{
  "abstract": "The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction $m/k$ and an integer $t\\ge 2$. We show that the Farey sum (mediant) $(m_1+m_2)/(k_1+k_2)$ of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the $(m_1+m_2,k_1+k_2)$-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition -- no non-adjacent parameters ever splice -- the cut position being $k_1=m^{-1}\\bmod k$, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are $tk_1$ and $tk_2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01270",
  "author_total": 1,
  "authors": [
    {
      "name": "Mikihiro Fujiwara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/98p9m6q0qqaye.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01270",
  "fulltext_url": null,
  "id": "98p9m6q0qqaye",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01270",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7206189573",
  "pdf_url": "https://arxiv.org/pdf/2609.01270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/98p9m6q0qqaye.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.01270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge",
  "title_text": "Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/98p9m6q0qqaye",
  "venue": "arXiv",
  "versions": []
}