Sonsam and Lahdeli: the Delian Bond on the Twinned Octet, with Ian Dury, Blockhead and Keyed Switch Points

In short. This paper is about webs of bonds: when pieces fill space, each shared face is a bond, and the bonds together make a network that rewires as the pieces change (#52, #53). Here we follow that network on a second stacking, and show that keys change the pieces but never the network. Stack layers of balls in the cubic way (ABCABC…) and the gaps between them are octahedra and tetrahedra: the octet. Stack them in the hexagonal way (ABAB…), each layer a mirror image of its neighbours across it, and the gaps are the same octahedra and tetrahedra, joined across the mirror planes: the twinned octet. Paper #53 slid one number, a weight, between the two kinds of gap and recorded every way space splits on the way. On the cubic octet, at one exact weight, λ = 3/8 − ∛4/4 (the cube root of 2, squared: the number of the Delian problem, doubling the cube), the two pieces have equal volume: DICTO's Samson and Delilah. We follow the same walk on the twinned octet. It has its own equal-volume point, at the single root of an irreducible cubic; there the two pieces are new solids, which DICTO named Sonsam (19 corners, one face a nine-sided polygon lying on the mirror plane) and Lahdeli (12 corners), the syllables swapped as the stacking is mirrored. Further on, the twinned walk makes a stop the cubic walk does not have: Ian Dury (16 corners) with the classical 60° rhombohedron; and where the cubic walk ends in cubes the twinned one ends in Blockheads (10 corners), the twin of the cube. Every volume is exact: √2/6, 13√2/54, √2/54, √2/4. Moving a cap from one piece into its neighbour keeps the tiling and its web of bonds; done with the new pairs this gives DICTO's keyed forms Sonsam-K and Lahdeli-K, and Ian Dury-K with Rhombo-K, a six-pointed star. Finally we read DICTO's Sunstar lattice (#42) in the same light: it is rock salt's switch point (#53) keyed with Euclid's roofs. The two new walks' pieces and the keyed forms were not found in the sources checked. The ideas are DICTO's: that further bonds may be "hiding in plain sight" in the census networks; other rhythms of volume (doubling, tripling); a different but cohesive bond network beside the golden and Delian ones; keyed versions of the new pairs; and the names Sonsam, Lahdeli, Ian Dury, Blockhead and the -K forms (25 October 2026). Computation, exact proofs and drafting with Claude (Anthropic). Exact studies and checks attached (commit ca73df8). Licence CC BY-SA 4.0. More at dictospheres.com.