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              9,
              8
            ],
            [
              5,
              20,
              24,
              9
            ],
            [
              4,
              8,
              23,
              19
            ],
            [
              21,
              25,
              26,
              22
            ],
            [
              6,
              7,
              11,
              10
            ],
            [
              10,
              11,
              26,
              25
            ],
            [
              7,
              22,
              26,
              11
            ],
            [
              6,
              10,
              25,
              21
            ],
            [
              23,
              27,
              28,
              24
            ],
            [
              8,
              9,
              13,
              12
            ],
            [
              12,
              13,
              28,
              27
            ],
            [
              8,
              12,
              27,
              23
            ],
            [
              9,
              10,
              14,
              13
            ],
            [
              13,
              14,
              29,
              28
            ],
            [
              9,
              24,
              25,
              10
            ],
            [
              10,
              25,
              29,
              14
            ],
            [
              31,
              33,
              34,
              32
            ],
            [
              20,
              21,
              25,
              24
            ],
            [
              20,
              31,
              32,
              21
            ],
            [
              21,
              32,
              34,
              25
            ],
            [
              20,
              24,
              33,
              31
            ],
            [
              33,
              36,
              37,
              34
            ],
            [
              28,
              29,
              37,
              36
            ],
            [
              24,
              28,
              36,
              33
            ],
            [
              34,
              37,
              38,
              35
            ],
            [
              25,
              26,
              30,
              29
            ],
            [
              29,
              30,
              38,
              37
            ],
            [
              25,
              34,
              35,
              26
            ],
            [
              26,
              35,
              38,
              30
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          1
        ],
        [
          0,
          0,
          2
        ],
        [
          0,
          1,
          0
        ],
        [
          0,
          1,
          2
        ],
        [
          0,
          2,
          0
        ],
        [
          0,
          2,
          1
        ],
        [
          1,
          1,
          1
        ],
        [
          1,
          2,
          1
        ],
        [
          1,
          2,
          2
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "nonacube_cross",
      "name": "Nonacube cross · two-unit arms",
      "group": "Research",
      "note": "Nine cubes in a planar cross. Exactly one complete lattice corona is possible: a first corona is verified and a checked SAT proof rules out two. Integer translations, cubic rotations, full face/edge/vertex surrounds; no topological-ball requirement. Watch the complete recorded search in the evidence viewer.",
      "evidence": "docs/projects/nonacube-search-tree.html",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "Nonacube cross · two-unit arms",
          "vertices": [
            [
              -2,
              0,
              0
            ],
            [
              -2,
              0,
              1
            ],
            [
              -2,
              1,
              0
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            [
              -2,
              1,
              1
            ],
            [
              -1,
              0,
              0
            ],
            [
              -1,
              0,
              1
            ],
            [
              -1,
              1,
              0
            ],
            [
              -1,
              1,
              1
            ],
            [
              0,
              -2,
              0
            ],
            [
              0,
              -2,
              1
            ],
            [
              0,
              -1,
              0
            ],
            [
              0,
              -1,
              1
            ],
            [
              0,
              0,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              1,
              1
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              2,
              1
            ],
            [
              0,
              3,
              0
            ],
            [
              0,
              3,
              1
            ],
            [
              1,
              -2,
              0
            ],
            [
              1,
              -2,
              1
            ],
            [
              1,
              -1,
              0
            ],
            [
              1,
              -1,
              1
            ],
            [
              1,
              0,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              1,
              1
            ],
            [
              1,
              2,
              0
            ],
            [
              1,
              2,
              1
            ],
            [
              1,
              3,
              0
            ],
            [
              1,
              3,
              1
            ],
            [
              2,
              0,
              0
            ],
            [
              2,
              0,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              1,
              1
            ],
            [
              3,
              0,
              0
            ],
            [
              3,
              0,
              1
            ],
            [
              3,
              1,
              0
            ],
            [
              3,
              1,
              1
            ]
          ],
          "faces": [
            [
              13,
              25,
              27,
              15
            ],
            [
              12,
              14,
              26,
              24
            ],
            [
              0,
              1,
              3,
              2
            ],
            [
              2,
              3,
              7,
              6
            ],
            [
              0,
              4,
              5,
              1
            ],
            [
              1,
              5,
              7,
              3
            ],
            [
              0,
              2,
              6,
              4
            ],
            [
              6,
              7,
              15,
              14
            ],
            [
              4,
              12,
              13,
              5
            ],
            [
              5,
              13,
              15,
              7
            ],
            [
              4,
              6,
              14,
              12
            ],
            [
              26,
              27,
              35,
              34
            ],
            [
              24,
              32,
              33,
              25
            ],
            [
              25,
              33,
              35,
              27
            ],
            [
              24,
              26,
              34,
              32
            ],
            [
              36,
              38,
              39,
              37
            ],
            [
              34,
              35,
              39,
              38
            ],
            [
              32,
              36,
              37,
              33
            ],
            [
              33,
              37,
              39,
              35
            ],
            [
              32,
              34,
              38,
              36
            ],
            [
              20,
              22,
              23,
              21
            ],
            [
              8,
              9,
              11,
              10
            ],
            [
              8,
              20,
              21,
              9
            ],
            [
              9,
              21,
              23,
              11
            ],
            [
              8,
              10,
              22,
              20
            ],
            [
              22,
              24,
              25,
              23
            ],
            [
              10,
              11,
              13,
              12
            ],
            [
              11,
              23,
              25,
              13
            ],
            [
              10,
              12,
              24,
              22
            ],
            [
              26,
              28,
              29,
              27
            ],
            [
              14,
              15,
              17,
              16
            ],
            [
              15,
              27,
              29,
              17
            ],
            [
              14,
              16,
              28,
              26
            ],
            [
              28,
              30,
              31,
              29
            ],
            [
              16,
              17,
              19,
              18
            ],
            [
              18,
              19,
              31,
              30
            ],
            [
              17,
              29,
              31,
              19
            ],
            [
              16,
              18,
              30,
              28
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          -2,
          0,
          0
        ],
        [
          -1,
          0,
          0
        ],
        [
          0,
          -2,
          0
        ],
        [
          0,
          -1,
          0
        ],
        [
          0,
          0,
          0
        ],
        [
          0,
          1,
          0
        ],
        [
          0,
          2,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          2,
          0,
          0
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "chair44_relief",
      "name": "Chair44 · centered relief",
      "group": "Research",
      "note": "Exact chamber occupancy and rational cube volumes. Centered apexes and joined dents; no angle rounding. Translations by whole original small cubes. Aperiodicity of this geometric version is unproved.",
      "evidence": "docs/projects/chair44-lattice-export.md",
      "categories": [],
      "components": []
    },
    {
      "id": "a2_hat_prism",
      "name": "Hat prism",
      "group": "Research",
      "note": "Single slab on the index-3 A₂ sublattice. Cap weights are doubled: interior t = 1, rim values are planar angles. Six in-plane rotations; reflected tiles are optional.",
      "categories": [
        "A2 Layered Solids"
      ],
      "components": [
        {
          "name": "A2 Hat Prism",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              0,
              -1
            ],
            [
              1,
              1,
              -2
            ],
            [
              3,
              0,
              -3
            ],
            [
              4,
              1,
              -5
            ],
            [
              3,
              2,
              -5
            ],
            [
              3,
              3,
              -6
            ],
            [
              1,
              4,
              -5
            ],
            [
              0,
              6,
              -6
            ],
            [
              -1,
              6,
              -5
            ],
            [
              -1,
              5,
              -4
            ],
            [
              -1,
              4,
              -3
            ],
            [
              0,
              3,
              -3
            ],
            [
              -1,
              2,
              -1
            ],
            [
              1,
              1,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              2,
              -1
            ],
            [
              4,
              1,
              -2
            ],
            [
              5,
              2,
              -4
            ],
            [
              4,
              3,
              -4
            ],
            [
              4,
              4,
              -5
            ],
            [
              2,
              5,
              -4
            ],
            [
              1,
              7,
              -5
            ],
            [
              0,
              7,
              -4
            ],
            [
              0,
              6,
              -3
            ],
            [
              0,
              5,
              -2
            ],
            [
              1,
              4,
              -2
            ],
            [
              0,
              3,
              0
            ]
          ],
          "faces": [
            [
              13,
              12,
              11,
              10,
              9,
              8,
              7,
              6,
              5,
              4,
              3,
              2,
              1,
              0
            ],
            [
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27
            ],
            [
              0,
              1,
              15,
              14
            ],
            [
              1,
              2,
              16,
              15
            ],
            [
              2,
              3,
              17,
              16
            ],
            [
              3,
              4,
              18,
              17
            ],
            [
              4,
              5,
              19,
              18
            ],
            [
              5,
              6,
              20,
              19
            ],
            [
              6,
              7,
              21,
              20
            ],
            [
              7,
              8,
              22,
              21
            ],
            [
              8,
              9,
              23,
              22
            ],
            [
              9,
              10,
              24,
              23
            ],
            [
              10,
              11,
              25,
              24
            ],
            [
              11,
              12,
              26,
              25
            ],
            [
              12,
              13,
              27,
              26
            ],
            [
              13,
              0,
              14,
              27
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "a2_turtle_prism",
      "name": "Turtle prism",
      "group": "Research",
      "note": "Single slab using the Turtle demo’s index-3 point domain. Each cap retains 11 boundary vertices and 2 interior points. Interior t = 1; rim values are planar angles.",
      "categories": [
        "A2 Layered Solids"
      ],
      "components": [
        {
          "name": "A2 Turtle Prism",
          "vertices": [
            [
              3,
              -2,
              -1
            ],
            [
              2,
              0,
              -2
            ],
            [
              0,
              1,
              -1
            ],
            [
              0,
              2,
              -2
            ],
            [
              -1,
              3,
              -2
            ],
            [
              -2,
              2,
              0
            ],
            [
              -1,
              0,
              1
            ],
            [
              -2,
              0,
              2
            ],
            [
              -2,
              -1,
              3
            ],
            [
              0,
              -2,
              2
            ],
            [
              1,
              -4,
              3
            ],
            [
              2,
              -4,
              2
            ],
            [
              3,
              -5,
              2
            ],
            [
              4,
              -4,
              0
            ],
            [
              4,
              -1,
              0
            ],
            [
              3,
              1,
              -1
            ],
            [
              1,
              2,
              0
            ],
            [
              1,
              3,
              -1
            ],
            [
              0,
              4,
              -1
            ],
            [
              -1,
              3,
              1
            ],
            [
              0,
              1,
              2
            ],
            [
              -1,
              1,
              3
            ],
            [
              -1,
              0,
              4
            ],
            [
              1,
              -1,
              3
            ],
            [
              2,
              -3,
              4
            ],
            [
              3,
              -3,
              3
            ],
            [
              4,
              -4,
              3
            ],
            [
              5,
              -3,
              1
            ]
          ],
          "faces": [
            [
              13,
              12,
              11,
              10,
              9,
              8,
              7,
              6,
              5,
              4,
              3,
              2,
              1,
              0
            ],
            [
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27
            ],
            [
              0,
              1,
              15,
              14
            ],
            [
              1,
              2,
              16,
              15
            ],
            [
              2,
              3,
              17,
              16
            ],
            [
              3,
              4,
              18,
              17
            ],
            [
              4,
              5,
              19,
              18
            ],
            [
              5,
              6,
              20,
              19
            ],
            [
              6,
              7,
              21,
              20
            ],
            [
              7,
              8,
              22,
              21
            ],
            [
              8,
              9,
              23,
              22
            ],
            [
              9,
              10,
              24,
              23
            ],
            [
              10,
              11,
              25,
              24
            ],
            [
              11,
              12,
              26,
              25
            ],
            [
              12,
              13,
              27,
              26
            ],
            [
              13,
              0,
              14,
              27
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "buckled_ring",
      "name": "Buckled ring",
      "group": "Periodic stress control",
      "note": "The reference screen found small periodic and isohedral point certificates. Retained as a nonconvex search stress control, not a nonperiodic example.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "BuckledRing",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              2
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              1,
              1
            ],
            [
              0,
              1,
              2
            ],
            [
              0,
              2,
              1
            ],
            [
              0,
              2,
              2
            ],
            [
              1,
              0,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              1,
              0,
              2
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              1,
              1
            ],
            [
              1,
              1,
              2
            ],
            [
              1,
              2,
              0
            ],
            [
              1,
              2,
              1
            ],
            [
              1,
              2,
              2
            ],
            [
              2,
              0,
              0
            ],
            [
              2,
              0,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              1,
              1
            ],
            [
              2,
              1,
              2
            ],
            [
              2,
              2,
              0
            ],
            [
              2,
              2,
              1
            ],
            [
              2,
              2,
              2
            ]
          ],
          "faces": [
            [
              0,
              1,
              4,
              3
            ],
            [
              3,
              4,
              12,
              11
            ],
            [
              0,
              8,
              9,
              1
            ],
            [
              0,
              3,
              11,
              8
            ],
            [
              17,
              19,
              20,
              18
            ],
            [
              8,
              17,
              18,
              9
            ],
            [
              9,
              18,
              20,
              12
            ],
            [
              8,
              11,
              19,
              17
            ],
            [
              19,
              22,
              23,
              20
            ],
            [
              11,
              12,
              15,
              14
            ],
            [
              14,
              15,
              23,
              22
            ],
            [
              11,
              14,
              22,
              19
            ],
            [
              20,
              23,
              24,
              21
            ],
            [
              15,
              16,
              24,
              23
            ],
            [
              12,
              20,
              21,
              13
            ],
            [
              13,
              21,
              24,
              16
            ],
            [
              4,
              5,
              7,
              6
            ],
            [
              6,
              7,
              16,
              15
            ],
            [
              5,
              13,
              16,
              7
            ],
            [
              4,
              6,
              15,
              12
            ],
            [
              9,
              12,
              13,
              10
            ],
            [
              1,
              2,
              5,
              4
            ],
            [
              1,
              9,
              10,
              2
            ],
            [
              2,
              10,
              13,
              5
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          1
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ],
        [
          1,
          1,
          1
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "twisted_h",
      "name": "Twisted H",
      "group": "Periodic stress control",
      "note": "The reference screen found small periodic and isohedral point certificates. Retained as a search stress control, not a nonperiodic example.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "TwistedH",
          "vertices": [
            [
              -1,
              0,
              1
            ],
            [
              -1,
              0,
              2
            ],
            [
              -1,
              1,
              1
            ],
            [
              -1,
              1,
              2
            ],
            [
              0,
              -1,
              -1
            ],
            [
              0,
              -1,
              0
            ],
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              0,
              4
            ],
            [
              7,
              0,
              8
            ],
            [
              9,
              0,
              7
            ],
            [
              1,
              10,
              11,
              3
            ],
            [
              1,
              12,
              10
            ],
            [
              2,
              12,
              1
            ],
            [
              2,
              0,
              6,
              12
            ],
            [
              4,
              13,
              12,
              6
            ],
            [
              4,
              14,
              13
            ],
            [
              5,
              14,
              4
            ],
            [
              5,
              0,
              9,
              14
            ],
            [
              7,
              15,
              14,
              9
            ],
            [
              7,
              11,
              15
            ],
            [
              8,
              11,
              7
            ],
            [
              8,
              0,
              3,
              11
            ],
            [
              12,
              13,
              14,
              15,
              11,
              10
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 24,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "cube",
      "name": "Cube",
      "group": "Easy control",
      "note": "Easy periodic control. Useful for checking the protocol and measuring the overhead of extra search machinery.",
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        "Fedorov Solids",
        "Polycubes"
      ],
      "components": [
        {
          "name": "Cube",
          "vertices": [
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              0,
              0,
              0
            ],
            [
              0,
              0,
              1
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              0,
              1,
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              0,
              1,
              1
            ],
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              1,
              0,
              0
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            [
              1,
              0,
              1
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              1,
              1
            ]
          ],
          "faces": [
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              4,
              6,
              7,
              5
            ],
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              0,
              1,
              3,
              2
            ],
            [
              2,
              3,
              7,
              6
            ],
            [
              0,
              4,
              5,
              1
            ],
            [
              1,
              5,
              7,
              3
            ],
            [
              0,
              2,
              6,
              4
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          0,
          0,
          0
        ]
      ],
      "voxelCornerReplay": true
    },
    {
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      "name": "A2 Hexagonal Prism",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "A2 Layered Solids"
      ],
      "components": [
        {
          "name": "A2 Hexagonal Prism",
          "vertices": [
            [
              0,
              2,
              -2
            ],
            [
              -2,
              2,
              0
            ],
            [
              -2,
              0,
              2
            ],
            [
              0,
              -2,
              2
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            [
              2,
              -2,
              0
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              2,
              0,
              -2
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            [
              1,
              3,
              -1
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            [
              -1,
              3,
              1
            ],
            [
              -1,
              1,
              3
            ],
            [
              1,
              -1,
              3
            ],
            [
              3,
              -1,
              1
            ],
            [
              3,
              1,
              -1
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          ],
          "faces": [
            [
              5,
              4,
              3,
              2,
              1,
              0
            ],
            [
              6,
              7,
              8,
              9,
              10,
              11
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            [
              0,
              1,
              7,
              6
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              1,
              2,
              8,
              7
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              2,
              3,
              9,
              8
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            [
              3,
              4,
              10,
              9
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              4,
              5,
              11,
              10
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            [
              5,
              0,
              6,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "scd_conway",
      "name": "Schmitt–Conway–Danzer Biprism",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Aperiodic Monotiles"
      ],
      "components": [
        {
          "name": "Conway Biprism",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              10,
              0,
              0
            ],
            [
              6,
              8,
              0
            ],
            [
              16,
              8,
              0
            ],
            [
              3,
              4,
              2
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            [
              13,
              4,
              2
            ],
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              5,
              0,
              -2
            ],
            [
              11,
              8,
              -2
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          ],
          "faces": [
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              4,
              0,
              1,
              5
            ],
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              1,
              0,
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              0,
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              0,
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              3
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              3,
              1,
              6,
              7
            ],
            [
              5,
              3,
              2,
              4
            ],
            [
              7,
              2,
              3
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "1_cross",
      "name": "1-Cross (Heptacube)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "1-Cross",
          "vertices": [
            [
              -1,
              0,
              0
            ],
            [
              -1,
              0,
              1
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              -1,
              1,
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              -1,
              1,
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              0,
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              0,
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              0,
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              0,
              0,
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              1,
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              0,
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              1,
              0,
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              1,
              0,
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              1,
              1,
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              1,
              1,
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            ],
            [
              1,
              1,
              1
            ],
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              1,
              1,
              2
            ],
            [
              1,
              2,
              0
            ],
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              1,
              2,
              1
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              2,
              0,
              0
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              2,
              0,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              1,
              1
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          ],
          "faces": [
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              28,
              30,
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              29
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            [
              23,
              24,
              31,
              30
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              19,
              28,
              29,
              20
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              20,
              29,
              31,
              24
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              19,
              23,
              30,
              28
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              0,
              1,
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              2,
              3,
              12,
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              0,
              7,
              8,
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              1,
              8,
              12,
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              0,
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              23,
              26,
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              24
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              11,
              12,
              15,
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              14,
              15,
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              26
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              12,
              24,
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              11,
              14,
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              16,
              19,
              20,
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              4,
              5,
              8,
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              4,
              16,
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              17,
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              24,
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              9,
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              8,
              20,
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              21,
              25,
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              22,
              23,
              19
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              7,
              11,
              10
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              10,
              11,
              23,
              22
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              6,
              18,
              19,
              7
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            [
              6,
              10,
              22,
              18
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          ],
          "solidAngle": {
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        }
      ],
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          0,
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          0,
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          1,
          0
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          1,
          0,
          0
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "2_cross",
      "name": "2-Cross (Tridecacube)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "2-Cross",
          "vertices": [
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              -2,
              0,
              0
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              0,
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              1,
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              1,
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              0,
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              0,
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              0,
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              1,
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              0,
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          "faces": [
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              51,
              55,
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              52,
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              49
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              53,
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              48,
              50,
              54,
              52
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              45,
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              11,
              15,
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              35,
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              30
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              30,
              31,
              29
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              10
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              28
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              33,
              34,
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              39,
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              13,
              19,
              18
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              19,
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              32,
              33,
              13
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              12,
              18,
              38,
              32
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          ],
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          }
        }
      ],
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          0
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    },
    {
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      "name": "3-Cross (Nonadecacube)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "3-Cross",
          "vertices": [
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              0,
              0
            ],
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              0,
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        "Fedorov Solids"
      ],
      "components": [
        {
          "name": "RhombicDod",
          "vertices": [
            [
              2,
              0,
              0
            ],
            [
              -2,
              0,
              0
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              -2,
              0
            ],
            [
              0,
              0,
              2
            ],
            [
              0,
              0,
              -2
            ],
            [
              -1,
              -1,
              -1
            ],
            [
              -1,
              -1,
              1
            ],
            [
              -1,
              1,
              -1
            ],
            [
              -1,
              1,
              1
            ],
            [
              1,
              -1,
              -1
            ],
            [
              1,
              -1,
              1
            ],
            [
              1,
              1,
              -1
            ],
            [
              1,
              1,
              1
            ]
          ],
          "faces": [
            [
              2,
              13,
              0,
              12
            ],
            [
              10,
              0,
              11,
              3
            ],
            [
              11,
              0,
              13,
              4
            ],
            [
              5,
              12,
              0,
              10
            ],
            [
              8,
              1,
              9,
              2
            ],
            [
              7,
              1,
              6,
              3
            ],
            [
              9,
              1,
              7,
              4
            ],
            [
              6,
              1,
              8,
              5
            ],
            [
              4,
              13,
              2,
              9
            ],
            [
              8,
              2,
              12,
              5
            ],
            [
              7,
              3,
              11,
              4
            ],
            [
              10,
              3,
              6,
              5
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "elongated_dod",
      "name": "Elongated Dodecahedron",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Fedorov Solids"
      ],
      "components": [
        {
          "name": "ElongatedDod",
          "vertices": [
            [
              2,
              0,
              0
            ],
            [
              -2,
              0,
              0
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              -2,
              0
            ],
            [
              0,
              0,
              4
            ],
            [
              0,
              0,
              -4
            ],
            [
              -1,
              -1,
              -2
            ],
            [
              -1,
              -1,
              2
            ],
            [
              -1,
              1,
              -2
            ],
            [
              -1,
              1,
              2
            ],
            [
              1,
              -1,
              -2
            ],
            [
              1,
              -1,
              2
            ],
            [
              1,
              1,
              -2
            ],
            [
              1,
              1,
              2
            ]
          ],
          "faces": [
            [
              2,
              13,
              0,
              12
            ],
            [
              10,
              0,
              11,
              3
            ],
            [
              11,
              0,
              13,
              4
            ],
            [
              5,
              12,
              0,
              10
            ],
            [
              8,
              1,
              9,
              2
            ],
            [
              7,
              1,
              6,
              3
            ],
            [
              9,
              1,
              7,
              4
            ],
            [
              6,
              1,
              8,
              5
            ],
            [
              4,
              13,
              2,
              9
            ],
            [
              8,
              2,
              12,
              5
            ],
            [
              7,
              3,
              11,
              4
            ],
            [
              10,
              3,
              6,
              5
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "trunc_oct",
      "name": "Truncated Octahedron",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Fedorov Solids"
      ],
      "components": [
        {
          "name": "TruncOct",
          "vertices": [
            [
              0,
              -1,
              -2
            ],
            [
              0,
              1,
              -2
            ],
            [
              0,
              -1,
              2
            ],
            [
              0,
              1,
              2
            ],
            [
              -1,
              0,
              -2
            ],
            [
              1,
              0,
              -2
            ],
            [
              -1,
              0,
              2
            ],
            [
              1,
              0,
              2
            ],
            [
              0,
              -2,
              -1
            ],
            [
              0,
              -2,
              1
            ],
            [
              0,
              2,
              -1
            ],
            [
              0,
              2,
              1
            ],
            [
              -1,
              -2,
              0
            ],
            [
              1,
              -2,
              0
            ],
            [
              -1,
              2,
              0
            ],
            [
              1,
              2,
              0
            ],
            [
              -2,
              0,
              -1
            ],
            [
              -2,
              0,
              1
            ],
            [
              2,
              0,
              -1
            ],
            [
              2,
              0,
              1
            ],
            [
              -2,
              -1,
              0
            ],
            [
              -2,
              1,
              0
            ],
            [
              2,
              -1,
              0
            ],
            [
              2,
              1,
              0
            ]
          ],
          "faces": [
            [
              5,
              0,
              4,
              1
            ],
            [
              16,
              4,
              0,
              8,
              12,
              20
            ],
            [
              13,
              8,
              0,
              5,
              18,
              22
            ],
            [
              14,
              10,
              1,
              4,
              16,
              21
            ],
            [
              18,
              5,
              1,
              10,
              15,
              23
            ],
            [
              6,
              2,
              7,
              3
            ],
            [
              12,
              9,
              2,
              6,
              17,
              20
            ],
            [
              19,
              7,
              2,
              9,
              13,
              22
            ],
            [
              17,
              6,
              3,
              11,
              14,
              21
            ],
            [
              15,
              11,
              3,
              7,
              19,
              23
            ],
            [
              12,
              8,
              13,
              9
            ],
            [
              11,
              15,
              10,
              14
            ],
            [
              21,
              16,
              20,
              17
            ],
            [
              22,
              18,
              23,
              19
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "twist",
      "name": "Twist (Tetracube)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "Twist",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              1,
              1
            ],
            [
              1,
              0,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              1,
              1
            ],
            [
              1,
              1,
              2
            ],
            [
              1,
              2,
              0
            ],
            [
              1,
              2,
              1
            ],
            [
              1,
              2,
              2
            ],
            [
              2,
              0,
              0
            ],
            [
              2,
              0,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              1,
              1
            ],
            [
              2,
              1,
              2
            ],
            [
              2,
              2,
              0
            ],
            [
              2,
              2,
              1
            ],
            [
              2,
              2,
              2
            ]
          ],
          "faces": [
            [
              0,
              1,
              3,
              2
            ],
            [
              2,
              3,
              7,
              6
            ],
            [
              0,
              4,
              5,
              1
            ],
            [
              1,
              5,
              7,
              3
            ],
            [
              0,
              2,
              6,
              4
            ],
            [
              12,
              14,
              15,
              13
            ],
            [
              4,
              12,
              13,
              5
            ],
            [
              5,
              13,
              15,
              7
            ],
            [
              4,
              6,
              14,
              12
            ],
            [
              14,
              17,
              18,
              15
            ],
            [
              6,
              7,
              10,
              9
            ],
            [
              9,
              10,
              18,
              17
            ],
            [
              6,
              9,
              17,
              14
            ],
            [
              15,
              18,
              19,
              16
            ],
            [
              7,
              8,
              11,
              10
            ],
            [
              10,
              11,
              19,
              18
            ],
            [
              7,
              15,
              16,
              8
            ],
            [
              8,
              16,
              19,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          0,
          0,
          0
        ],
        [
          1,
          0,
          0
        ],
        [
          1,
          1,
          0
        ],
        [
          1,
          1,
          1
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "knuckle",
      "name": "Knuckle (Pentacube)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Polycubes"
      ],
      "components": [
        {
          "name": "Knuckle",
          "vertices": [
            [
              -1,
              0,
              0
            ],
            [
              -1,
              0,
              1
            ],
            [
              -1,
              1,
              0
            ],
            [
              -1,
              1,
              1
            ],
            [
              0,
              0,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              2
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              1,
              1
            ],
            [
              0,
              1,
              2
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              2,
              1
            ],
            [
              1,
              0,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              1,
              0,
              2
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              1,
              1
            ],
            [
              1,
              1,
              2
            ],
            [
              1,
              2,
              0
            ],
            [
              1,
              2,
              1
            ],
            [
              2,
              0,
              0
            ],
            [
              2,
              0,
              1
            ],
            [
              2,
              1,
              0
            ],
            [
              2,
              1,
              1
            ]
          ],
          "faces": [
            [
              4,
              12,
              13,
              5
            ],
            [
              4,
              7,
              15,
              12
            ],
            [
              20,
              22,
              23,
              21
            ],
            [
              15,
              16,
              23,
              22
            ],
            [
              12,
              20,
              21,
              13
            ],
            [
              13,
              21,
              23,
              16
            ],
            [
              12,
              15,
              22,
              20
            ],
            [
              0,
              1,
              3,
              2
            ],
            [
              2,
              3,
              8,
              7
            ],
            [
              0,
              4,
              5,
              1
            ],
            [
              1,
              5,
              8,
              3
            ],
            [
              0,
              2,
              7,
              4
            ],
            [
              15,
              18,
              19,
              16
            ],
            [
              7,
              8,
              11,
              10
            ],
            [
              10,
              11,
              19,
              18
            ],
            [
              8,
              16,
              19,
              11
            ],
            [
              7,
              10,
              18,
              15
            ],
            [
              13,
              16,
              17,
              14
            ],
            [
              5,
              6,
              9,
              8
            ],
            [
              8,
              9,
              17,
              16
            ],
            [
              5,
              13,
              14,
              6
            ],
            [
              6,
              14,
              17,
              9
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 8,
            "symbols": []
          }
        }
      ],
      "voxels": [
        [
          -1,
          0,
          0
        ],
        [
          0,
          0,
          0
        ],
        [
          0,
          0,
          1
        ],
        [
          0,
          1,
          0
        ],
        [
          1,
          0,
          0
        ]
      ],
      "voxelCornerReplay": true
    },
    {
      "id": "tet_oct",
      "name": "Tetrahedron + Octahedron",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Platonic Solids"
      ],
      "components": [
        {
          "name": "Tetrahedron",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              1,
              1
            ]
          ],
          "faces": [
            [
              0,
              1,
              2
            ],
            [
              0,
              2,
              3
            ],
            [
              0,
              3,
              1
            ],
            [
              1,
              3,
              2
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        },
        {
          "name": "Octahedron",
          "vertices": [
            [
              1,
              0,
              0
            ],
            [
              -1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              -1,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              -1
            ]
          ],
          "faces": [
            [
              0,
              2,
              4
            ],
            [
              2,
              1,
              4
            ],
            [
              1,
              3,
              4
            ],
            [
              3,
              0,
              4
            ],
            [
              0,
              5,
              2
            ],
            [
              2,
              5,
              1
            ],
            [
              1,
              5,
              3
            ],
            [
              3,
              5,
              0
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        }
      ]
    },
    {
      "id": "tetragonal_disphenoid",
      "name": "Tetragonal Disphenoid (B₃ alcove)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "Disphenoid",
          "vertices": [
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              -1
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              -1,
              0
            ]
          ],
          "faces": [
            [
              3,
              2,
              0
            ],
            [
              2,
              3,
              1
            ],
            [
              1,
              3,
              0
            ],
            [
              2,
              1,
              0
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "corner_tetra",
      "name": "Corner Tetrahedron",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "CornerTetra",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              0,
              1
            ]
          ],
          "faces": [
            [
              0,
              2,
              1
            ],
            [
              0,
              1,
              3
            ],
            [
              0,
              3,
              2
            ],
            [
              1,
              2,
              3
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "big_corner_tetra",
      "name": "Big Corner Tetrahedron",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "BigCornerTetra",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              2,
              0,
              0
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              0,
              2
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              1,
              1
            ],
            [
              1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              0,
              1
            ]
          ],
          "faces": [
            [
              8,
              7,
              0
            ],
            [
              4,
              1,
              7
            ],
            [
              2,
              4,
              8
            ],
            [
              8,
              4,
              7
            ],
            [
              7,
              9,
              0
            ],
            [
              5,
              3,
              9
            ],
            [
              1,
              5,
              7
            ],
            [
              7,
              5,
              9
            ],
            [
              9,
              8,
              0
            ],
            [
              6,
              2,
              8
            ],
            [
              3,
              6,
              9
            ],
            [
              9,
              6,
              8
            ],
            [
              1,
              4,
              5
            ],
            [
              4,
              2,
              6
            ],
            [
              5,
              6,
              3
            ],
            [
              4,
              6,
              5
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "diamond_lattice",
      "name": "Double FCC Lattice (Diamond)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Platonic Solids"
      ],
      "components": [
        {
          "name": "Tetrahedron",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              1,
              1
            ]
          ],
          "faces": [
            [
              0,
              1,
              2
            ],
            [
              0,
              2,
              3
            ],
            [
              0,
              3,
              1
            ],
            [
              1,
              3,
              2
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        },
        {
          "name": "Octahedron",
          "vertices": [
            [
              1,
              0,
              0
            ],
            [
              -1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              -1,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              -1
            ]
          ],
          "faces": [
            [
              0,
              2,
              4
            ],
            [
              2,
              1,
              4
            ],
            [
              1,
              3,
              4
            ],
            [
              3,
              0,
              4
            ],
            [
              0,
              5,
              2
            ],
            [
              2,
              5,
              1
            ],
            [
              1,
              5,
              3
            ],
            [
              3,
              5,
              0
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        },
        {
          "name": "CornerTetra",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              0,
              1
            ]
          ],
          "faces": [
            [
              0,
              2,
              1
            ],
            [
              0,
              1,
              3
            ],
            [
              0,
              3,
              2
            ],
            [
              1,
              2,
              3
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "perovskite",
      "name": "Perovskite (Cuboctahedron + Octa)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Platonic Solids"
      ],
      "components": [
        {
          "name": "Cuboctahedron",
          "vertices": [
            [
              -1,
              -1,
              0
            ],
            [
              -1,
              1,
              0
            ],
            [
              1,
              -1,
              0
            ],
            [
              1,
              1,
              0
            ],
            [
              -1,
              0,
              -1
            ],
            [
              -1,
              0,
              1
            ],
            [
              1,
              0,
              -1
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              -1,
              -1
            ],
            [
              0,
              -1,
              1
            ],
            [
              0,
              1,
              -1
            ],
            [
              0,
              1,
              1
            ]
          ],
          "faces": [
            [
              4,
              0,
              5,
              1
            ],
            [
              9,
              0,
              8,
              2
            ],
            [
              8,
              0,
              4
            ],
            [
              5,
              0,
              9
            ],
            [
              10,
              1,
              11,
              3
            ],
            [
              4,
              1,
              10
            ],
            [
              11,
              1,
              5
            ],
            [
              3,
              7,
              2,
              6
            ],
            [
              6,
              2,
              8
            ],
            [
              9,
              2,
              7
            ],
            [
              10,
              3,
              6
            ],
            [
              7,
              3,
              11
            ],
            [
              8,
              4,
              10,
              6
            ],
            [
              7,
              11,
              5,
              9
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        },
        {
          "name": "Octahedron",
          "vertices": [
            [
              1,
              0,
              0
            ],
            [
              -1,
              0,
              0
            ],
            [
              0,
              1,
              0
            ],
            [
              0,
              -1,
              0
            ],
            [
              0,
              0,
              1
            ],
            [
              0,
              0,
              -1
            ]
          ],
          "faces": [
            [
              0,
              2,
              4
            ],
            [
              2,
              1,
              4
            ],
            [
              1,
              3,
              4
            ],
            [
              3,
              0,
              4
            ],
            [
              0,
              5,
              2
            ],
            [
              2,
              5,
              1
            ],
            [
              1,
              5,
              3
            ],
            [
              3,
              5,
              0
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        }
      ]
    },
    {
      "id": "orthoscheme",
      "name": "Orthoscheme (B̃₃ alcove)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "Orthoscheme",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              2,
              0,
              0
            ],
            [
              2,
              2,
              0
            ],
            [
              2,
              2,
              2
            ],
            [
              1,
              1,
              0
            ],
            [
              2,
              1,
              1
            ]
          ],
          "faces": [
            [
              4,
              1,
              0
            ],
            [
              4,
              2,
              1
            ],
            [
              1,
              2,
              5
            ],
            [
              2,
              3,
              5
            ],
            [
              0,
              1,
              3
            ],
            [
              3,
              2,
              0
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "elongated_sq_bipyramid",
      "name": "Elongated Bipyramid (J15)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "Johnson15",
          "vertices": [
            [
              -1,
              -1,
              -1
            ],
            [
              -1,
              -1,
              1
            ],
            [
              -1,
              1,
              -1
            ],
            [
              -1,
              1,
              1
            ],
            [
              1,
              -1,
              -1
            ],
            [
              1,
              -1,
              1
            ],
            [
              1,
              1,
              -1
            ],
            [
              1,
              1,
              1
            ],
            [
              0,
              0,
              2
            ],
            [
              0,
              0,
              -2
            ]
          ],
          "faces": [
            [
              2,
              0,
              1,
              3
            ],
            [
              1,
              0,
              4,
              5
            ],
            [
              9,
              0,
              2
            ],
            [
              4,
              0,
              9
            ],
            [
              3,
              1,
              8
            ],
            [
              8,
              1,
              5
            ],
            [
              6,
              2,
              3,
              7
            ],
            [
              9,
              2,
              6
            ],
            [
              7,
              3,
              8
            ],
            [
              5,
              4,
              6,
              7
            ],
            [
              6,
              4,
              9
            ],
            [
              8,
              5,
              7
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "gyrobifastigium",
      "name": "Gyrobifastigium (J26)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "Johnson26",
          "vertices": [
            [
              1,
              1,
              0
            ],
            [
              1,
              -1,
              0
            ],
            [
              -1,
              -1,
              0
            ],
            [
              -1,
              1,
              0
            ],
            [
              0,
              1,
              2
            ],
            [
              0,
              -1,
              2
            ],
            [
              1,
              0,
              -2
            ],
            [
              -1,
              0,
              -2
            ]
          ],
          "faces": [
            [
              1,
              0,
              4,
              5
            ],
            [
              6,
              0,
              1
            ],
            [
              4,
              0,
              3
            ],
            [
              3,
              0,
              6,
              7
            ],
            [
              2,
              1,
              5
            ],
            [
              6,
              1,
              2,
              7
            ],
            [
              3,
              2,
              5,
              4
            ],
            [
              7,
              2,
              3
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "laves_c15",
      "name": "Laves C15 (Truncated Tetra + Tetra))",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Platonic Solids"
      ],
      "components": [
        {
          "name": "TruncTetra",
          "vertices": [
            [
              1,
              1,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              1,
              1
            ],
            [
              2,
              2,
              0
            ],
            [
              3,
              2,
              1
            ],
            [
              2,
              3,
              1
            ],
            [
              2,
              0,
              2
            ],
            [
              3,
              1,
              2
            ],
            [
              2,
              1,
              3
            ],
            [
              0,
              2,
              2
            ],
            [
              1,
              3,
              2
            ],
            [
              1,
              2,
              3
            ]
          ],
          "faces": [
            [
              2,
              0,
              1
            ],
            [
              6,
              1,
              0,
              3,
              4,
              7
            ],
            [
              5,
              3,
              0,
              2,
              9,
              10
            ],
            [
              9,
              2,
              1,
              6,
              8,
              11
            ],
            [
              4,
              3,
              5
            ],
            [
              8,
              7,
              4,
              5,
              10,
              11
            ],
            [
              8,
              6,
              7
            ],
            [
              10,
              9,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        },
        {
          "name": "Tetra",
          "vertices": [
            [
              0,
              0,
              0
            ],
            [
              1,
              1,
              0
            ],
            [
              1,
              0,
              1
            ],
            [
              0,
              1,
              1
            ]
          ],
          "faces": [
            [
              0,
              1,
              2
            ],
            [
              0,
              2,
              3
            ],
            [
              0,
              3,
              1
            ],
            [
              1,
              3,
              2
            ]
          ],
          "solidAngle": {
            "kind": "symbolic",
            "max_value": 48,
            "symbols": [
              "α = (3 arccos(1/3) - π)/(4π)"
            ]
          }
        }
      ]
    },
    {
      "id": "escher_compound",
      "name": "Escher Solid",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Space Fillers"
      ],
      "components": [
        {
          "name": "EscherSolid",
          "vertices": [
            [
              2,
              0,
              0
            ],
            [
              -2,
              0,
              0
            ],
            [
              0,
              2,
              0
            ],
            [
              0,
              -2,
              0
            ],
            [
              0,
              0,
              2
            ],
            [
              0,
              0,
              -2
            ],
            [
              -1,
              -1,
              -1
            ],
            [
              -1,
              -1,
              1
            ],
            [
              -1,
              1,
              -1
            ],
            [
              -1,
              1,
              1
            ],
            [
              1,
              -1,
              -1
            ],
            [
              1,
              -1,
              1
            ],
            [
              1,
              1,
              -1
            ],
            [
              1,
              1,
              1
            ],
            [
              2,
              2,
              0
            ],
            [
              2,
              -2,
              0
            ],
            [
              2,
              0,
              2
            ],
            [
              2,
              0,
              -2
            ],
            [
              -2,
              2,
              0
            ],
            [
              -2,
              -2,
              0
            ],
            [
              -2,
              0,
              2
            ],
            [
              -2,
              0,
              -2
            ],
            [
              0,
              2,
              2
            ],
            [
              0,
              2,
              -2
            ],
            [
              0,
              -2,
              2
            ],
            [
              0,
              -2,
              -2
            ]
          ],
          "faces": [
            [
              2,
              13,
              14
            ],
            [
              13,
              0,
              14
            ],
            [
              0,
              12,
              14
            ],
            [
              12,
              2,
              14
            ],
            [
              10,
              0,
              15
            ],
            [
              0,
              11,
              15
            ],
            [
              11,
              3,
              15
            ],
            [
              3,
              10,
              15
            ],
            [
              11,
              0,
              16
            ],
            [
              0,
              13,
              16
            ],
            [
              13,
              4,
              16
            ],
            [
              4,
              11,
              16
            ],
            [
              5,
              12,
              17
            ],
            [
              12,
              0,
              17
            ],
            [
              0,
              10,
              17
            ],
            [
              10,
              5,
              17
            ],
            [
              8,
              1,
              18
            ],
            [
              1,
              9,
              18
            ],
            [
              9,
              2,
              18
            ],
            [
              2,
              8,
              18
            ],
            [
              7,
              1,
              19
            ],
            [
              1,
              6,
              19
            ],
            [
              6,
              3,
              19
            ],
            [
              3,
              7,
              19
            ],
            [
              9,
              1,
              20
            ],
            [
              1,
              7,
              20
            ],
            [
              7,
              4,
              20
            ],
            [
              4,
              9,
              20
            ],
            [
              6,
              1,
              21
            ],
            [
              1,
              8,
              21
            ],
            [
              8,
              5,
              21
            ],
            [
              5,
              6,
              21
            ],
            [
              4,
              13,
              22
            ],
            [
              13,
              2,
              22
            ],
            [
              2,
              9,
              22
            ],
            [
              9,
              4,
              22
            ],
            [
              8,
              2,
              23
            ],
            [
              2,
              12,
              23
            ],
            [
              12,
              5,
              23
            ],
            [
              5,
              8,
              23
            ],
            [
              7,
              3,
              24
            ],
            [
              3,
              11,
              24
            ],
            [
              11,
              4,
              24
            ],
            [
              4,
              7,
              24
            ],
            [
              10,
              3,
              25
            ],
            [
              3,
              6,
              25
            ],
            [
              6,
              5,
              25
            ],
            [
              5,
              10,
              25
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "fcc_pure",
      "name": "FCC (Pure Rhombic Dodecahedron)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Sphere Packings"
      ],
      "components": [
        {
          "name": "Rhombic_Dodecahedron_(FCC)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              -3,
              -3,
              -3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              -6,
              0,
              0
            ],
            [
              0,
              -6,
              0
            ],
            [
              0,
              0,
              -6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              -3,
              -3,
              3
            ],
            [
              -3,
              3,
              -3
            ],
            [
              3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              3,
              0,
              2,
              8
            ],
            [
              2,
              0,
              4,
              9
            ],
            [
              4,
              0,
              3,
              10
            ],
            [
              5,
              1,
              6,
              11
            ],
            [
              7,
              1,
              5,
              12
            ],
            [
              6,
              1,
              7,
              13
            ],
            [
              13,
              2,
              9,
              6
            ],
            [
              7,
              8,
              2,
              13
            ],
            [
              5,
              10,
              3,
              12
            ],
            [
              12,
              3,
              8,
              7
            ],
            [
              11,
              4,
              10,
              5
            ],
            [
              6,
              9,
              4,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "hcp_pure",
      "name": "HCP (Pure Trapezo-Rhombic Dodecahedron)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Sphere Packings"
      ],
      "components": [
        {
          "name": "Trapezo_Rhombic_Dodecahedron_(HCP)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              1,
              1,
              -5
            ],
            [
              1,
              -5,
              1
            ],
            [
              -5,
              1,
              1
            ],
            [
              2,
              -4,
              -4
            ],
            [
              -4,
              2,
              -4
            ],
            [
              -4,
              -4,
              2
            ],
            [
              -3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              2,
              0,
              1,
              4
            ],
            [
              1,
              0,
              3,
              5
            ],
            [
              3,
              0,
              2,
              6
            ],
            [
              7,
              4,
              1,
              10
            ],
            [
              10,
              1,
              5,
              8
            ],
            [
              11,
              2,
              4,
              7
            ],
            [
              9,
              6,
              2,
              11
            ],
            [
              8,
              5,
              3,
              12
            ],
            [
              12,
              3,
              6,
              9
            ],
            [
              11,
              7,
              10,
              13
            ],
            [
              10,
              8,
              12,
              13
            ],
            [
              12,
              9,
              11,
              13
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "barlow_fcc",
      "name": "Barlow Packing (Root: FCC)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Sphere Packings"
      ],
      "components": [
        {
          "name": "Rhombic_Dodecahedron_(FCC)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              -3,
              -3,
              -3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              -6,
              0,
              0
            ],
            [
              0,
              -6,
              0
            ],
            [
              0,
              0,
              -6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              -3,
              -3,
              3
            ],
            [
              -3,
              3,
              -3
            ],
            [
              3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              3,
              0,
              2,
              8
            ],
            [
              2,
              0,
              4,
              9
            ],
            [
              4,
              0,
              3,
              10
            ],
            [
              5,
              1,
              6,
              11
            ],
            [
              7,
              1,
              5,
              12
            ],
            [
              6,
              1,
              7,
              13
            ],
            [
              13,
              2,
              9,
              6
            ],
            [
              7,
              8,
              2,
              13
            ],
            [
              5,
              10,
              3,
              12
            ],
            [
              12,
              3,
              8,
              7
            ],
            [
              11,
              4,
              10,
              5
            ],
            [
              6,
              9,
              4,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        },
        {
          "name": "Trapezo_Rhombic_Dodecahedron_(HCP)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              1,
              1,
              -5
            ],
            [
              1,
              -5,
              1
            ],
            [
              -5,
              1,
              1
            ],
            [
              2,
              -4,
              -4
            ],
            [
              -4,
              2,
              -4
            ],
            [
              -4,
              -4,
              2
            ],
            [
              -3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              2,
              0,
              1,
              4
            ],
            [
              1,
              0,
              3,
              5
            ],
            [
              3,
              0,
              2,
              6
            ],
            [
              7,
              4,
              1,
              10
            ],
            [
              10,
              1,
              5,
              8
            ],
            [
              11,
              2,
              4,
              7
            ],
            [
              9,
              6,
              2,
              11
            ],
            [
              8,
              5,
              3,
              12
            ],
            [
              12,
              3,
              6,
              9
            ],
            [
              11,
              7,
              10,
              13
            ],
            [
              10,
              8,
              12,
              13
            ],
            [
              12,
              9,
              11,
              13
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    },
    {
      "id": "barlow_hcp",
      "name": "Barlow Packing (Root: HCP)",
      "group": "Catalog",
      "note": "Legacy catalog geometry. V2 accepts only exact integer point weights; structural status is independent of the finite-window result.",
      "categories": [
        "Sphere Packings"
      ],
      "components": [
        {
          "name": "Trapezo_Rhombic_Dodecahedron_(HCP)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              1,
              1,
              -5
            ],
            [
              1,
              -5,
              1
            ],
            [
              -5,
              1,
              1
            ],
            [
              2,
              -4,
              -4
            ],
            [
              -4,
              2,
              -4
            ],
            [
              -4,
              -4,
              2
            ],
            [
              -3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              2,
              0,
              1,
              4
            ],
            [
              1,
              0,
              3,
              5
            ],
            [
              3,
              0,
              2,
              6
            ],
            [
              7,
              4,
              1,
              10
            ],
            [
              10,
              1,
              5,
              8
            ],
            [
              11,
              2,
              4,
              7
            ],
            [
              9,
              6,
              2,
              11
            ],
            [
              8,
              5,
              3,
              12
            ],
            [
              12,
              3,
              6,
              9
            ],
            [
              11,
              7,
              10,
              13
            ],
            [
              10,
              8,
              12,
              13
            ],
            [
              12,
              9,
              11,
              13
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        },
        {
          "name": "Rhombic_Dodecahedron_(FCC)",
          "vertices": [
            [
              3,
              3,
              3
            ],
            [
              -3,
              -3,
              -3
            ],
            [
              6,
              0,
              0
            ],
            [
              0,
              6,
              0
            ],
            [
              0,
              0,
              6
            ],
            [
              -6,
              0,
              0
            ],
            [
              0,
              -6,
              0
            ],
            [
              0,
              0,
              -6
            ],
            [
              3,
              3,
              -3
            ],
            [
              3,
              -3,
              3
            ],
            [
              -3,
              3,
              3
            ],
            [
              -3,
              -3,
              3
            ],
            [
              -3,
              3,
              -3
            ],
            [
              3,
              -3,
              -3
            ]
          ],
          "faces": [
            [
              3,
              0,
              2,
              8
            ],
            [
              2,
              0,
              4,
              9
            ],
            [
              4,
              0,
              3,
              10
            ],
            [
              5,
              1,
              6,
              11
            ],
            [
              7,
              1,
              5,
              12
            ],
            [
              6,
              1,
              7,
              13
            ],
            [
              13,
              2,
              9,
              6
            ],
            [
              7,
              8,
              2,
              13
            ],
            [
              5,
              10,
              3,
              12
            ],
            [
              12,
              3,
              8,
              7
            ],
            [
              11,
              4,
              10,
              5
            ],
            [
              6,
              9,
              4,
              11
            ]
          ],
          "solidAngle": {
            "kind": "rational",
            "max_value": 48,
            "symbols": []
          }
        }
      ]
    }
  ],
  "figures": [
    {
      "id": "polycube_p10_346304::0",
      "name": "p10-346304 · 10 cubes",
      "mode": "polycube_p10_346304",
      "index": 0,
      "systems": [
        "p10-346304 · 10 cubes"
      ]
    },
    {
      "id": "polycube_p9_42947::0",
      "name": "p9-42947 · 9 cubes",
      "mode": "polycube_p9_42947",
      "index": 0,
      "systems": [
        "p9-42947 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p10_054782::0",
      "name": "p10-054782 · 10 cubes",
      "mode": "polycube_p10_054782",
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      "systems": [
        "p10-054782 · 10 cubes"
      ]
    },
    {
      "id": "polycube_p10_055695::0",
      "name": "p10-055695 · 10 cubes",
      "mode": "polycube_p10_055695",
      "index": 0,
      "systems": [
        "p10-055695 · 10 cubes"
      ]
    },
    {
      "id": "polycube_p10_290795::0",
      "name": "p10-290795 · 10 cubes",
      "mode": "polycube_p10_290795",
      "index": 0,
      "systems": [
        "p10-290795 · 10 cubes"
      ]
    },
    {
      "id": "polycube_p9_02127::0",
      "name": "p9-02127 · 9 cubes",
      "mode": "polycube_p9_02127",
      "index": 0,
      "systems": [
        "p9-02127 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p9_08203::0",
      "name": "p9-08203 · 9 cubes",
      "mode": "polycube_p9_08203",
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      "systems": [
        "p9-08203 · 9 cubes"
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    },
    {
      "id": "polycube_p9_08219::0",
      "name": "p9-08219 · 9 cubes",
      "mode": "polycube_p9_08219",
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      "systems": [
        "p9-08219 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p9_20656::0",
      "name": "p9-20656 · 9 cubes",
      "mode": "polycube_p9_20656",
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      "systems": [
        "p9-20656 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p9_24025::0",
      "name": "p9-24025 · 9 cubes",
      "mode": "polycube_p9_24025",
      "index": 0,
      "systems": [
        "p9-24025 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p9_48258::0",
      "name": "p9-48258 · 9 cubes",
      "mode": "polycube_p9_48258",
      "index": 0,
      "systems": [
        "p9-48258 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p9_43172::0",
      "name": "p9-43172 · 9 cubes",
      "mode": "polycube_p9_43172",
      "index": 0,
      "systems": [
        "p9-43172 · 9 cubes"
      ]
    },
    {
      "id": "polycube_p10_052588::0",
      "name": "p10-052588 · 10 cubes",
      "mode": "polycube_p10_052588",
      "index": 0,
      "systems": [
        "p10-052588 · 10 cubes"
      ]
    },
    {
      "id": "polycube_p10_052670::0",
      "name": "p10-052670 · 10 cubes",
      "mode": "polycube_p10_052670",
      "index": 0,
      "systems": [
        "p10-052670 · 10 cubes"
      ]
    },
    {
      "id": "mathematica_16_vertex::0",
      "name": "16-vertex lattice tile",
      "mode": "mathematica_16_vertex",
      "index": 0,
      "systems": [
        "16-vertex lattice tile"
      ]
    },
    {
      "id": "a2_hexagonal_prism::0",
      "name": "A2 Hexagonal Prism",
      "mode": "a2_hexagonal_prism",
      "index": 0,
      "systems": [
        "A2 Hexagonal Prism"
      ]
    },
    {
      "id": "a2_hat_prism::0",
      "name": "A2 Hat Prism",
      "mode": "a2_hat_prism",
      "index": 0,
      "systems": [
        "A2 Hat Prism"
      ]
    },
    {
      "id": "a2_turtle_prism::0",
      "name": "A2 Turtle Prism",
      "mode": "a2_turtle_prism",
      "index": 0,
      "systems": [
        "A2 Turtle Prism"
      ]
    },
    {
      "id": "scd_conway::0",
      "name": "Conway Biprism",
      "mode": "scd_conway",
      "index": 0,
      "systems": [
        "Schmitt–Conway–Danzer Biprism"
      ]
    },
    {
      "id": "nonacube_cross::0",
      "name": "Nonacube cross · two-unit arms",
      "mode": "nonacube_cross",
      "index": 0,
      "systems": [
        "Nonacube cross · two-unit arms"
      ]
    },
    {
      "id": "1_cross::0",
      "name": "1-Cross",
      "mode": "1_cross",
      "index": 0,
      "systems": [
        "1-Cross (Heptacube)"
      ]
    },
    {
      "id": "2_cross::0",
      "name": "2-Cross",
      "mode": "2_cross",
      "index": 0,
      "systems": [
        "2-Cross (Tridecacube)"
      ]
    },
    {
      "id": "3_cross::0",
      "name": "3-Cross",
      "mode": "3_cross",
      "index": 0,
      "systems": [
        "3-Cross (Nonadecacube)"
      ]
    },
    {
      "id": "t_cross::0",
      "name": "T-Cross",
      "mode": "t_cross",
      "index": 0,
      "systems": [
        "T-Cross"
      ]
    },
    {
      "id": "1_semicross::0",
      "name": "1-Semi",
      "mode": "1_semicross",
      "index": 0,
      "systems": [
        "1-Semicross (Tripod)"
      ]
    },
    {
      "id": "2_semicross::0",
      "name": "2-Semi",
      "mode": "2_semicross",
      "index": 0,
      "systems": [
        "2-Semicross (Corner)"
      ]
    },
    {
      "id": "buckled_ring::0",
      "name": "Buckled Ring",
      "mode": "buckled_ring",
      "index": 0,
      "systems": [
        "Buckled Ring"
      ]
    },
    {
      "id": "large_buckled_ring::0",
      "name": "Large Buckled Ring",
      "mode": "large_buckled_ring",
      "index": 0,
      "systems": [
        "Large Buckled Ring"
      ]
    },
    {
      "id": "double_ring::0",
      "name": "Double Buckled Ring",
      "mode": "double_ring",
      "index": 0,
      "systems": [
        "Double Buckled Ring"
      ]
    },
    {
      "id": "tuning_fork::0",
      "name": "Tuning Fork",
      "mode": "tuning_fork",
      "index": 0,
      "systems": [
        "Tuning Fork (Reinhardt)"
      ]
    },
    {
      "id": "twisted_h::0",
      "name": "Letter H",
      "mode": "twisted_h",
      "index": 0,
      "systems": [
        "Letter H (Twisted)"
      ]
    },
    {
      "id": "cube::0",
      "name": "Cube",
      "mode": "cube",
      "index": 0,
      "systems": [
        "Cube"
      ]
    },
    {
      "id": "letter_o::0",
      "name": "Letter O",
      "mode": "letter_o",
      "index": 0,
      "systems": [
        "Letter O"
      ]
    },
    {
      "id": "1_cross_plus::0",
      "name": "1-Cross + 1",
      "mode": "1_cross_plus",
      "index": 0,
      "systems": [
        "1-Cross + 1"
      ]
    },
    {
      "id": "hex_prism::0",
      "name": "Hexagonal Prism",
      "mode": "hex_prism",
      "index": 0,
      "systems": [
        "Hexagonal Prism"
      ]
    },
    {
      "id": "rhombic::0",
      "name": "Rhombic Dodecahedron",
      "mode": "rhombic",
      "index": 0,
      "systems": [
        "Rhombic Dodecahedron"
      ]
    },
    {
      "id": "elongated_dod::0",
      "name": "Elongated Dodecahedron",
      "mode": "elongated_dod",
      "index": 0,
      "systems": [
        "Elongated Dodecahedron"
      ]
    },
    {
      "id": "trunc_oct::0",
      "name": "Truncated Octahedron",
      "mode": "trunc_oct",
      "index": 0,
      "systems": [
        "Truncated Octahedron"
      ]
    },
    {
      "id": "twist::0",
      "name": "Twist",
      "mode": "twist",
      "index": 0,
      "systems": [
        "Twist (Tetracube)"
      ]
    },
    {
      "id": "knuckle::0",
      "name": "Knuckle",
      "mode": "knuckle",
      "index": 0,
      "systems": [
        "Knuckle (Pentacube)"
      ]
    },
    {
      "id": "tet_oct::0",
      "name": "Tetrahedron",
      "mode": "tet_oct",
      "index": 0,
      "systems": [
        "Tetrahedron + Octahedron",
        "Double FCC Lattice (Diamond)",
        "Laves C15 (Truncated Tetra + Tetra))"
      ]
    },
    {
      "id": "tet_oct::1",
      "name": "Octahedron",
      "mode": "tet_oct",
      "index": 1,
      "systems": [
        "Tetrahedron + Octahedron",
        "Double FCC Lattice (Diamond)",
        "Perovskite (Cuboctahedron + Octa)"
      ]
    },
    {
      "id": "tetragonal_disphenoid::0",
      "name": "Tetragonal Disphenoid",
      "mode": "tetragonal_disphenoid",
      "index": 0,
      "systems": [
        "Tetragonal Disphenoid (B₃ alcove)"
      ]
    },
    {
      "id": "corner_tetra::0",
      "name": "Corner Tetrahedron",
      "mode": "corner_tetra",
      "index": 0,
      "systems": [
        "Corner Tetrahedron",
        "Double FCC Lattice (Diamond)"
      ]
    },
    {
      "id": "big_corner_tetra::0",
      "name": "Big Corner Tetrahedron",
      "mode": "big_corner_tetra",
      "index": 0,
      "systems": [
        "Big Corner Tetrahedron"
      ]
    },
    {
      "id": "perovskite::0",
      "name": "Cuboctahedron",
      "mode": "perovskite",
      "index": 0,
      "systems": [
        "Perovskite (Cuboctahedron + Octa)"
      ]
    },
    {
      "id": "orthoscheme::0",
      "name": "Orthoscheme",
      "mode": "orthoscheme",
      "index": 0,
      "systems": [
        "Orthoscheme (B̃₃ alcove)"
      ]
    },
    {
      "id": "elongated_sq_bipyramid::0",
      "name": "Elongated Bipyramid",
      "mode": "elongated_sq_bipyramid",
      "index": 0,
      "systems": [
        "Elongated Bipyramid (J15)"
      ]
    },
    {
      "id": "gyrobifastigium::0",
      "name": "Gyrobifastigium",
      "mode": "gyrobifastigium",
      "index": 0,
      "systems": [
        "Gyrobifastigium (J26)"
      ]
    },
    {
      "id": "laves_c15::0",
      "name": "Truncated Tetrahedron",
      "mode": "laves_c15",
      "index": 0,
      "systems": [
        "Laves C15 (Truncated Tetra + Tetra))"
      ]
    },
    {
      "id": "escher_compound::0",
      "name": "Escher Solid",
      "mode": "escher_compound",
      "index": 0,
      "systems": [
        "Escher Solid"
      ]
    },
    {
      "id": "fcc_pure::0",
      "name": "Rhombic Dodecahedron",
      "mode": "fcc_pure",
      "index": 0,
      "systems": [
        "FCC (Pure Rhombic Dodecahedron)",
        "Barlow Packing (Root: FCC)",
        "Barlow Packing (Root: HCP)"
      ]
    },
    {
      "id": "hcp_pure::0",
      "name": "Trapezo-Rhombic Dodecahedron",
      "mode": "hcp_pure",
      "index": 0,
      "systems": [
        "HCP (Pure Trapezo-Rhombic Dodecahedron)",
        "Barlow Packing (Root: FCC)",
        "Barlow Packing (Root: HCP)"
      ]
    }
  ]
}