Plato's Five in the Octet: All Five Platonic Solids Nested in Every Cell, and a Golden Ladder of Crystals to the Quasilattice

Theorem 1 (Five in the octet). In the octet of edge 1 every octahedron holds the nest octahedron > icosahedron (Pacioli, edge √2/φ²) > dodecahedron (edge √2/(3φ)) > cube (edge √2/3) > tetrahedron (edge 2/3); the nest's cube is exactly the octahedron's dual cube, so the dodecahedron touches the octahedron at its eight face centres. Every tetrahedron of the octet holds the same nest at half size through its midpoint octahedron, with the truncated tetrahedron (edge 1/3) between. All five Platonic solids sit in every cell of a structure periodic in three directions. Theorem 2 (For ever). The nest repeats inward and outward by exactly 1/3 per turn, keeping its orientation: the golden factors cancel; the shorter loop cube > icosahedron > dodecahedron > cube repeats by φ/3. Exact in Q(√2, √5). We list the twelve classical inscriptions, the fourteen chains through all five (eighteen with the truncated tetrahedron), the gap pieces (two of them new shapes) and supercells from the 120-cell, the 600-cell and the middle slice; the Dodeca-13 fits inside its own central dodecahedron at exactly 1/3. Part 2. On the cubic approximants of the icosahedral quasilattice (Elser and Henley 1985), rungs 1/1 to 13/8, every cell is tiled by golden rhombohedra, and Theorem 3 gives points = tiles per cell by an exact count (16, 136, 576, 1220, 10 336, 43 784); the icosahedral seats are the triacontahedral vertices, at least √3 φ² apart, and each carries a nest. Diffraction measures the climb: ten-fold score 0.916 at 1/1, 0.9976 at 13/8, 0.9977 for the quasilattice. New pieces named by DICTO: the Sheriff (the pentagonal cap of the icosahedron minus its dual dodecahedron) and the Lone-star (the wedge). DICTO proposed putting all five Platonic solids into the octet with pieces of the solids themselves between them, nesting them in several combinations and at several scales, the supercells from the 120-cell, the 600-cell and the middle slice (his UNITY), filling the gaps with harmonics, waves and golden skin, and the ladder to a universal lattice. Computation, proofs and drafting with Claude (Anthropic); an independent review (Claude) checked the prior art and supplied the proofs of Theorem 3.