Granular Matter and Particle Shape: A Gravity-Relative Closure Theory from Contact Geometry to Macroscopic Heap Form

This working paper examines how particle shape influences the transition from microscopic contact geometry to macroscopic granular form. It asks how different deposition histories, particle orientations and terminal contact networks can nevertheless produce similar heap geometries. The analysis considers rigid, compact, convex, dry and cohesionless particles under uniform gravity. Spheres, triangular prisms and regular tetrahedra serve as canonical reference shapes. Gravity-relative closure is defined by the absence of an admissible gravity-descending motion, with compression-only contact equilibrium as its force-dual representation. Particle geometry determines accessible translations, rotations and contact transitions; formation history selects the microscopic path; and Structural Closure retains configurations that can persist. The resulting heap form is interpreted as a geometry–closure characteristic rather than a direct reproduction of particle outline. The paper continues the development begun in From Free Bodies to Static Structure and From Space-Filling Particles to a Porous Heap. No universal heap angle or terminal porosity is assumed.