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                1
              ],
              "normalized_value": 357.3333333333333,
              "id": "67:0",
              "constructed_multiplicity": 1
            },
            {
              "q": 73,
              "trig": "cos",
              "representative": [
                -9,
                -8
              ],
              "frequencies_axial": [
                [
                  -9,
                  -8
                ],
                [
                  -9,
                  -1
                ],
                [
                  -8,
                  -9
                ],
                [
                  -8,
                  1
                ],
                [
                  -1,
                  -9
                ],
                [
                  -1,
                  8
                ],
                [
                  1,
                  -8
                ],
                [
                  1,
                  9
                ],
                [
                  8,
                  -1
                ],
                [
                  8,
                  9
                ],
                [
                  9,
                  1
                ],
                [
                  9,
                  8
                ]
              ],
              "coefficients": [
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1
              ],
              "normalized_value": 389.3333333333333,
              "id": "73:0",
              "constructed_multiplicity": 1
            },
            {
              "q": 75,
              "trig": "cos",
              "representative": [
                -10,
                -5
              ],
              "frequencies_axial": [
                [
                  -10,
                  -5
                ],
                [
                  -5,
                  -10
                ],
                [
                  -5,
                  5
                ],
                [
                  5,
                  -5
                ],
                [
                  5,
                  10
                ],
                [
                  10,
                  5
                ]
              ],
              "coefficients": [
                1,
                1,
                1,
                1,
                1,
                1
              ],
              "normalized_value": 400.0,
              "id": "75:0",
              "constructed_multiplicity": 1
            },
            {
              "q": 76,
              "trig": "cos",
              "representative": [
                -10,
                -6
              ],
              "frequencies_axial": [
                [
                  -10,
                  -6
                ],
                [
                  -10,
                  -4
                ],
                [
                  -6,
                  -10
                ],
                [
                  -6,
                  4
                ],
                [
                  -4,
                  -10
                ],
                [
                  -4,
                  6
                ],
                [
                  4,
                  -6
                ],
                [
                  4,
                  10
                ],
                [
                  6,
                  -4
                ],
                [
                  6,
                  10
                ],
                [
                  10,
                  4
                ],
                [
                  10,
                  6
                ]
              ],
              "coefficients": [
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1
              ],
              "normalized_value": 405.3333333333333,
              "id": "76:0",
              "constructed_multiplicity": 1
            },
            {
              "q": 79,
              "trig": "cos",
              "representative": [
                -10,
                -7
              ],
              "frequencies_axial": [
                [
                  -10,
                  -7
                ],
                [
                  -10,
                  -3
                ],
                [
                  -7,
                  -10
                ],
                [
                  -7,
                  3
                ],
                [
                  -3,
                  -10
                ],
                [
                  -3,
                  7
                ],
                [
                  3,
                  -7
                ],
                [
                  3,
                  10
                ],
                [
                  7,
                  -3
                ],
                [
                  7,
                  10
                ],
                [
                  10,
                  3
                ],
                [
                  10,
                  7
                ]
              ],
              "coefficients": [
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1,
                1
              ],
              "normalized_value": 421.3333333333333,
              "id": "79:0",
              "constructed_multiplicity": 1
            }
          ]
        }
      ],
      "tiling_status": "The decorated Socolar\u2013Taylor hexagon forces aperiodicity through nearest- and next-nearest-neighbor rules. Here the spectrum is that of the underlying regular hexagon; those rules have not been encoded in its boundary operator.",
      "source_url": "https://arxiv.org/abs/1003.4279",
      "matching_source_url": null
    },
    {
      "id": "pinwheel",
      "label": "Pinwheel triangle",
      "polygon": [
        [
          0,
          0
        ],
        [
          2,
          0
        ],
        [
          0,
          1
        ]
      ],
      "area": 1.0,
      "lattice_basis": [
        [
          1,
          0
        ],
        [
          0,
          1
        ]
      ],
      "lattice_vertices": null,
      "boundary_certificates": [],
      "source_polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          0,
          1
        ]
      ],
      "families": [
        {
          "bc": "N",
          "label": "Universal constant only",
          "source_kind": "square",
          "constant_only": true,
          "source_boundary": [
            "N",
            "N",
            "N",
            "N"
          ],
          "boundary_edges": [
            "N",
            "N",
            "N"
          ],
          "modes": [
            {
              "id": "0:0",
              "m": 0,
              "n": 0,
              "q": 0,
              "normalized_value": 0,
              "x_kind": "cos",
              "y_kind": "cos",
              "constructed_multiplicity": 1
            }
          ]
        }
      ],
      "tiling_status": "Radin\u2019s pinwheel substitution gives nonperiodic tilings with infinitely many tile orientations. The bare right triangle also tiles periodically by half-turn pairs.",
      "source_url": "https://annals.math.princeton.edu/1994/139-3/p05",
      "matching_source_url": null
    },
    {
      "id": "penrose-thick",
      "label": "Penrose thick rhombus",
      "polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1.3090169943749475,
          0.9510565162951535
        ],
        [
          0.30901699437494745,
          0.9510565162951535
        ]
      ],
      "area": 0.9510565162951535,
      "lattice_basis": [
        [
          1,
          0
        ],
        [
          0,
          1
        ]
      ],
      "lattice_vertices": null,
      "boundary_certificates": [],
      "source_polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          0,
          1
        ]
      ],
      "families": [
        {
          "bc": "N",
          "label": "Universal constant only",
          "source_kind": "square",
          "constant_only": true,
          "source_boundary": [
            "N",
            "N",
            "N",
            "N"
          ],
          "boundary_edges": [
            "N",
            "N",
            "N",
            "N"
          ],
          "modes": [
            {
              "id": "0:0",
              "m": 0,
              "n": 0,
              "q": 0,
              "normalized_value": 0,
              "x_kind": "cos",
              "y_kind": "cos",
              "constructed_multiplicity": 1
            }
          ]
        }
      ],
      "tiling_status": "The thick and thin rhombi with Penrose matching rules form an aperiodic pair. Each bare rhombus is a parallelogram and tiles periodically; this operator does not yet encode the arrows.",
      "source_url": "https://www.math.brown.edu/~res/M272/pentagrid.pdf",
      "matching_source_url": null
    },
    {
      "id": "penrose-thin",
      "label": "Penrose thin rhombus",
      "polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1.8090169943749475,
          0.5877852522924731
        ],
        [
          0.8090169943749475,
          0.5877852522924731
        ]
      ],
      "area": 0.5877852522924731,
      "lattice_basis": [
        [
          1,
          0
        ],
        [
          0,
          1
        ]
      ],
      "lattice_vertices": null,
      "boundary_certificates": [],
      "source_polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          0,
          1
        ]
      ],
      "families": [
        {
          "bc": "N",
          "label": "Universal constant only",
          "source_kind": "square",
          "constant_only": true,
          "source_boundary": [
            "N",
            "N",
            "N",
            "N"
          ],
          "boundary_edges": [
            "N",
            "N",
            "N",
            "N"
          ],
          "modes": [
            {
              "id": "0:0",
              "m": 0,
              "n": 0,
              "q": 0,
              "normalized_value": 0,
              "x_kind": "cos",
              "y_kind": "cos",
              "constructed_multiplicity": 1
            }
          ]
        }
      ],
      "tiling_status": "The thick and thin rhombi with Penrose matching rules form an aperiodic pair. Each bare rhombus is a parallelogram and tiles periodically; this operator does not yet encode the arrows.",
      "source_url": "https://www.math.brown.edu/~res/M272/pentagrid.pdf",
      "matching_source_url": null
    },
    {
      "id": "ammann-beenker",
      "label": "Ammann\u2013Beenker rhombus",
      "polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1.7071067811865475,
          0.7071067811865476
        ],
        [
          0.7071067811865476,
          0.7071067811865476
        ]
      ],
      "area": 0.7071067811865475,
      "lattice_basis": [
        [
          1,
          0
        ],
        [
          0,
          1
        ]
      ],
      "lattice_vertices": null,
      "boundary_certificates": [],
      "source_polygon": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          0,
          1
        ]
      ],
      "families": [
        {
          "bc": "N",
          "label": "Universal constant only",
          "source_kind": "square",
          "constant_only": true,
          "source_boundary": [
            "N",
            "N",
            "N",
            "N"
          ],
          "boundary_edges": [
            "N",
            "N",
            "N",
            "N"
          ],
          "modes": [
            {
              "id": "0:0",
              "m": 0,
              "n": 0,
              "q": 0,
              "normalized_value": 0,
              "x_kind": "cos",
              "y_kind": "cos",
              "constructed_multiplicity": 1
            }
          ]
        }
      ],
      "tiling_status": "Beenker\u2019s tetragrid construction gives nonperiodic tilings by a unit square and a rhombus with an acute angle of forty-five degrees. The bare rhombus tiles periodically. His simple arrow conditions alone do not force every tiling to be nonperiodic.",
      "source_url": "https://pure.tue.nl/ws/files/4269555/253166.pdf",
      "matching_source_url": null
    }
  ],
  "scope": "Bare polygon D/N spectra. Nonperiodicity belongs to the cited substitution, matching rules or cut-and-project construction. No scalar spectrum here is an aperiodicity test."
}