{
  "schemaVersion": "1.0",
  "discourse": {
    "id": "euclid-elements-book-xiii",
    "name": "Euclid's Elements, Book XIII",
    "subject": "regular_solids",
    "variant": "classical",
    "description": "Regular solids: tetrahedron, octahedron, cube, icosahedron, dodecahedron. 18 propositions. Depends on Books I, IV, VI, X, XI. Source: David E. Joyce.",
    "structure": {
      "books": 13,
      "propositions": 18,
      "foundationTypes": [
        "foundation"
      ]
    }
  },
  "metadata": {
    "created": "2026-03-18",
    "lastUpdated": "2026-03-18",
    "version": "1.0.0",
    "license": "CC BY 4.0",
    "authors": [
      "Welz, G."
    ],
    "methodology": "Programming Framework",
    "citation": "Welz, G. (2026). Euclid's Elements Book XIII Dependency Graph. Programming Framework.",
    "keywords": [
      "Euclid",
      "Elements",
      "Book XIII",
      "regular solids",
      "Platonic",
      "tetrahedron",
      "octahedron",
      "cube",
      "icosahedron",
      "dodecahedron"
    ]
  },
  "sources": [
    {
      "id": "joyce",
      "type": "digital",
      "authors": "Joyce, David E.",
      "title": "Euclid's Elements, Book XIII",
      "year": "1996",
      "url": "https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html",
      "notes": "Clark University"
    }
  ],
  "nodes": [
    {
      "id": "BookI",
      "type": "foundation",
      "label": "Book I — Plane geometry",
      "shortLabel": "Book I",
      "short": "Foundation",
      "book": 1,
      "colorClass": "foundation"
    },
    {
      "id": "BookIV",
      "type": "foundation",
      "label": "Book IV — Inscribed figures",
      "shortLabel": "Book IV",
      "short": "Foundation",
      "book": 4,
      "colorClass": "foundation"
    },
    {
      "id": "BookVI",
      "type": "foundation",
      "label": "Book VI — Similar figures",
      "shortLabel": "Book VI",
      "short": "Foundation",
      "book": 6,
      "colorClass": "foundation"
    },
    {
      "id": "BookX",
      "type": "foundation",
      "label": "Book X — Incommensurables",
      "shortLabel": "Book X",
      "short": "Foundation",
      "book": 10,
      "colorClass": "foundation"
    },
    {
      "id": "BookXI",
      "type": "foundation",
      "label": "Book XI — Solid geometry",
      "shortLabel": "Book XI",
      "short": "Foundation",
      "book": 11,
      "colorClass": "foundation"
    },
    {
      "id": "Prop1",
      "type": "proposition",
      "label": "Line cut in extreme and mean ratio: square on greater plus half whole equals five times square on half",
      "shortLabel": "Prop. XIII.1",
      "short": "Extreme and mean: square on greater",
      "book": 13,
      "number": 1,
      "colorClass": "proposition"
    },
    {
      "id": "Prop2",
      "type": "proposition",
      "label": "If square on line five times square on segment: double segment cut in extreme and mean, greater is remainder",
      "shortLabel": "Prop. XIII.2",
      "short": "Square five times: extreme and mean",
      "book": 13,
      "number": 2,
      "colorClass": "proposition"
    },
    {
      "id": "Prop3",
      "type": "proposition",
      "label": "Line cut in extreme and mean: square on lesser + half greater equals five times square on half",
      "shortLabel": "Prop. XIII.3",
      "short": "Extreme and mean: sum of segments",
      "book": 13,
      "number": 3,
      "colorClass": "proposition"
    },
    {
      "id": "Prop4",
      "type": "proposition",
      "label": "Line cut in extreme and mean: sum of squares on whole and lesser triple square on greater",
      "shortLabel": "Prop. XIII.4",
      "short": "Extreme and mean: sum of squares",
      "book": 13,
      "number": 4,
      "colorClass": "proposition"
    },
    {
      "id": "Prop5",
      "type": "proposition",
      "label": "Line cut in extreme and mean, add greater: whole cut in extreme and mean, original is greater",
      "shortLabel": "Prop. XIII.5",
      "short": "Extreme and mean: add greater",
      "book": 13,
      "number": 5,
      "colorClass": "proposition"
    },
    {
      "id": "Prop6",
      "type": "proposition",
      "label": "Rational line cut in extreme and mean ratio: each segment is apotome",
      "shortLabel": "Prop. XIII.6",
      "short": "Rational cut: apotome",
      "book": 13,
      "number": 6,
      "colorClass": "proposition"
    },
    {
      "id": "Prop7",
      "type": "proposition",
      "label": "Equilateral pentagon: if three angles equal (order or not), pentagon equiangular",
      "shortLabel": "Prop. XIII.7",
      "short": "Equilateral pentagon: three angles",
      "book": 13,
      "number": 7,
      "colorClass": "proposition"
    },
    {
      "id": "Prop8",
      "type": "proposition",
      "label": "Equilateral equiangular pentagon: diagonals subtending two angles cut in extreme and mean ratio",
      "shortLabel": "Prop. XIII.8",
      "short": "Pentagon: diagonals in extreme and mean",
      "book": 13,
      "number": 8,
      "colorClass": "proposition"
    },
    {
      "id": "Prop9",
      "type": "proposition",
      "label": "Side of hexagon + decagon in same circle: cut in extreme and mean, greater is hexagon",
      "shortLabel": "Prop. XIII.9",
      "short": "Hexagon + decagon: extreme and mean",
      "book": 13,
      "number": 9,
      "colorClass": "proposition"
    },
    {
      "id": "Prop10",
      "type": "proposition",
      "label": "Equilateral pentagon in circle: square on side equals sum of squares on hexagon and decagon",
      "shortLabel": "Prop. XIII.10",
      "short": "Pentagon: square equals hexagon + decagon",
      "book": 13,
      "number": 10,
      "colorClass": "proposition"
    },
    {
      "id": "Prop11",
      "type": "proposition",
      "label": "Equilateral pentagon in circle with rational diameter: side is minor",
      "shortLabel": "Prop. XIII.11",
      "short": "Pentagon in rational circle: minor",
      "book": 13,
      "number": 11,
      "colorClass": "proposition"
    },
    {
      "id": "Prop12",
      "type": "proposition",
      "label": "Equilateral triangle in circle: square on side triple square on radius",
      "shortLabel": "Prop. XIII.12",
      "short": "Equilateral triangle: side triple radius",
      "book": 13,
      "number": 12,
      "colorClass": "proposition"
    },
    {
      "id": "Prop13",
      "type": "proposition",
      "label": "To construct pyramid (tetrahedron) in given sphere; diameter squared 1.5 times side squared",
      "shortLabel": "Prop. XIII.13",
      "short": "Construct tetrahedron in sphere",
      "book": 13,
      "number": 13,
      "colorClass": "proposition"
    },
    {
      "id": "Prop14",
      "type": "proposition",
      "label": "To construct octahedron in sphere; diameter squared double side squared",
      "shortLabel": "Prop. XIII.14",
      "short": "Construct octahedron in sphere",
      "book": 13,
      "number": 14,
      "colorClass": "proposition"
    },
    {
      "id": "Prop15",
      "type": "proposition",
      "label": "To construct cube in sphere; diameter squared triple side squared",
      "shortLabel": "Prop. XIII.15",
      "short": "Construct cube in sphere",
      "book": 13,
      "number": 15,
      "colorClass": "proposition"
    },
    {
      "id": "Prop16",
      "type": "proposition",
      "label": "To construct icosahedron in sphere; side is minor",
      "shortLabel": "Prop. XIII.16",
      "short": "Construct icosahedron in sphere",
      "book": 13,
      "number": 16,
      "colorClass": "proposition"
    },
    {
      "id": "Prop17",
      "type": "proposition",
      "label": "To construct dodecahedron in sphere; side is apotome",
      "shortLabel": "Prop. XIII.17",
      "short": "Construct dodecahedron in sphere",
      "book": 13,
      "number": 17,
      "colorClass": "proposition"
    },
    {
      "id": "Prop18",
      "type": "proposition",
      "label": "To set out sides of five figures and compare them; no other such figure exists",
      "shortLabel": "Prop. XIII.18",
      "short": "Compare five regular solids",
      "book": 13,
      "number": 18,
      "colorClass": "proposition"
    }
  ],
  "edges": [
    {
      "from": "BookI",
      "to": "Prop1"
    },
    {
      "from": "BookIV",
      "to": "Prop1"
    },
    {
      "from": "BookVI",
      "to": "Prop1"
    },
    {
      "from": "BookX",
      "to": "Prop1"
    },
    {
      "from": "BookXI",
      "to": "Prop1"
    },
    {
      "from": "BookI",
      "to": "Prop2"
    },
    {
      "from": "BookIV",
      "to": "Prop2"
    },
    {
      "from": "BookVI",
      "to": "Prop2"
    },
    {
      "from": "BookX",
      "to": "Prop2"
    },
    {
      "from": "BookXI",
      "to": "Prop2"
    },
    {
      "from": "BookI",
      "to": "Prop3"
    },
    {
      "from": "BookIV",
      "to": "Prop3"
    },
    {
      "from": "BookVI",
      "to": "Prop3"
    },
    {
      "from": "BookX",
      "to": "Prop3"
    },
    {
      "from": "BookXI",
      "to": "Prop3"
    },
    {
      "from": "BookI",
      "to": "Prop4"
    },
    {
      "from": "BookIV",
      "to": "Prop4"
    },
    {
      "from": "BookVI",
      "to": "Prop4"
    },
    {
      "from": "BookX",
      "to": "Prop4"
    },
    {
      "from": "BookXI",
      "to": "Prop4"
    },
    {
      "from": "BookI",
      "to": "Prop5"
    },
    {
      "from": "BookIV",
      "to": "Prop5"
    },
    {
      "from": "BookVI",
      "to": "Prop5"
    },
    {
      "from": "BookX",
      "to": "Prop5"
    },
    {
      "from": "BookXI",
      "to": "Prop5"
    },
    {
      "from": "BookI",
      "to": "Prop6"
    },
    {
      "from": "BookIV",
      "to": "Prop6"
    },
    {
      "from": "BookVI",
      "to": "Prop6"
    },
    {
      "from": "BookX",
      "to": "Prop6"
    },
    {
      "from": "BookXI",
      "to": "Prop6"
    },
    {
      "from": "BookI",
      "to": "Prop7"
    },
    {
      "from": "BookIV",
      "to": "Prop7"
    },
    {
      "from": "BookVI",
      "to": "Prop7"
    },
    {
      "from": "BookX",
      "to": "Prop7"
    },
    {
      "from": "BookXI",
      "to": "Prop7"
    },
    {
      "from": "BookI",
      "to": "Prop8"
    },
    {
      "from": "BookIV",
      "to": "Prop8"
    },
    {
      "from": "BookVI",
      "to": "Prop8"
    },
    {
      "from": "BookX",
      "to": "Prop8"
    },
    {
      "from": "BookXI",
      "to": "Prop8"
    },
    {
      "from": "BookI",
      "to": "Prop9"
    },
    {
      "from": "BookIV",
      "to": "Prop9"
    },
    {
      "from": "BookVI",
      "to": "Prop9"
    },
    {
      "from": "BookX",
      "to": "Prop9"
    },
    {
      "from": "BookXI",
      "to": "Prop9"
    },
    {
      "from": "BookI",
      "to": "Prop10"
    },
    {
      "from": "BookIV",
      "to": "Prop10"
    },
    {
      "from": "BookVI",
      "to": "Prop10"
    },
    {
      "from": "BookX",
      "to": "Prop10"
    },
    {
      "from": "BookXI",
      "to": "Prop10"
    },
    {
      "from": "BookI",
      "to": "Prop11"
    },
    {
      "from": "BookIV",
      "to": "Prop11"
    },
    {
      "from": "BookVI",
      "to": "Prop11"
    },
    {
      "from": "BookX",
      "to": "Prop11"
    },
    {
      "from": "BookXI",
      "to": "Prop11"
    },
    {
      "from": "BookI",
      "to": "Prop12"
    },
    {
      "from": "BookIV",
      "to": "Prop12"
    },
    {
      "from": "BookVI",
      "to": "Prop12"
    },
    {
      "from": "BookX",
      "to": "Prop12"
    },
    {
      "from": "BookXI",
      "to": "Prop12"
    },
    {
      "from": "BookI",
      "to": "Prop13"
    },
    {
      "from": "BookIV",
      "to": "Prop13"
    },
    {
      "from": "BookVI",
      "to": "Prop13"
    },
    {
      "from": "BookX",
      "to": "Prop13"
    },
    {
      "from": "BookXI",
      "to": "Prop13"
    },
    {
      "from": "BookI",
      "to": "Prop14"
    },
    {
      "from": "BookIV",
      "to": "Prop14"
    },
    {
      "from": "BookVI",
      "to": "Prop14"
    },
    {
      "from": "BookX",
      "to": "Prop14"
    },
    {
      "from": "BookXI",
      "to": "Prop14"
    },
    {
      "from": "BookI",
      "to": "Prop15"
    },
    {
      "from": "BookIV",
      "to": "Prop15"
    },
    {
      "from": "BookVI",
      "to": "Prop15"
    },
    {
      "from": "BookX",
      "to": "Prop15"
    },
    {
      "from": "BookXI",
      "to": "Prop15"
    },
    {
      "from": "BookI",
      "to": "Prop16"
    },
    {
      "from": "BookIV",
      "to": "Prop16"
    },
    {
      "from": "BookVI",
      "to": "Prop16"
    },
    {
      "from": "BookX",
      "to": "Prop16"
    },
    {
      "from": "BookXI",
      "to": "Prop16"
    },
    {
      "from": "BookI",
      "to": "Prop17"
    },
    {
      "from": "BookIV",
      "to": "Prop17"
    },
    {
      "from": "BookVI",
      "to": "Prop17"
    },
    {
      "from": "BookX",
      "to": "Prop17"
    },
    {
      "from": "BookXI",
      "to": "Prop17"
    },
    {
      "from": "BookI",
      "to": "Prop18"
    },
    {
      "from": "BookIV",
      "to": "Prop18"
    },
    {
      "from": "BookVI",
      "to": "Prop18"
    },
    {
      "from": "BookX",
      "to": "Prop18"
    },
    {
      "from": "BookXI",
      "to": "Prop18"
    }
  ],
  "colorScheme": {
    "foundation": {
      "fill": "#95a5a6",
      "stroke": "#7f8c8d"
    },
    "proposition": {
      "fill": "#1abc9c",
      "stroke": "#16a085"
    }
  }
}