{
  "abstract": "In this paper, We prove that: (A) If $m\\ge 4,$ there exists a Platonic geodesic tree having $m$ edges (geodesic segments), which form $m$ platonic angles with measure $120^{o}$ and $(\\frac{m(m-1)}{2}-m)$ platonic angles with measure $90^{o}.$ (Existence of a Fermat Coordinate system) (B) If $m=2,$ there exists a Platonic geodesic tree with three vertices and 2 edges, which meet at $120^{\\circ}$ or $150^{\\circ}$ or $180^{\\circ}$ (Minimal spanning geodesic tree). (C) If $m=3,$ there exists a Platonic tree and the Steiner point is identical with the Fermat point, having three edges, which meet at $120^{\\circ}$ (Hexagonal coordinate system). (D) If $m$ is even and $m \\ge 4,$ there exists a Platonic geodesic tree having $m$ edges , which form platonic angles with only two distinct measures $90^{o}$ and $180^{\\circ}.$ (Existence of a Cartesian coordinate system). (E) If $m\\ge 4$, there does not exist a Platonic geodesic tree having $m$ edges , which form only platonic angles with measure $120^{o}$ (Non existence of a Fermat coordinate system). (F) If $m\\ge 3,$ there does not exist a Platonic geodesic tree having $m$ edges, which form platonic angles with only one measure $150^{o},$ or two distinct measures $120^{\\circ}$ and $150^{\\circ}$ or two distinct measures $90^{\\circ}$ and $150^{\\circ}$ (Non existence of a Fermat coordinate system). A classification of platonic geodesic trees with boundary vertex set $V$ and added vertex set $Q$ in the $K-$Space is given by conditions (A), (B), (C), (D), (E), (F), which characterize each vertex in $V\\cup Q.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13950",
  "author_total": 1,
  "authors": [
    {
      "name": "Anastasios N. Zachos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6thznvpwrk77b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13950",
  "fulltext_url": null,
  "id": "6thznvpwrk77b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13950",
  "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.MG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7221816618",
  "pdf_url": "https://arxiv.org/pdf/2610.13950",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6thznvpwrk77b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.13950",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Platonic geodesic trees in the $K-$Space",
  "title_text": "Platonic geodesic trees in the K−Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6thznvpwrk77b",
  "venue": "arXiv",
  "versions": []
}