The DICTO Ogdoecontahedron on the Gyrated Octet

In a companion paper the DICTO Ogdoecontahedron (OG) sits in every octahedron of the octet truss, its eight joint hexagons on the eight faces. DICTO asked whether it can also live in the gyrated octet, the tetrahedral–octahedral honeycomb stacked A B A B, in which octahedron meets octahedron and tetrahedron meets tetrahedron across every layer. It can: the OGs on both sides of a layer meet on the same joint hexagon, but each is the mirror image of the other, and since the OG has no such mirror the layers are twin boundaries; exactly, the twin is the OG turned 60° about the joint three-fold axis, the spinel law of crystallography. Cutting each octahedron into the OG and eight chiral caps, and keeping the tetrahedra whole, gives one exact kit for the octet, the gyrated octet and every stacking between; at a twin layer a left and a right cap join into a twin cap and two tetrahedra into a triangular bipyramid. The facing-star middle piece of the octet fails at twin layers, and a mirror middle piece of volume (424 − 88√5)/4617 takes its place. The trait runs in the family: the OCD Stella twins in the same way, meeting its twin on a whole joint triangle; the golden envelope does not fit as it is, but cut flat at two opposite three-fold corners it becomes a solid of its own, the DICTO Belted Triacontahedron (DBT), which holds the OG and fills the gyrated octet's octahedra pole to pole without overlap. The DICTO Jewel, framed by cubes, does not twin. All claims are exact in ℚ(√5) except where marked float. Companion to The DICTO Ogdoecontahedron on the Octet Truss (#37), DICTO DOI 10.5281/zenodo.23457016. Finding #37a on dictospheres.com.