{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T04:53:34Z","timestamp":1774500814519,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2012,10,22]],"date-time":"2012-10-22T00:00:00Z","timestamp":1350864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.<\/jats:p>","DOI":"10.3390\/sym4040581","type":"journal-article","created":{"date-parts":[[2012,10,22]],"date-time":"2012-10-22T11:27:55Z","timestamp":1350905275000},"page":"581-602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Hexagonal Inflation Tilings and Planar Monotiles"],"prefix":"10.3390","volume":"4","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany"}]},{"given":"Franz","family":"G\u00e4hler","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Bielefeld, Postfach 100131, Bielefeld 33501, Germany"}]},{"given":"Uwe","family":"Grimm","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK"}]}],"member":"1968","published-online":{"date-parts":[[2012,10,22]]},"reference":[{"key":"ref_1","unstructured":"Gr\u00fcnbaum, B., and Shephard, G.C. 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