Catalog of Isohedral Tilings

From the article "One Corona is Enough for the Euclidean Plane."Authors:
Doris Schattschneider
Moravian College
1200 Main St.
Bethlehem, PA 180186650 USA
<schattdo@moravian.edu>
Nikolai Dolbilin
Steklov Mathematical Institute
Gubkin 8, 117966 Moscow
GSP1 Russia
<nikolai@dolbilin.mian.su>Published in "Quasicrystals and Discrete Geometry," J. Patera, editor, The Fields Institute for Research in Mathematical Sciences Monograph Series, Vol. 10, AMS, Providence, RI, 1998, pp. 207246.
In the article cited above, tilings* of the Euclidean plane by a single polygon are considered, and it is shown that such a tiling is isohedral if and only if each polygon is "surrounded in the same way," or, more technically, the centered coronas of tiles are pairwise congruent.
(*In these tilings, at each vertex of each polygon, three or more polygons must meet.)
When the polygon tile is asymmetric, the proof of the result is a consequence of the local theorem for tilings, a general result that holds for tilings in ddimensional space (see N. Dolbilin and D. Schattschneider, "The Local Theorem for Tilings," in the volume mentioned above, pp. 193199). Therefore the result needs to be verified only for symmetric polygons. The proof is casebycase, constructing all possible tilings with a single symmetric polygon tile that satisfies the condition on pairwise congruent coronas. All are shown to be isohedral. One consequence of the proof is the production of this complete catalog of isohedral tilings by symmetric polygon tiles.
About onequarter of the 42 different types of tilings are rigid; that is, they are unique up to similarity. However, the remaining tilings are highly flexible, with the shape of the tile able to be deformed in a continuous manner to assume a variety of distinct shapes. The tilings were constructed using The Geometer's Sketchpad, and the dynamic action of that program allows you to explore the range of shapes of these tiles and their associated tilings by simply dragging on a vertex of a tile.
This web site allows you to view the whole catalog, and also to view the deformation of the flexible tilings interactively, using the JavaSketchpad version of the tilings. You can also download the Sketchpad files to your computer if you wish to explore them further.
Each sketch shows a patch of a tiling with a central tile surrounded by shaded copies of the tile. The set consisting of the central tile and the shaded surrounding tiles is called a centered corona of the tiling. (Formally, the corona of a tile is the set of all tiles that have nonempty intersection with the tile.) The Laves net of the tiling is given, and any constraints on the tile are described. When the tile (and hence its tiling) is flexible, a "play" button is shown. This link leads to the interactive JavaSketchpad version of the sketch in which one or more vertices is highlighted. Drag on a highlighted vertex to deform the tile and its tiling and view the wide range of shapes that the tile can assume. In many cases, one degree of freedom has been fixed, so the range of shapes does not show all possible sizes.
Further details on each of the tilings (as well as their construction) can be found in the article cited at the top of this page.
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