Reading the Voids: Spherical Waves, Harmonics and Golden Bricks in the Gaps of a Packing

A packing of solids leaves gaps. Most of the time we only measure how much. Here we look at them: we send waves through the packing and the gaps show up dark; we read the dark places like a chemist reads the cloud of electrons in a crystal, as rooms joined by windows; we listen to each room's shape; and we grow golden bricks into it until it is full. One thing no brick can ever fill, and an old invariant from 1901 tells us exactly what it is. Abstract. Grown from DICTO's work on harmonics and resonance, where single solids were heard as tones, we let spherical waves run through whole packings and assemble a toolkit for their gaps and try it on DICTO's Ogdoecontahedron (OG), its pentagon-cut form (the cut OG), and DICTO's Bull-horns OG adaptor. Theorem 1 (exact): the OG and the cut OG have nonzero Dehn invariants; they share their 120 hexagon-rim edges, and the obstruction lives only on the edges. Theorem 2 (exact): the room adaptor of every lattice packing of either solid, and DICTO's Bull-horns OG adaptor, carry a nonzero Dehn invariant (minus the OG's, or a third of it), so none of them can be cut into golden rhombohedra or any pieces whose dihedral angles are rational multiples of π. Then the tools (float): spherical waves find the gaps of the densest lattice packing of the cut OG, never mapped before; reading the wave field by its critical points gives, per copy, one atom, five bonds, five rings and one cage: one gap room opening through eight windows; the room closes with one piece, the DICTO Void Keeper, so the cut OG and the Void Keeper fill space by translation. Golden rhombohedra grown in φ-steps fill 84% of the room; one coherent golden tiling completes it with whole and trimmed bricks; around the original OG a complete fill by the OG, the DICTO Golden Skin and one golden rhombohedron makes the skin as thin as wished (2.9% of space), and the Bull-horns adaptor is completed with 66% whole golden bricks. Golden tetrahedra, whose Dehn invariants are not zero, cannot rescue a completely golden fill either: their angles are of another kind than the OG's (strong numerical evidence).