The Golden Split: Icosahedra, Unborns and AXE Become Two Convex Space-Fillers

In short. Icosahedra cannot fill space on their own. Set them in a cubic grid, edge to edge, and two kinds of gap are left: a squeezed icosahedron (the Unborn) in the middle of each cube, and thin tetrahedra (the AXE) between them. Cut each AXE into four pyramids from the point 1/φ of the way along its axis and give each pyramid to the piece it touches: the icosahedra and the Unborns become two convex truncated octahedra that fill space. Main theorem. In the simple cubic lattice of regular icosahedra of edge 1 and spacing φ, the golden split of every AXE makes the icosahedron with its 12 pyramids, the Icosaxes, and the Unborn with its 12, the Unborn Axes, convex truncated octahedra with full octahedral symmetry, the same equiangular hexagon (sides 1/√2 and 1/(φ√2)), volumes 1 + 2√5/3 and 1 + √5/3, filling space face to face. Theorem 2: for any cut points inside the AXEs, both padded pieces are convex only if every AXE is cut at its golden point. Theorem 3: in the Lotus Jewel network the same cut gives the keyed pair Lotus Jewel Axes (volume √5) and Dodecaxes (volume 2), and every keying fills space. Theorem 4 (factor x): the pair belongs to a one-parameter family of mixed space-fillers, the truncated octahedra T(x) and T(1/x), from Kelvin's cell (x = 1) to the golden pair (x = τ). A remark places the octet (tetrahedra and octahedra) in a similar family between the rhombic dodecahedron and the cube. Exact in ℚ(√5). The two smooth shapes are known (Koca et al. 2015, x = τ and x = 1/τ); the route from the icosahedron lattice, the golden split and its uniqueness, and the keyed pair were not found in the sources checked. Paper #48 continues DICTO's search for mixed Platonic space-fillers (#47), answering his conjecture that a more elegant and simpler filling was within reach. The ideas and the names Unborn, AXE, Lotus Jewel, Icosaxes, Unborn Axes, Lotus Jewel Axes and Dodecaxes are DICTO's; computation, exact proofs and drafting with Claude (Anthropic). More at dictospheres.com.