A Fibonacci Spine in a Five-fold Chorus of Ogdoecontahedron Packings

Abstract

The DICTO Ogdoecontahedron is a convex, centrally symmetric icosahedral solid with 20 regular hexagons and 60 kites. Its hexagon directions contain five inscribed cubes, and each cube gives a periodic (body-centred cubic) packing in which copies meet face to face on hexagons. We place the five packings ("voices") in five 72° wedges round a five-fold axis, which is the icosahedral irrational twin, and remove clashing copies in whole five-fold rings. Checked exactly in ℚ(√5), the result is overlap-free to radius 10 (2756 sites, 385 removed, 7465 close pairs). Every copy lies at a golden height (m + nφ)u along the axis. Apart from one ring, the removed copies nearest the axis lie in one plane on a rank-2 lattice with axis slope −φ; together with the central copy, which is kept, they are exactly the points of the canonical singular Fibonacci cut-and-project chain with window [0, W), W² = 64/81 + 128√5/405. The window has a mechanism: a ring of five plane copies overlaps itself exactly when it is closer to the axis than tc, tc² = 8/9 + 8√5/45. Since tc < W and no plane lattice point lies between them for |h| < 12.81, the two windows select the same points there. At h = 12.81 the core copy overlaps one other copy, which overlaps two more; on finite patches (integer program, float clash test) keeping the core copy is cheaper by one ring, here and at the next three such points. Minimal removal therefore thins the spine to a three-gap chain beyond |h| = 12.81, and staying pure Fibonacci costs at least one extra ring at each such point. Thus aperiodic order appears where periodic packings meet: the lattice of the chain is a plane section of one packing, its internal coordinate is the distance from the axis, and its window is fixed by the ring clash at tc. Irrational interfaces as one-dimensional quasicrystals, model sets near irrational lines and hard-core windows are known; we did not find this selection along a five-fold junction line, its self-clash window or the departure from it, and present them as a new exact instance of known ideas. Version 2 adds a discussion of the chain as a link between periodic and aperiodic order (abstract, §1, §9), checked for overstatement by an independent review; the results are unchanged from Version 1. Data, scripts and the exact check outputs are included; the full study is at https://github.com/DICTOR-Master/kaleidohedra (branch research, studies/2026-10-16-ocd-pyrochlore). Computation, exact arithmetic and checking with Claude (Anthropic); independent pre-publication checks by Claude Fable. Explore it in Kaleidohedra: https://kaleidohedra.dictospheres.com/?open=bodhiPyro:chorus