Editing Polyomino (section)

===Tiling the plane with copies of a single polyomino===
[[File:Conway criterion false negative nonominoes.svg|thumb|180px|The two tiling nonominoes not satisfying the Conway criterion.]]
Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes<ref>{{cite journal |last=Gardner |first=Martin |date=July 1965 |title=On the relation between mathematics and the ordered patterns of Op art |journal=[[Scientific American]] |volume=213 |issue=1 |pages=100–104|doi=10.1038/scientificamerican1265-100 }}</ref> and all but four heptominoes tile the plane.<ref>{{cite journal |last=Gardner |first=Martin |date=August 1965 |title=Thoughts on the task of communication with intelligent organisms on other worlds |journal=[[Scientific American]] |volume=213 |issue=2 |pages=96–100|doi=10.1038/scientificamerican0865-96 }}</ref> It was then established by David Bird that all but 26 octominoes tile the plane.<ref>{{cite journal |last=Gardner |first=Martin |date=August 1975 |title=More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes |journal=[[Scientific American]] |volume=233 |issue=2 |pages=112–115|doi=10.1038/scientificamerican0875-112 }}</ref> Rawsthorne found that all but 235 polyominoes of size 9 tile,<ref>{{cite journal |last=Rawsthorne |first=Daniel A. |year=1988 |title=Tiling complexity of small ''n''-ominoes<br/>(''n''<10) |journal=Discrete Mathematics |volume=70 |pages=71–75 |doi=10.1016/0012-365X(88)90081-7|doi-access=free }}</ref> and such results have been extended to higher area by Rhoads (to size 14)<ref>{{cite book |last=Rhoads |first=Glenn C. |title=Planar Tilings and the Search for an Aperiodic Prototile |year=2003 |publisher=PhD dissertation, Rutgers University}}</ref> and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.<ref>Grünbaum and Shephard, section 9.4</ref><ref>{{cite journal |last=Keating |first=K. |author2=Vince, A. |year=1999 |title=Isohedral Polyomino Tiling of the Plane |journal=[[Discrete & Computational Geometry]] |volume=21 |pages=615–630 |doi=10.1007/PL00009442 |issue=4|doi-access=free }}</ref>

The study of which polyominoes can tile the plane has been facilitated using the [[Conway criterion]]: except for two nonominoes, all tiling polyominoes up to size 9 form a patch of at least one tile satisfying it, with higher-size exceptions more frequent.<ref>{{cite journal |last=Rhoads |first=Glenn C. |year=2005 |title=Planar tilings by polyominoes, polyhexes, and polyiamonds |journal=Journal of Computational and Applied Mathematics |volume=174 |pages=329–353 |doi=10.1016/j.cam.2004.05.002 |issue=2|bibcode=2005JCoAM.174..329R |doi-access=free }}</ref>

Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a [[rep-tile]] tiling of the plane. For instance, for every positive integer {{mvar|n}}, it is possible to combine {{math|''n''<sup>2</sup>}} copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.<ref>{{citation
 | last = Niţică | first = Viorel
 | contribution = Rep-tiles revisited
 | location = Providence, RI
 | mr = 2027179
 | pages = 205–217
 | publisher = American Mathematical Society
 | title = MASS selecta
 | year = 2003
 | mode = cs1}}</ref>
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