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Wang tiles (or Wang dominoes), first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems. They are modelled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, without rotating or reflecting them.

The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern.

Domino problem

In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a periodic tiling, which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane.^[1]^[2] The idea of constraining adjacent tiles to match each other occurs in the game of dominoes, so Wang tiles are also known as Wang dominoes.^[3] The algorithmic problem of determining whether a tile set can tile the plane became known as the domino problem.^[4]

According to Wang's student, Robert Berger,^[4]

The Domino Problem deals with the class of all domino sets. It consists of deciding, for each domino set, whether or not it is solvable. We say that the Domino Problem is decidable or undecidable according to whether there exists or does not exist an algorithm which, given the specifications of an arbitrary domino set, will decide whether or not the set is solvable.

In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets.

In 1966, Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem.^[4]

Aperiodic sets of tiles

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 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.^[5]^[6]^[7]^[8] For example, a set of 13 aperiodic tiles was published by Karel Culik II in 1996.^[6]

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.^[8] This set, shown above in the title image, can be examined more closely at File:Wang 11 tiles.svg.

Generalizations

Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense. For example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes.^[9] Winfree et al. have demonstrated the feasibility of creating molecular "tiles" made from DNA (deoxyribonucleic acid) that can act as Wang tiles.^[10] Mittal et al. have shown that these tiles can also be composed of peptide nucleic acid (PNA), a stable artificial mimic of DNA.^[11]

Applications

Wang tiles have been used for procedural synthesis of textures, heightfields, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity. In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly.^[12]^[13]^[14]^[15]^[16]

Wang tiles have also been used in cellular automata theory decidability proofs.^[17]

In popular culture

The short story "Wang's Carpets", later expanded to the novel Diaspora, by Greg Egan, postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules.^[18]

References

^ Wang, Hao (1961), "Proving theorems by pattern recognition—II", Bell System Technical Journal, 40 (1): 1–41, doi:10.1002/j.1538-7305.1961.tb03975.x. Wang proposes his tiles, and conjectures that there are no aperiodic sets.
^ Wang, Hao (November 1965), "Games, logic and computers", Scientific American, 213 (5): 98–106, doi:10.1038/scientificamerican1165-98. Presents the domino problem for a popular audience.
^ Renz, Peter (1981), "Mathematical proof: What it is and what it ought to be", The Two-Year College Mathematics Journal, 12 (2): 83–103, doi:10.2307/3027370, JSTOR 3027370.
^ ^a ^b ^c Berger, Robert (1966), "The undecidability of the domino problem", Memoirs of the American Mathematical Society, 66: 72, MR 0216954. Berger coins the term "Wang tiles", and demonstrates the first aperiodic set of them.
^ Robinson, Raphael M. (1971), "Undecidability and nonperiodicity for tilings of the plane", Inventiones Mathematicae, 12 (3): 177–209, Bibcode:1971InMat..12..177R, doi:10.1007/bf01418780, MR 0297572, S2CID 14259496.
^ ^a ^b Culik, Karel II (1996), "An aperiodic set of 13 Wang tiles", Discrete Mathematics, 160 (1–3): 245–251, doi:10.1016/S0012-365X(96)00118-5, MR 1417576. (Showed an aperiodic set of 13 tiles with 5 colors.)
^ Kari, Jarkko (1996), "A small aperiodic set of Wang tiles", Discrete Mathematics, 160 (1–3): 259–264, doi:10.1016/0012-365X(95)00120-L, MR 1417578.
^ ^a ^b Jeandel, Emmanuel; Rao, Michaël (2021), "An aperiodic set of 11 Wang tiles", Advances in Combinatorics: 1:1–1:37, arXiv:1506.06492, doi:10.19086/aic.18614, MR 4210631, S2CID 13261182. (Showed an aperiodic set of 11 tiles with 4 colors, and proved that fewer tiles or fewer colors cannot be aperiodic.)
^ Culik, Karel II; Kari, Jarkko (1995), "An aperiodic set of Wang cubes", Journal of Universal Computer Science, 1 (10): 675–686, doi:10.1007/978-3-642-80350-5_57, MR 1392428.
^ Winfree, E.; Liu, F.; Wenzler, L.A.; Seeman, N.C. (1998), "Design and self-assembly of two-dimensional DNA crystals", Nature, 394 (6693): 539–544, Bibcode:1998Natur.394..539W, doi:10.1038/28998, PMID 9707114, S2CID 4385579.
^ Lukeman, P.; Seeman, N.; Mittal, A. (2002), "Hybrid PNA/DNA nanosystems", 1st International Conference on Nanoscale/Molecular Mechanics (N-M2-I), Outrigger Wailea Resort, Maui, Hawaii.
^ Stam, Jos (1997), Aperiodic Texture Mapping (PDF). Introduces the idea of using Wang tiles for texture variation, with a deterministic substitution system.
^ Neyret, Fabrice; Cani, Marie-Paule (1999), "Pattern-Based Texturing Revisited", Proceedings of the 26th annual conference on Computer graphics and interactive techniques - SIGGRAPH '99 (PDF), Los Angeles, United States: ACM, pp. 235–242, doi:10.1145/311535.311561, ISBN 0-201-48560-5, S2CID 11247905. Introduces stochastic tiling.
^ Cohen, Michael F.; Shade, Jonathan; Hiller, Stefan; Deussen, Oliver (2003), "Wang tiles for image and texture generation", ACM SIGGRAPH 2003 Papers on - SIGGRAPH '03 (PDF), New York, NY, USA: ACM, pp. 287–294, doi:10.1145/1201775.882265, ISBN 1-58113-709-5, S2CID 207162102, archived from the original (PDF) on 2006-03-18.
^ Wei, Li-Yi (2004), "Tile-based texture mapping on graphics hardware", Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Conference on Graphics Hardware (HWWS '04), New York, NY, USA: ACM, pp. 55–63, doi:10.1145/1058129.1058138, ISBN 3-905673-15-0, S2CID 53224612. Applies Wang Tiles for real-time texturing on a GPU.
^ . Kopf, Johannes; Cohen-Or, Daniel; Deussen, Oliver; Lischinski, Dani (2006), "Recursive Wang tiles for real-time blue noise", ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06, New York, NY, USA: ACM, pp. 509–518, doi:10.1145/1179352.1141916, ISBN 1-59593-364-6, S2CID 11007853. Shows advanced applications.
^ Kari, Jarkko (1990), "Reversibility of 2D cellular automata is undecidable", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, vol. 45, pp. 379–385, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882.
^ Burnham, Karen (2014), Greg Egan, Modern Masters of Science Fiction, University of Illinois Press, pp. 72–73, ISBN 978-0-252-09629-7.

External links

Steven Dutch's page including many pictures of aperiodic tilings
Animated demonstration of a naïve Wang tiling method - requires Javascript and HTML5

[1] Wang, Hao (1961), "Proving theorems by pattern recognition—II", Bell System Technical Journal, 40 (1): 1–41, doi:10.1002/j.1538-7305.1961.tb03975.x. Wang proposes his tiles, and conjectures that there are no aperiodic sets.

[2] Wang, Hao (November 1965), "Games, logic and computers", Scientific American, 213 (5): 98–106, doi:10.1038/scientificamerican1165-98. Presents the domino problem for a popular audience.

[3] Renz, Peter (1981), "Mathematical proof: What it is and what it ought to be", The Two-Year College Mathematics Journal, 12 (2): 83–103, doi:10.2307/3027370, JSTOR 3027370.

[berger-4] Berger, Robert (1966), "The undecidability of the domino problem", Memoirs of the American Mathematical Society, 66: 72, MR 0216954. Berger coins the term "Wang tiles", and demonstrates the first aperiodic set of them.

[5] Robinson, Raphael M. (1971), "Undecidability and nonperiodicity for tilings of the plane", Inventiones Mathematicae, 12 (3): 177–209, Bibcode:1971InMat..12..177R, doi:10.1007/bf01418780, MR 0297572, S2CID 14259496.

[culik-6] Culik, Karel II (1996), "An aperiodic set of 13 Wang tiles", Discrete Mathematics, 160 (1–3): 245–251, doi:10.1016/S0012-365X(96)00118-5, MR 1417576. (Showed an aperiodic set of 13 tiles with 5 colors.)

[7] Kari, Jarkko (1996), "A small aperiodic set of Wang tiles", Discrete Mathematics, 160 (1–3): 259–264, doi:10.1016/0012-365X(95)00120-L, MR 1417578.

[jeandel-8] Jeandel, Emmanuel; Rao, Michaël (2021), "An aperiodic set of 11 Wang tiles", Advances in Combinatorics: 1:1–1:37, arXiv:1506.06492, doi:10.19086/aic.18614, MR 4210631, S2CID 13261182. (Showed an aperiodic set of 11 tiles with 4 colors, and proved that fewer tiles or fewer colors cannot be aperiodic.)

[9] Culik, Karel II; Kari, Jarkko (1995), "An aperiodic set of Wang cubes", Journal of Universal Computer Science, 1 (10): 675–686, doi:10.1007/978-3-642-80350-5_57, MR 1392428.

[10] Winfree, E.; Liu, F.; Wenzler, L.A.; Seeman, N.C. (1998), "Design and self-assembly of two-dimensional DNA crystals", Nature, 394 (6693): 539–544, Bibcode:1998Natur.394..539W, doi:10.1038/28998, PMID 9707114, S2CID 4385579.

[11] Lukeman, P.; Seeman, N.; Mittal, A. (2002), "Hybrid PNA/DNA nanosystems", 1st International Conference on Nanoscale/Molecular Mechanics (N-M2-I), Outrigger Wailea Resort, Maui, Hawaii.

[12] Stam, Jos (1997), Aperiodic Texture Mapping (PDF). Introduces the idea of using Wang tiles for texture variation, with a deterministic substitution system.

[13] Neyret, Fabrice; Cani, Marie-Paule (1999), "Pattern-Based Texturing Revisited", Proceedings of the 26th annual conference on Computer graphics and interactive techniques - SIGGRAPH '99 (PDF), Los Angeles, United States: ACM, pp. 235–242, doi:10.1145/311535.311561, ISBN 0-201-48560-5, S2CID 11247905. Introduces stochastic tiling.

[14] Cohen, Michael F.; Shade, Jonathan; Hiller, Stefan; Deussen, Oliver (2003), "Wang tiles for image and texture generation", ACM SIGGRAPH 2003 Papers on - SIGGRAPH '03 (PDF), New York, NY, USA: ACM, pp. 287–294, doi:10.1145/1201775.882265, ISBN 1-58113-709-5, S2CID 207162102, archived from the original (PDF) on 2006-03-18.

[15] Wei, Li-Yi (2004), "Tile-based texture mapping on graphics hardware", Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Conference on Graphics Hardware (HWWS '04), New York, NY, USA: ACM, pp. 55–63, doi:10.1145/1058129.1058138, ISBN 3-905673-15-0, S2CID 53224612. Applies Wang Tiles for real-time texturing on a GPU.

[16] . Kopf, Johannes; Cohen-Or, Daniel; Deussen, Oliver; Lischinski, Dani (2006), "Recursive Wang tiles for real-time blue noise", ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06, New York, NY, USA: ACM, pp. 509–518, doi:10.1145/1179352.1141916, ISBN 1-59593-364-6, S2CID 11007853. Shows advanced applications.

[17] Kari, Jarkko (1990), "Reversibility of 2D cellular automata is undecidable", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, vol. 45, pp. 379–385, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882.

[18] Burnham, Karen (2014), Greg Egan, Modern Masters of Science Fiction, University of Illinois Press, pp. 72–73, ISBN 978-0-252-09629-7.

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@@ Line 1: / Line 1: @@
+{{Short description|Square tiles with a color on each edge}}
 [[File:Wang 11 tiles.svg|thumb|This set of 11 Wang tiles will tile the plane but only [[aperiodic tiling|aperiodically]].]]
 {{Commons category|Wang tiles}}
@@ Line 8: / Line 9: @@
 ==Domino problem==
 [[Image:Wang tesselation.svg|thumb|400px|Example of Wang tessellation with 13 tiles.]]
-In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there exists also a [[tessellation|''periodic'' tiling]], i.e., a tiling that is invariant under translations by vectors in a 2-dimensional lattice, like a wallpaper pattern.  He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane.<ref>{{citation
+In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there also exists a [[tessellation|''periodic'' tiling]], which, mathematically, is a tiling that is invariant under translations by vectors in a 2-dimensional lattice. This can be likened to the periodic tiling in a wallpaper pattern, where the overall pattern is a repetition of some smaller pattern. He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane.<ref>{{citation
  | last = Wang | first = Hao | author-link = Hao Wang (academic)
  | issue = 1
@@ Line 49: / Line 50: @@
 ==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, increasingly smaller sets were found.<ref>{{citation
+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]]
@@ Line 57: / Line 58: @@
  | volume = 12
  | year = 1971
- | issue = 3 | doi=10.1007/bf01418780| bibcode = 1971InMat..12..177R}}.</ref><ref name="culik">{{citation
+ | issue = 3 | doi=10.1007/bf01418780| bibcode = 1971InMat..12..177R| s2cid = 14259496 }}.</ref><ref name="culik">{{citation
- | last = Culik | first = Karel, II
+ | last = Culik | first = Karel II
  | doi = 10.1016/S0012-365X(96)00118-5
  | issue = 1–3
@@ Line 67: / Line 68: @@
  | volume = 160
  | year = 1996| doi-access = free
- }}. (Showed an aperiodic set of 13 tiles with 5 colors).</ref><ref>{{citation
+ }}. (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
@@ Line 78: / Line 79: @@
  | year = 1996| doi-access = free
  }}.</ref><ref name="jeandel">{{citation
- | last1 = Jeandel | first1 =  Emmanuel
+ | last1 = Jeandel | first1 = Emmanuel
- | last2 = Rao | first2 = Michael
+ | last2 = Rao | first2 = Michaël
+ | arxiv = 1506.06492
- | journal = [[Computing Research Repository|CoRR]]
- | title = An aperiodic set of 11 Wang tiles
- | year = 2021
  | doi = 10.19086/aic.18614
+ | journal = Advances in Combinatorics
- | arxiv = 1506.06492| bibcode = 2015arXiv150606492J}}. (Showed an aperiodic set of 11 tiles with 4 colors)}</ref> For example, a set of 13 aperiodic tiles was published by Karel Culik II in 1996.<ref name="culik"/>
+ | 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]].
@@ Line 91: / Line 95: @@
 Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense.  For example, ''Wang cubes'' are equal-sized cubes with colored faces and side colors can be matched on any polygonal [[tessellation]].
 Culik and Kari have demonstrated aperiodic sets of Wang cubes.<ref>{{citation
- | last1 = Culik | first1 = Karel, II
+ | last1 = Culik | first1 = Karel II
  | last2 = Kari | first2 = Jarkko | author2-link = Jarkko Kari
  | issue = 10
@@ Line 111: / Line 115: @@
  | year = 1998
  | issue = 6693
- | doi = 10.1038/28998 | pmid=9707114| bibcode = 1998Natur.394..539W}}.</ref> Mittal et al. have shown that these tiles can also be composed of [[peptide nucleic acid]] (PNA), a stable artificial mimic of DNA.<ref>{{citation
+ | doi = 10.1038/28998 | pmid=9707114| bibcode = 1998Natur.394..539W| s2cid = 4385579
+ }}.</ref> Mittal et al. have shown that these tiles can also be composed of [[peptide nucleic acid]] (PNA), a stable artificial mimic of DNA.<ref>{{citation
  | last1 = Lukeman | first1 = P.
  | last2 = Seeman | first2 = N.
@@ Line 120: / Line 125: @@
 ==Applications==
-Wang tiles have recently become a popular tool for [[procedural synthesis]] of textures, [[heightfield]]s, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity.
+Wang tiles have been used for [[procedural synthesis]] of [[procedural texture|textures]], [[heightfield]]s, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity.
 In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly.<ref>{{citation
  | last = Stam | first = Jos
@@ Line 135: / Line 140: @@
  | publisher = ACM
  | url = https://hal.inria.fr/inria-00537511/file/patternTexture.pdf
- | year = 1999| isbn = 0201485605
+ | year = 1999| isbn = 0-201-48560-5
+ | s2cid = 11247905
  }}. Introduces stochastic tiling.</ref><ref>{{citation
  | last1 = Cohen | first1 = Michael F. |author1-link=Michael F. Cohen
@@ Line 147: / Line 153: @@
  | pages = 287–294
  | publisher = ACM
- | url = http://research.microsoft.com/~cohen/WangFinal.pdf | archive-url = https://web.archive.org/web/20060318064425/http://research.microsoft.com/~cohen/WangFinal.pdf | url-status = dead | archive-date = 2006-03-18 | year = 2003}}. </ref><ref>{{citation
+ | url = http://research.microsoft.com/~cohen/WangFinal.pdf | archive-url = https://web.archive.org/web/20060318064425/http://research.microsoft.com/~cohen/WangFinal.pdf  | archive-date = 2006-03-18 | year = 2003| s2cid = 207162102 }}. </ref><ref>{{citation
  | last = Wei | first = Li-Yi
  | contribution = Tile-based texture mapping on graphics hardware
@@ Line 157: / Line 163: @@
  | title = Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Conference on Graphics Hardware (HWWS '04)
  | url = http://graphics.stanford.edu/papers/tile_mapping_gh2004/
- | year = 2004}}. Applies Wang Tiles for real-time texturing on a GPU.</ref><ref>. {{citation
+ | year = 2004| s2cid = 53224612
+ }}. Applies Wang Tiles for real-time texturing on a GPU.</ref><ref>. {{citation
  | last1 = Kopf | first1 = Johannes
  | last2 = Cohen-Or | first2 = Daniel
@@ Line 170: / Line 177: @@
  | publisher = ACM
  | url = http://johanneskopf.de/publications/blue_noise
- | year = 2006}}. Shows advanced applications.</ref>
+ | year = 2006| s2cid = 11007853
+ }}. Shows advanced applications.</ref>
 Wang tiles have also been used in [[cellular automaton|cellular automata theory]] [[decision problem|decidability]] proofs.<ref>{{citation
@@ Line 186: / Line 194: @@
 ==In popular culture==
-The short story ''Wang's Carpets'', later expanded to the novel ''[[Diaspora (novel)|Diaspora]]'', by [[Greg Egan]], postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules.<ref>{{citation
+The short story "[[Wang's Carpets]]", later expanded to the novel ''[[Diaspora (novel)|Diaspora]]'', by [[Greg Egan]], postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules.<ref>{{citation
  | last = Burnham | first = Karen
- | isbn = 9780252096297
+ | isbn = 978-0-252-09629-7
  | pages = 72–73
  | publisher = University of Illinois Press
@@ Line 198: / Line 206: @@
 ==See also==
 * [[Edge-matching puzzle]]
-* [[Eternity II puzzle]]
-* [[Percy Alexander MacMahon]]
 ==References==
@@ Line 217: / Line 223: @@
 [[Category:Euclidean tilings]]
 [[Category:Theory of computation]]
+[[Category:Undecidable problems]]