Capturing the In-betweens: a Study of Transitional Polyhedral Shells

Our Chain of Karma, Spheres of Dharma and Bodhi Seed are exact icosahedral dissections: each grows a large polyhedron from a small core by adding rings of congruent pieces. Between the familiar solids at the start and end of each step lie the in-between stages, the transitional shells. Taking each stage as one solid, with its pieces fused, the builds pass through celebrated polyhedra on the way: Icosahedron + 20 CLEO is Kepler's great stellated dodecahedron. Adding 30 AXE turns it into the excavated dodecahedron, a stellation of the icosahedron, with all 32 corners kept. Adding 12 caps closes the dodecahedron: the Dodecahedral Chain walks from the icosahedron to its dual through two star solids. The Bodhi Seed's centre + 60 Vajra (fine) is the rhombic hexecontahedron. The Spheres of Dharma's seventh stage is a convex solid with 102 faces and 80 corners: 12 regular pentagons, 30 rhombi and 60 triangles, its rhombi and triangles tangent to one sphere. Not found in the catalogues we searched, we name it the DICTO Truncated-Icosahedral Hecatodihedron. Every identification is checked against an independent construction to within 10⁻⁴. The paper describes all 21 piece types (with their names, including Stupa and Monkey Staff), shows 45 further convex stages, and gives cut-out nets so the in-betweens can be built by hand. The transitional shells have their own group, DICTO Transitional Shells, in the Polyhedraverse app. Files: the paper (PDF), cut-out nets for four shells (A4 PDF), and the data, scripts, figures and checks (zip). Computation and checking were done with Claude (Anthropic) in the open-source library krp-core. DICTO · dictospheres.com Version 2 (2026-10-14): the paper is written in the first person throughout; no other changes. Version 3 corrects Section 4.6: only the 30 rhombi and 60 triangles of the DICTO Truncated-Icosahedral Hecatodihedron touch one sphere (radius 1.5); its 12 pentagons lie at 1.4384, so it has no insphere. Version 2 said all 102 faces were tangent.