Capturing the In-betweens: a Study of Transitional Polyhedral Shells
- 1Independent artist, Osaka, Japan
Abstract
DICTO's 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, every face tangent to one sphere. Not found in the catalogues searched, it is named 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 DICTO's 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
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BibTeX
@misc{dicto2026capturing,
title = {{Capturing the In-betweens: a Study of Transitional Polyhedral Shells}},
author = {DICTO},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23345502},
url = {https://doi.org/10.5281/zenodo.23345502},
note = {CC BY-SA},
}
RIS
TY - UNPB
TI - Capturing the In-betweens: a Study of Transitional Polyhedral Shells
AU - DICTO
PY - 2026
DA - 2026/10/14
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23345502
UR - https://doi.org/10.5281/zenodo.23345502
AB - DICTO's 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, every face tangent to one sphere. Not found in the catalogues searched, it is named 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 DICTO's 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
ER -