Double Vision – Both Cycles in One Space

Abstract

The Spheres of Dharma nest six icosahedral solids, with the Hecatodihedron as an in-between stage; their polar reciprocal is a second exact cycle in which every nesting is reversed (Inverted Shells). We put both cycles in one space. The sphere of reciprocation is then the one free choice, and polarity with the cycle's φ³ repeat gives mirror spheres every φ3/2 in radius. At ρ² = φ²/4 the two cycles share the icosahedron and dodecahedron; at ρ² = 3φ²/4 they share the icosidodecahedron and triacontahedron; no sphere gives both, because the two settings differ by a factor 3, which is not a power of φ, and the loose ends of the cycle cannot be bent far enough to close the gap. Cutting every piece of one cycle by every piece of the other gives one solid of convex pieces, exact in ℚ(√5) with remainder 0 in every piece and layer: per turn, 536 pieces of 12 kinds in the first setting and 502 pieces of 11 kinds in the second. The setting moves where the two cycles cut each other: the first cuts the outer roofs hardest and leaves the dodecahedron columns whole; the second leaves the roofs, pyramids and tetrahedra whole and cuts the columns. The innermost layer is cut in both. Data, scripts and exact check outputs are included (see README.md); the full studies are at https://github.com/DICTOR-Master/kaleidohedra (branch research: studies/2026-10-16-interlock, studies/2026-10-16-polar-dharma, studies/_lib). Computation, exact arithmetic and checking with Claude (Anthropic); two independent pre-publication reviews by Claude Fable. Explore DICTO's worlds in the apps: https://dictospheres.com Version 2. Version 2 changes one phrase (the Ogdoecontahedron as the cycle's one non-classical solid, not "new"), following the Hecatodihedron paper, Version 2.