The DICTO OG Family on Every Stacking

One exact kit of pieces builds the DICTO Ogdoecontahedron (OG) into the octet, the gyrated octet, the classical polytypes between them and a Fibonacci crystal that never repeats; the OG family stacks like a real mineral, twinning by the spinel law. Two short rules say why. The tetrahedral–octahedral honeycomb falls into layers, and each layer can be laid on the one below in two ways: as in the octet truss (c) or as its mirror image in the shared plane (h). Any word in c and h is a stacking, exactly as for close-packed crystals, whose polytypes 3C, 2H, 4H, 6H, 9R, 15R … are words of this kind. The OG sits exactly in the octahedra of the octet [2] and of the gyrated octet [3]; here we place it, and the solids of its family, in every stacking. The twin rule: if a solid is centrosymmetric and has a three-fold axis at right angles to the layers, its mirror image in a layer plane is the same solid turned 60° about that axis; unless it also has a mirror at right angles to the axis, this is a true twin. The fit rule: a solid that lies in its own layer and does not overlap its copies within the layer fits every stacking, whatever the word. The OG, the DICTO OTT and the OCD Stella lie in their octahedra, and [3] showed that they fit every stacking; the rules explain why. The DICTO Belted Triacontahedron (DBT) leaves its octahedron but not its layer, so it too fits every stacking (shown in [3] only for the gyrated octet); exactly in ℚ(√5), the golden envelope (a rhombic triacontahedron) it was cut from fits the octet but overlaps at every h plane by exactly the volume of the two tips the DBT removes. The shares of space do not depend on the stacking: OG 1025/114 − 145√5/38 ≈ 45.89%, OTT 176/3 − 26√5 ≈ 52.89%, DBT 493/6 − 73√5/2 ≈ 55.02%. For the OG we give the kit per cell for 3C, 2H, 4H, 6H, 9R and 15R, and for the Fibonacci stacking (c → ch, h → c), which never repeats: in the limit exactly (3 − √5)/2 of the planes between layers are h planes (twin planes), and a cell needs 5 + √5 caps, (3 − √5)/2 twin caps, √5 − 1 tetrahedra and (3 − √5)/2 bipyramids, all exact. Companion to The DICTO Ogdoecontahedron on the Gyrated Octet (#37a), DICTO DOI 10.5281/zenodo.23484012. Finding #40 on dictospheres.com. Scripts, exact checks and data: https://github.com/DICTOR-Master/kaleidohedra/tree/b7c6307/studies/2026-10-20-og-polytypes, also attached here (og-polytypes-study-b7c6307.zip).