Article Instance

HTTP 200 OK
Allow: GET
Content-Type: application/json
Vary: Accept

{
    "pk": 64930,
    "title": "An aperiodic monotile",
    "subtitle": null,
    "abstract": "A longstanding open problem asks for an aperiodic monotile, also known as an \"einstein\": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the \"hat\" polykite, can form clusters called \"metatiles\", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical--and hence aperiodic--tilings.\n \nMathematics Subject Classifications: 05B45, 52C20, 05B50\n \nKeywords: Tilings, aperiodic order, polyforms",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Tilings"
        },
        {
            "word": "aperiodic order"
        },
        {
            "word": "polyforms"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3317z9z9",
    "frozenauthors": [
        {
            "first_name": "David",
            "middle_name": "",
            "last_name": "Smith",
            "name_suffix": "",
            "institution": "Yorkshire, U.K.",
            "department": ""
        },
        {
            "first_name": "Joseph",
            "middle_name": "Samuel",
            "last_name": "Myers",
            "name_suffix": "",
            "institution": "Cambridge, U.K.",
            "department": ""
        },
        {
            "first_name": "Craig",
            "middle_name": "S.",
            "last_name": "Kaplan",
            "name_suffix": "",
            "institution": "School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada",
            "department": ""
        },
        {
            "first_name": "Chaim",
            "middle_name": "",
            "last_name": "Goodman-Strauss",
            "name_suffix": "",
            "institution": "National Museum of Mathematics, New York, New York, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2024-07-01T11:34:59+02:00",
    "date_accepted": "2024-07-01T11:34:59+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64930/galley/49740/download/"
        }
    ]
}