Inverted Shells: The Spheres of Dharma Turned Inside Out

Abstract

The Spheres of Dharma nest six icosahedral solids exactly, from an icosahedron out to the icosahedron φ³ larger; between the truncated icosahedron and the triacontahedron, the layer closes part-way into the DICTO Hecatodihedron. DICTO asked what the cycle looks like turned inside out. Turning each layer's pieces over does not work: no layer closes into a shell. Taking the six solids and the Hecatodihedron stage as seven stations and replacing each by its polar reciprocal reverses the nesting and gives a new exact cycle: dodecahedron ⊂ icosidodecahedron ⊂ Ogdoecontahedron ⊂ pentakis dodecahedron ⊂ icosahedron ⊂ rhombic triacontahedron ⊂ dodecahedron φ³ larger (volume ratio φ⁹). Its one new station, the dual of the Hecatodihedron, is the DICTO Ogdoecontahedron (DICTO DOI: 10.5281/zenodo.23404688). Cutting each layer by cones over the faces of its inner solid gives one kind of piece per layer (12, 12, 20, 60, 20, 30), every volume exact in ℚ(√5) with remainder 0. The cycle inherits two loose ends. Tightening them gives an evenly stepped nest, still one kind per layer and exact. Left loose, the large outer gap holds the great stellated dodecahedron at exactly one size: its 20 points on the outer dodecahedron's corners, its 12 valleys on the triacontahedron's tips, cutting the gap into 60 tetrahedra and 60 pentagonal pyramids. The inner gap admits no star and no locked dual pair among those tested, but the truncated icosidodecahedron pair fits in reverse order (as polarity demands), with play exactly √5·φ³/9. The golden rectangles that frame the whole are the outer dodecahedron's own small stellation, and the cycle repeats inward without end. We also record the checks this study forced on us, because three of our first tables were wrong. 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-polar-dharma, studies/2026-10-16-flip-test, studies/_lib). Computation, exact arithmetic and checking with Claude (Anthropic); an independent pre-publication check by Claude Fable. Explore DICTO's worlds in the apps: https://dictospheres.com Version 2. Version 2 corrects one sentence: the Ogdoecontahedron is not called a new solid; it is the canonical pentakis dodecahedron cut at the hexagon planes, an unnamed solid derived from a known one (see the Hecatodihedron paper, Version 2, DICTO DOI: 10.5281/zenodo.23360857).