Four-Dimensional Solids Whose Every Cell Is a Golden Cut: Dual Pairs through the Dimensions

Put a solid and its dual side by side in a space of twice the dimension and cut that space along a tilted flat. What you see in the cut is the solid trimmed by its dual, a solid with its corners cut, and the depth of the cut is set by the tilt. As the tilt turns, the cut passes through the whole family, from the dual to the solid itself, and every golden cut is a station on the way. Do the same with the four-dimensional cube and its dual, the 16-cell, and two of the cuts are four-dimensional solids whose every cell is a golden cut: Icosaxes beside the Lesser Golden Cut Tetrahedron, and Unborn Axes beside the Greater. The dodecahedron's pair carries DICTO's Ogdoecontahedron and the whole polar Dharma cycle as stations of the same sweep, and new cycles close by the golden ratio. DICTO proposed following each Platonic solid with its dual through the dimensions. For a regular polyhedron P and its polar P*, the flats y = x/μ of the product P × P* cut out S(μ) = P ∩ μP*, and we prove that every corner cut of P or P* is such a section, so all ten golden cuts of #88 lie on the three sweeps D·I, C·O and T·T. Our main theorem is four-dimensional: in the tesseract K trimmed by its polar 16-cell, K ∩ μK*, every cell is a golden cut exactly when μ = φ² or μ = 2 + 1/φ²; the 24 cells are then 8 Icosaxes and 16 Lesser Golden Cut Tetrahedra, or 8 Unborn Axes and 16 Greater Golden Cut Tetrahedra, because the golden heights of the cube cells and of the tetrahedral cells coincide; these two are DICTO All-Golden Polytopes, four-dimensional solids all of whose cells are golden cuts. In the 600-cell × 120-cell family, by a linear cell rule checked exactly at three points, the bitruncated stage gives DICTO All-Golden Polytopes at μ = 2 − √5/2 and (19 + 9√5)/44. In three dimensions the D·I sweep lies entirely in Q(√5) but no lattice carries its symmetry; the tesseract's slices along a long diagonal are the T·T sweep in a Möbius coordinate; and the DICTO Ogdoecontahedron is exactly OG = H(μTI) ∩ φ²D*, the hull of the dodecahedron and the scaled icosahedron at the truncated-icosahedron level cut by icosahedron planes at the rectification level, so that the four middle stations of the polar Dharma cycle are sections and hulls of the six-dimensional product, unscaled and touching; at the golden levels √5 and (13 + 6√5)/11 the same construction gives DICTO's Lesser and Greater Golden OG, with OG's combinatorics. Tight cycles that close at a power of φ, the DICTO Golden Tight Cycles, occur at φ for D·I and C·O and at φ⁶ for T·T; slices of the 24-cell honeycomb give Icosaxes + Unborn Axes at 1/φ² and Kelvin's truncated octahedra at 1/2. On the cube side √2 selects the silver ratio throughout. All results are exact in Q(√2, √5) except where stated. The proposal to follow the Platonic solids and their duals through the dimensions, the follow-up proposals (OG and the dodecahedron–icosahedron pair, new golden cycles, hyper-lattice cycles, cycles of the rhombic dodecahedron and truncated octahedron, the golden cut rhombic dodecahedra as four-dimensional slices), the golden cuts, OG, the polar Dharma cycle the names of the solids, among them Icosaxes, Unborn Axes and the Lesser and Greater Golden OG, and the names DICTO All-Golden Polytopes and DICTO Golden Tight Cycles are DICTO's. Computation, proofs, exact checks and drafting with Claude (Anthropic).