation exists: the rearrangements required to reach that arrangement may become inaccessible during formation. For an ideal three-support spherical surface cell, the directional closure domain is derived exactly. The isotropic limit is 19.4712°, and the most favourable directional limit is 35.2644°. These are conditional reference values for the specified contact cell, not universal angles of repose. Under spatially stationary and macroscopically isotropic formation, a constant closure-selected inclination produces a conical mean surface. The paper develops a falsifiable framework for relating particle geometry, contact topology, formation history and boundary support to the emergence of persistent granular structures. Numerical closure domains for compact triangular prisms and regular tetrahedra remain open pending the construction of formation-accessible free-surface cells. The work provides a geometrical foundation for subsequent investigations of structural closure under weak gravitational conditions.