The Chain of Karma: the icosahedron and the dodecahedron as endless shells of each other

We show that the regular icosahedron and the regular dodecahedron can each be filled exactly as a shell around the other, and that the two shells alternate without end. A dodecahedron of edge 1 is an icosahedron of edge 1/φ inside a shell of 12 pentagonal pyramids, 30 AXE and 20 CLEO; an icosahedron of edge φ² (the Lotus seed) is that dodecahedron inside a shell of 12 J11 (each an icosahedron with one pyramid cap removed), 30 AXE and 20 FUJI. Thus the nesting I(1/φ) ⊂ D(1) ⊂ I(φ²) ⊂ D(φ³) ⊂ … grows by φ³ per cycle, and each shell has one piece for every vertex, edge and face of the dual pair (12, 30, 20). The pieces are exact, at edge 1: AXE is a tetrahedron with five edges 1 and one 1/φ (two equilateral faces, two golden triangles 1, 1, 1/φ); CLEO is a triangular pyramid on an equilateral base of side 1/φ with side edges 1; FUJI is a pyramid of height 1/√3 on a flat-ended (truncated) triangle with sides 1 and 1/φ alternating and all angles 120°, with side edges 1, three equilateral and three golden faces, and a cube corner at its apex; the pentagonal pyramid has all edges 1. Every dihedral angle of the kit is, modulo 180°, an integer combination of two angles, α = arccos(√5/3) = 41.81° and δ = arctan(3 − √5) = 37.38°: the icosahedron's own is 180° − α = 138.19°, and FUJI's faces meet its base at α on the long sides and δ on the short. Because every icosahedron inside is again whole, each can be replaced by its own Lotus seed: the Chain of Karma, built and checked to 20,497 pieces. Only the icosahedron is ever subdivided: the gap pieces stay whole (they cannot be cut into the kit at one size, by the Dehn invariant) and occur at every scale φᵏ. A second exact cycle, through the icosidodecahedron, the fullerene cage and the rhombic triacontahedron, shares the dodecahedron with the Lotus seed and is described separately (Spheres of Dharma). The nesting of the dual pair at golden ratios goes back to Pacioli and Kepler and appears at every φ-power in Gray (2002); a different dodecahedral growth, with whole dodecahedra inside, is due to Al Raisi, Koca, Koca and Koc (2026). The exact shells and their endless alternation are new. Pieces and faces (edge 1). Icosahedron: 20 equilateral faces, dihedral angle 138.19° = 180° − arccos(√5/3). AXE: tetrahedron, five edges 1 and one 1/φ; 2 equilateral faces, 2 golden triangles (1, 1, 1/φ) (Mosseri–Sadoc t6 at 1/φ). CLEO: triangular pyramid, base equilateral 1/φ, side edges 1; 1 equilateral face, 3 golden triangles (Mosseri–Sadoc t4 at 1/φ). FUJI: pyramid of height 1/√3 on a flat-ended triangle (sides 1 and 1/φ alternating, angles 120°), side edges 1; 3 equilateral + 3 golden faces; apex a cube corner; faces meet the base at 41.81° (long sides) and 37.38° (short sides). Pentagonal pyramid (J2): all edges 1. J11: an icosahedron with one J2 cap removed. Every dihedral angle, modulo 180°, is a whole-number combination of α = arccos(√5/3) = 41.810° and δ = arctan(3 − √5) = 37.377°. The Lotus seed, the Chain of Karma, the pyramid tip + mirror rule and the names Lotus seed, AXE, FUJI and CLEO are DICTO's. Computation and checking were done with Claude (Anthropic) in the open-source library krp-core. Files: the paper (PDF) and its data, scripts, logs and renders (zip; also at https://github.com/DICTOR-Master/kaleidohedra). Version 2 (2026-10-12): the Chain is stated as an iterated (hierarchical) dissection in which only the icosahedron is subdivided; the earlier pyramid growth rule, whose volume and Dehn identities hold for any solid, and its figure have been removed.