{
  "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",
  "abstract_shown": true,
  "affiliations": [
    "Independent artist, Osaka, Japan"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "DICTO"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3r9039fpnsyp1.bib",
  "cited_by": 7,
  "doi": "10.5281/zenodo.23345502",
  "fulltext_url": null,
  "id": "3r9039fpnsyp1",
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  "landing_url": "https://doi.org/10.5281/zenodo.23345502",
  "language": "en",
  "licence": "CC-BY-SA",
  "licence_label": "CC BY-SA",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221832356",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/3r9039fpnsyp1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7221832356",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.23345502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Advanced Combinatorial Mathematics",
      "subfield": "combinatorics"
    }
  ],
  "title": "Capturing the In-betweens: a Study of Transitional Polyhedral Shells",
  "title_text": "Capturing the In-betweens: a Study of Transitional Polyhedral Shells",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/3r9039fpnsyp1",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}