Spheres of Dharma: a second endless golden cycle of the icosahedral solids, through the icosidodecahedron, the fullerene cage and the rhombic triacontahedron
- 1Independent artist, Osaka, Japan
Abstract
We describe an exact, endlessly repeating nesting of six icosahedral solids (the Ritsudō cycle), each layer filled by a few repeated pieces. With the dodecahedron at edge 1: an icosahedron of edge 1/φ, an icosidodecahedron of edge φ/2 (around it 12 pentagonal pyramids and 20 twisted triangular frusta), the dodecahedron (20 triangular pyramids), a truncated icosahedron of edge 1/φ, the fullerene cage (12 twisted pentagonal frusta and 30 Mosseri–Sadoc tetrahedra t1 at scale 1/φ), a rhombic triacontahedron of edge ½√(45 − 18√5) = 1.089814 (12 pentagonal and 20 hexagonal caps and 60 tetrahedral wedges), and an icosahedron of edge φ² (30 edge and 20 face pieces). The outer icosahedron is φ³ times the inner one, so the cycle repeats without end in both directions; one cycle has 237 pieces. All pieces of a cycle are checked to be free of overlap, with total volume exact. The construction rests on three exact relations: the icosahedron and the dodecahedron share the icosidodecahedron as their core, differing only in 12 pentagonal or 20 triangular corner pyramids; the fullerene cage wraps the dodecahedron at exactly one golden step, the dodecahedron's corners on the cage's hexagon centres; and the rhombic triacontahedron is the cage, with its 60 corners on the triacontahedron's faces, plus caps and wedges. The cycle shares its dodecahedron and its two icosahedra with the Lotus seed of the Chain of Karma, and each half of either structure fills its side of that dodecahedron exactly, so the two can be exchanged half by half at every scale. The cycle grows concentrically, one icosahedron per cycle (Nenrin growth), while the Lotus seed branches, 13 icosahedra per φ² step; the Lotus seed's 12 corner icosahedra straddle the shared dodecahedron, so they stay whole only when both halves are the Lotus seed's. The five cubes of the dodecahedron lie in the network's seams, and the five octahedra on the icosidodecahedron hold one icosahedron of edge 1 in Pacioli's golden position. Its outer shell, the rhombic triacontahedron, is the only other form found to branch like the Lotus seed (the Bodhi seed, described separately). The shells are the family of the Tsai-type cluster of icosahedral quasicrystals, here solid and exactly filled. Checks. All 237 pieces of one cycle are pairwise free of overlap (separating axes), lie inside the outer icosahedron, and their total volume equals I(φ²) to 1e-12. The triacontahedron's edge is exactly ½√(45 − 18√5) = 1.0898138. Terms: Ritsudō (律動, rhythm) for the six-shell cycle; Nenrin (年輪, tree rings) for its concentric growth. The icosidodecahedron as the twin form, the dual triangular structures, the triangles of rhombi, the fullerene transition, the last missing layer and all names are DICTO's. Computation and checking were done with Claude (Anthropic). Files: the paper (PDF) and its data, scripts, logs and renders (zip; also at https://github.com/DICTOR-Master/kaleidohedra). Version 2 (2026-10-14): the paper is written in the first person throughout; no other changes. More of DICTO's geometry: https://dictospheres.com View in the app: https://kaleidohedra.dictospheres.com/?open=chains:dharma
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BibTeX
@misc{dicto2026spheres,
title = {{Spheres of Dharma: a second endless golden cycle of the icosahedral solids, through the icosidodecahedron, the fullerene cage and the rhombic triacontahedron}},
author = {DICTO},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23310833},
url = {https://doi.org/10.5281/zenodo.23310833},
note = {CC BY-SA},
}
RIS
TY - UNPB
TI - Spheres of Dharma: a second endless golden cycle of the icosahedral solids, through the icosidodecahedron, the fullerene cage and the rhombic triacontahedron
AU - DICTO
PY - 2026
DA - 2026/10/12
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23310833
UR - https://doi.org/10.5281/zenodo.23310833
AB - We describe an exact, endlessly repeating nesting of six icosahedral solids (the Ritsudō cycle), each layer filled by a few repeated pieces. With the dodecahedron at edge 1: an icosahedron of edge 1/φ, an icosidodecahedron of edge φ/2 (around it 12 pentagonal pyramids and 20 twisted triangular frusta), the dodecahedron (20 triangular pyramids), a truncated icosahedron of edge 1/φ, the fullerene cage (12 twisted pentagonal frusta and 30 Mosseri–Sadoc tetrahedra t1 at scale 1/φ), a rhombic triacontahedron of edge ½√(45 − 18√5) = 1.089814 (12 pentagonal and 20 hexagonal caps and 60 tetrahedral wedges), and an icosahedron of edge φ² (30 edge and 20 face pieces). The outer icosahedron is φ³ times the inner one, so the cycle repeats without end in both directions; one cycle has 237 pieces. All pieces of a cycle are checked to be free of overlap, with total volume exact. The construction rests on three exact relations: the icosahedron and the dodecahedron share the icosidodecahedron as their core, differing only in 12 pentagonal or 20 triangular corner pyramids; the fullerene cage wraps the dodecahedron at exactly one golden step, the dodecahedron's corners on the cage's hexagon centres; and the rhombic triacontahedron is the cage, with its 60 corners on the triacontahedron's faces, plus caps and wedges. The cycle shares its dodecahedron and its two icosahedra with the Lotus seed of the Chain of Karma, and each half of either structure fills its side of that dodecahedron exactly, so the two can be exchanged half by half at every scale. The cycle grows concentrically, one icosahedron per cycle (Nenrin growth), while the Lotus seed branches, 13 icosahedra per φ² step; the Lotus seed's 12 corner icosahedra straddle the shared dodecahedron, so they stay whole only when both halves are the Lotus seed's. The five cubes of the dodecahedron lie in the network's seams, and the five octahedra on the icosidodecahedron hold one icosahedron of edge 1 in Pacioli's golden position. Its outer shell, the rhombic triacontahedron, is the only other form found to branch like the Lotus seed (the Bodhi seed, described separately). The shells are the family of the Tsai-type cluster of icosahedral quasicrystals, here solid and exactly filled. Checks. All 237 pieces of one cycle are pairwise free of overlap (separating axes), lie inside the outer icosahedron, and their total volume equals I(φ²) to 1e-12. The triacontahedron's edge is exactly ½√(45 − 18√5) = 1.0898138. Terms: Ritsudō (律動, rhythm) for the six-shell cycle; Nenrin (年輪, tree rings) for its concentric growth. The icosidodecahedron as the twin form, the dual triangular structures, the triangles of rhombi, the fullerene transition, the last missing layer and all names are DICTO's. Computation and checking were done with Claude (Anthropic). Files: the paper (PDF) and its data, scripts, logs and renders (zip; also at https://github.com/DICTOR-Master/kaleidohedra). Version 2 (2026-10-14): the paper is written in the first person throughout; no other changes. More of DICTO's geometry: https://dictospheres.com View in the app: https://kaleidohedra.dictospheres.com/?open=chains:dharma
ER -