From the Octet to Every Stacking: the Web of Bonds

In short. Pile up oranges and there are small gaps between them, of two shapes, arranged according to how the layers are stacked. Over ten papers (#40–#57) DICTO and collaborators asked how the space is shared out into solid pieces, and which pieces touch which. This series paper tells that story, from DICTO's PET-bottle-cap DICTOspheres and the Polyhedraverse app, through the Golden Split, the Delian Bond, the web of bonds and the named pairs David and Goliath and Sonsam and Lahdeli, to the rule of halves: however layers of equal balls are stacked, the space between and around them is shared out into pieces of only finitely many kinds, each borrowed, whole or in half, from the cubic and the hexagonal stackings. Each part ends with the exact statement proved in the paper it retells; no new results are claimed. The ideas and names are DICTO's. Telling and checking with Claude (Anthropic); a first draft and a referee reading by a local open model (gpt-oss-120b), checked line by line against the papers. Photographs of DICTO's sculptures: DICTO. Licence CC BY-SA 4.0. More at dictospheres.com. View in the app: https://kaleidohedra.dictospheres.com/?open=stackings:4H