Editing Wang tile (section)

==Aperiodic sets of tiles==
[[File:Wang_11_tiles_monochromatic.svg|thumb|Wang tiles made monochromatic by replacing edges of each quadrant with a shape corresponding on its colour &ndash; this set is isomorphic to Jeandel and Rao's minimal set above]]
Combining Berger's undecidability result with Wang's observation shows that there must exist a finite set of Wang tiles that tiles the plane, but only ''[[aperiodic tiling|aperiodically]]''.  This is similar to a [[Penrose tiling]], or the arrangement of atoms in a [[quasicrystal]].  Although Berger's original set contained 20,426 tiles, he conjectured that smaller sets would work, including subsets of his set, and in his unpublished Ph.D. thesis, he reduced the number of tiles to 104.  In later years, ever smaller sets were found.<ref>{{citation
 | last = Robinson | first = Raphael M. | author-link = Raphael Robinson
 | journal = [[Inventiones Mathematicae]]
 | mr = 0297572
 | pages = 177–209
 | title = Undecidability and non periodicity for tilings of the plane
 | volume = 12
 | year = 1971
 | issue = 3 | doi=10.1007/bf01418780| bibcode = 1971InMat..12..177R| s2cid = 14259496 }}.</ref><ref name="culik">{{citation
 | last = Culik | first = Karel II
 | doi = 10.1016/S0012-365X(96)00118-5
 | issue = 1–3
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 1417576
 | pages = 245–251
 | title = An aperiodic set of 13 Wang tiles
 | volume = 160
 | year = 1996| doi-access = free
 }}. (Showed an aperiodic set of 13 tiles with 5 colors.)</ref><ref>{{citation
 | last = Kari | first = Jarkko | author-link = Jarkko Kari
 | doi = 10.1016/0012-365X(95)00120-L
 | issue = 1–3
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 1417578
 | pages = 259–264
 | title = A small aperiodic set of Wang tiles
 | volume = 160
 | year = 1996| doi-access = free
 }}.</ref><ref name="jeandel">{{citation
 | last1 = Jeandel | first1 = Emmanuel
 | last2 = Rao | first2 = Michaël
 | arxiv = 1506.06492
 | doi = 10.19086/aic.18614
 | journal = Advances in Combinatorics
 | mr = 4210631
 | page = 1:1–1:37
 | title = An aperiodic set of 11 Wang tiles
 | year = 2021| s2cid = 13261182
 }}. (Showed an aperiodic set of 11 tiles with 4 colors, and proved that fewer tiles or fewer colors cannot be aperiodic.)</ref> For example, a set of 13 aperiodic tiles was published by Karel Culik II in 1996.<ref name="culik"/>

The smallest set of aperiodic tiles was discovered by Emmanuel Jeandel and Michael Rao in 2015, with 11 tiles and 4 colors. They used an exhaustive computer search to prove that 10 tiles or 3 colors are insufficient to force aperiodicity.<ref name="jeandel" /> This set, shown above in the title image, can be examined more closely at [[:File: Wang 11 tiles.svg]].