{
  "abstract": "FINDING: The silver ratio (1+√2) emerges as the defining constant of octagonal symmetry, linking metallic means to quasicrystalline order and the Ammann-Beenker tiling. | MATH: Silver ratio δ_S = 1+√2 ≈ 2.41421356; satisfies δ_S = 2 + 1/δ_S; related to golden ratio φ = (1+√5)/2 via metallic means family: δ_n = (n+√(n²+4))/2 (n=1→φ, n=2→δ_S). In octagon: diagonal/side = δ_S; cot(π/8) = 1+√2; tan(π/8) = √2−1 = 1/δ_S. | CONNECTION: Direct crystallographic link — octagonal quasicrystals (8-fold rotational symmetry, forbidden in periodic crystals) are generated by the Ammann-Beenker tiling, whose inflation factor is exactly δ_S. This is the 2D analogue of the golden ratio's role in icosahedral (5-fold) quasicrystals. The ratio 1/δ_S ≈ 0.4142, and δ_S−2 ≈ 0.4142; note δ_S/2 ≈ 1.2071, and 2/δ_S ≈ 0.8284 — none of the classical 0.382/0.618/0.786 appear here; this is a *distinct* harmonic family. | DEPTH: 8 — The silver ratio is not a mere curiosity: it is the unique algebraic integer (degree 2\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctxp8z6t60fbk.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23465036",
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  "id": "ctxp8z6t60fbk",
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  "landing_url": "https://zenodo.org/records/23465036",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctxp8z6t60fbk.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:49Z",
      "native_id": "23465036",
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      "url": "https://zenodo.org/records/23465036"
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  "subjects": [],
  "title": "Silver Ratio Defines Octagonal Symmetry and Ammann-Beenker Quasicrystals — E8 Intelligence Research",
  "title_text": "Silver Ratio Defines Octagonal Symmetry and Ammann-Beenker Quasicrystals — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-02T12:21:49Z",
  "url": "https://research.draconitic.com/paper/ctxp8z6t60fbk",
  "venue": "Zenodo",
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}