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the other two. Quasicrystal structures of Cd–Te appear to consist of atomic layers in which the atoms are arranged in a planar aperiodic pattern. Sometimes an energetical minimum or a maximum of entropy occur for such aperiodic structures. Steinhardt has shown that
Gummelt's overlapping decagons allow the application of an extremal principle and thus provide the link between the mathematics of aperiodic tiling and the structure of quasicrystals.
643:
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122:). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrarily large periodic parts.
884:
had already extended the
Penrose construction to a three-dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in
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originally consists of four prototiles together with some matching rules. One of the four tiles is a pentagon. One can replace this pentagon prototile by three distinct pentagonal shapes that have additional protrusions and indentations at the boundary making three distinct tiles. Together with the
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For a tiling congruent copies of the prototiles need to pave all of the
Euclidean plane without overlaps (except at boundaries) and without leaving uncovered pieces. Therefore the boundaries of the tiles forming a tiling need to match geometrically. This is generally true for all tilings, aperiodic
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There are a few constructions of aperiodic tilings known. Some constructions are based on infinite families of aperiodic sets of tiles. The tilings which have been found so far are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this,
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Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. (In many later examples, this structure can be described as a substitution tiling system; this is described below). No tiling admitted by such a set of tiles can be periodic, simply because no single
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Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic. The
Penrose tiles are the first and most famous example of
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tiling is simply one that is not fixed by any non-trivial translation. Sometimes the term described – implicitly or explicitly – a tiling generated by an aperiodic set of prototiles. Frequently the term aperiodic was just used vaguely to describe the structures under consideration, referring to
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is decidable – that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets that cannot tile the plane, and the tilesets that tile it periodically; by this he showed that such a decision
906:
has been used in a wide variety of ways in the mathematical literature on tilings (and in other mathematical fields as well, such as dynamical systems or graph theory, with altogether different meanings). With respect to tilings the term aperiodic was sometimes used synonymously with the term
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the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non-periodic, in precisely the same manner as described above, but the chair tile itself is not
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Robinson proves these tiles must form this structure inductively; in effect, the tiles must form blocks which themselves fit together as larger versions of the original tiles, and so on. This idea – of finding sets of tiles that can only admit hierarchical structures – has been used in the
605:
three other prototiles with suitably adapted boundaries one gets a set of six prototiles that essentially create the same aperiodic tilings as the original four tiles, but for the six tiles no additional matching rules are necessary, the geometric matching condition suffice.
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Any tiling by these tiles can only exhibit a hierarchy of square lattices: the centre of any orange square is also a corner of a larger orange square, ad infinitum. Any translation must be smaller than some size of square, and so cannot leave any such tiling invariant.
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Some tilings obtained by the cut and project method. The cutting planes are all parallel to the one which defines
Penrose tilings (the fourth tiling on the third line). These tilings are all in different local isomorphism classes, that is, they are locally
564:
Goodman-Straus proved that all tilings generated by substitution rules and satisfying a technical condition can be generated through matching rules. The technical condition is mild and usually satisfied in practice. The tiles are required to admit a set of
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found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable. This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and
608:
Also note that
Robinsion's protiles below come equipped with markings to make it easier to visually recognize the structure, but these markings do not put more matching rules on the tiles as are already in place through the geometric boundaries.
919:, for example: the Penrose rhombs admit infinitely many tilings (which cannot be distinguished locally). A common solution is to try to use the terms carefully in technical writing, but recognize the widespread use of the informal terms.
879:
announced the discovery of a phase of an aluminium-manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry – so it had to be a crystalline substance with icosahedral symmetry. In 1975
889:
have been observed to form large patches of aperiodic patterns. The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with
635:
For aperiodic tilings, whether additional matching rules are involved or not, the matching conditions forces some hierarchical structure on the tilings that in turn make period structures impossible.
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235:
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ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.
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To date, there is not a formal definition describing when a tiling has a hierarchical structure; nonetheless, it is clear that substitution tilings have them, as do the tilings of Berger,
804:. There is yet no complete (algebraic) characterization of cut and project tilings that can be enforced by matching rules, although numerous necessary or sufficient conditions are known.
407:
162:
96:, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints.
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In some cases it has been possible to replace matching rules by geometric matching conditions altogether by modifying the prototiles at their boundary. The
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tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure".
425:
429:
617:
489:. After the discovery of quasicrystals aperiodic tilings become studied intensively by physicists and mathematicians. The cut-and-project method of
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Substitution tiling systems provide a rich source of aperiodic tilings. A set of tiles that forces a substitution structure to emerge is said to
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For some tilings only one of the constructions is known to yield that tiling. Others can be constructed by all three classical methods, e.g. the
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subsequently found an aperiodic set requiring only 40 Wang tiles. A smaller set, of six aperiodic tiles (based on Wang tiles), was discovered by
126:
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gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to
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gave the first general construction, showing that every product of one-dimensional substitution systems can be enforced by matching rules.
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enforce the chair substitution structure—they can only admit tilings in which the chair substitution can be discerned and so are aperiodic.
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many different tilings unrelated by
Euclidean isometries, all of them necessarily nonperiodic, that can arise from the Robinsion's tiles.
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However, there are three principles of construction that have been predominantly used for finite sets of prototiles up until 2023:
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who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood.
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discovered several new sets in 1977. The number of tiles required was reduced to one in 2023 by David Smith, Joseph Samuel Myers,
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several different sets of tiles, were the first example based on explicitly forcing a substitution tiling structure to emerge.
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has found many alternative constructions of aperiodic sets of tiles, some in more exotic settings; for example in semi-simple
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For well-behaved tilings (e.g. substitution tilings with finitely many local patterns) holds: if a tiling is non-periodic and
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algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling. In 1964,
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with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or
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showed that local matching rules can be found to force any substitution tiling structure, subject to some mild conditions.
1519:
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Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023-05-28). "A chiral aperiodic monotile".
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aperiodic – it is easy to find periodic tilings by unmarked chair tiles that satisfy the geometric matching conditions.
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are required to hold. These usually involve colors or markings that have to match over several tiles across boundaries.
523:, which is however not connected into one piece. In 2023 a connected tile was discovered, using a shape termed a "hat".
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Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023-03-19). "An aperiodic monotile".
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However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic.
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The use of the word "tiling" is problematic as well, despite its straightforward definition. There is no single
519:, one stone) is an aperiodic tiling that uses only a single shape. The first such tile was discovered in 2010 -
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physical aperiodic solids, namely quasicrystals, or to something non-periodic with some kind of global order.
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is enough to force a tile set to be aperiodic, this is e.g. the case for
Robinsion's tilings discussed below.
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833:), with the aperiodicity mainly relying on the fact that 2/3 is never equal to 1 for any positive integers
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The aperiodic
Penrose tilings can be generated not only by an aperiodic set of prototiles, but also by a
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translation can leave the entire hierarchical structure invariant. Consider
Robinson's 1971 tiles:
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240:
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853:. Block and Weinberger used homological methods to construct aperiodic sets of tiles for all non-
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40:
28:
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in the local topology. In the local topology (resp. the corresponding metric) two tilings are
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discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and
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2001:
1944:
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Robinson, Raphael M. (1971). "Undecidability and Nonperiodicity for Tilings of the Plane".
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980:
861:. This generally leads to much smaller tile sets than the one derived from substitutions.
8:
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Edwards, W.; Fauve, S. (1993). "Parametrically excited quasicrystalline surface waves".
2085:
2005:
1948:
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The Mathematics Long Range Aperiodic Order, NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci
1716:
1363:
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984:
682:. A complete tiling of the plane constructed from Robinsion's tiles may or may not have
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2017:
1972:
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Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist
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around the origin (possibly after shifting one of the tilings by an amount less than
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93:
89:
19:
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Le, T.T.Q. (1997). "Local Rules for Quasiperiodic Tilings". In Moody, R.V. (ed.).
1461:
1091:"Metallic Phase with long-range orientational order and no translational symmetry"
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2445:
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1876:
1871:. NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci. Vol. 489. pp. 331–366.
1517:(1989). "Tilings, substitution systems and dynamical systems generated by them".
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A tiling is called aperiodic if its hull contains only non-periodic tilings. The
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The first specific occurrence of aperiodic tilings arose in 1961, when logician
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24:
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To give an even simpler example than above, consider a one-dimensional tiling
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for Penrose tilings eventually turned out to be an instance of the theory of
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104:
35:
857:. Joshua Socolar also gave another way to enforce aperiodicity, in terms of
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is not an aperiodic tiling, since its hull contains the periodic tiling ...
100:
51:
39:
An aperiodic tiling using a single shape and its reflection, discovered by
2172:
2013:
1732:
1703:
Socolar, J.E.S. (1989). "Simple octagonal and dodecagonal quasicrystals".
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928:
822:
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converges – in the local topology – to the periodic tiling consisting of
119:
2037:"Aperiodic tilings, positive scalar curvature and amenability of spaces"
1395:
Moody, R.V. (1997). "Meyer Sets and Their Duals". In Moody, R.V. (ed.).
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27:
is an example of an aperiodic tiling; every tiling it can produce lacks
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However, the tiling produced in this way is not unique, not even up to
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Only a few different kinds of constructions have been found. Notably,
178:, together with all tilings that can be approximated by translates of
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2587:
2563:
1579:
1157:
850:
807:
743:
594:
494:
55:
1827:
497:. Today there is a large amount of literature on aperiodic tilings.
88:, announced the proof that the tile discovered by David Smith is an
2455:
1746:
Penrose, R. (1997). "Remarks on Tiling: details of a 1 +
1489:
1430:"Mathematicians have finally discovered an elusive 'einstein' tile"
1071:
1047:
845:
to give a strongly aperiodic set of tiles in the hyperbolic plane.
719:
667:
118:
Consider a periodic tiling by unit squares (it looks like infinite
654:
628:. As with the term "aperiodic tiling" itself, the term "aperiodic
576:
945:
642:
694:
and there are additional choices that allow for the encoding of
2247:
1325:
1064:
1040:
1854:
Algebraic theory of Penrose's nonperiodic tilings of the plane
110:
Several methods for constructing aperiodic tilings are known.
663:
construction of most known aperiodic sets of tiles to date.
2381:
1900:
Kari, Jarkko (1996). "A small aperiodic set of Wang tiles".
1812:
Radin, Charles (1994). "The pinwheel tilings of the plane".
1638:
942: – Ordered chemical structure with no repeating pattern
2186:
Levy, J-C. S.; Mercier, D. (2006). "Stable quasicrystals".
1931:"A strongly aperiodic set of tiles in the hyperbolic plane"
1089:
Schechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. (1984).
1599:
1597:
1088:
1455:
1453:
1451:
1848:
N. G. de Bruijn, Nederl. Akad. Wetensch. Indag. Math.
1594:
1348:"Meyer's concept of quasicrystal and quasiregular sets"
950:
Pages displaying short descriptions of redirect targets
1448:
324:
represents an interval of length two. Thus the tiling
1632:
358:
291:
263:
243:
188:
135:
1194:(1966). "The undecidability of the domino problem".
2119:"A New Paradigm for the Structure of Quasicrystals"
2034:
829:made as the differences of consecutive elements of
99:Aperiodic tilings serve as mathematical models for
1010:
897:
432:way throughout the tiling), then it is aperiodic.
401:
297:
277:
249:
229:
156:
103:, physical solids that were discovered in 1982 by
1162:Aperiodic Order. Vol 1: A Mathematical Invitation
931: – Five tiles used in Islamic decorative art
230:{\displaystyle \{T+x\,:\,x\in \mathbb {R} ^{d}\}}
3341:
2209:. Morgan & Claypool Publishers. p. 55.
2207:Introductory Tiling Theory for Computer Graphics
1290:
1208:
73:are a well-known example of aperiodic tilings.
1928:
1603:
1509:
1507:
1459:
800:this, as first noted in the pioneering work of
577:Aperiodic hierarchical tilings through matching
1552:
1550:
702:Σ of up to four letters. In summary there are
113:
2263:
1869:The Mathematics of Long-Range Aperiodic Order
1772:
1397:The Mathematics of Long-Range Aperiodic Order
16:Form of plane tiling without repeats at scale
2142:
2041:Journal of the American Mathematical Society
1775:"A construction of inflation rules based on
1567:Notices of the American Mathematical Society
1504:
1196:Memoirs of the American Mathematical Society
1084:
1082:
336:(with centre 0, say). Now all translates of
224:
189:
2185:
1990:Mozes, Shahar (1997). "Aperiodic tilings".
1547:
486:
402:{\displaystyle 1,2,4,\ldots ,2^{n},\ldots }
62:if copies of these tiles can form only non-
2270:
2256:
1559:"Aperiodic Tilings, Order, and Randomness"
1155:
755:The Penrose tiles, and shortly thereafter
182:. Formally this is the closure of the set
2581:Dividing a square into similar rectangles
2093:
2052:
1966:
1956:
1796:
1623:
1479:
1462:"Matching rules and substitution tilings"
1371:
1304:. New York: W. H. Freeman. p. 528f.
1222:. New York: W. H. Freeman. p. 584f.
1114:
1079:
1070:
1046:
794:
658:A portion of tiling by the Robinson tiles
257:-close if they agree in a ball of radius
214:
205:
201:
157:{\displaystyle T\subset \mathbb {R} ^{d}}
144:
1668:
1662:
1345:
1242:
806:
742:
731:
653:
641:
597:usually require such additional rules.
34:
18:
2243:The Infinite Pattern That Never Repeats
2070:"Weak matching rules for quasicrystals"
2067:
1745:
1702:
1606:"A small aperiodic set of planar tiles"
1556:
1427:
1399:. NATO ASI Series C. pp. 403–441.
1036:
1034:
1005:
967:
3342:
2204:
2116:
2074:Communications in Mathematical Physics
1352:Communications in Mathematical Physics
1286:
1284:
1190:
971:(January 1977). "Mathematical Games".
690:) going off to infinity in up to four
348:s else. The sequence of tilings where
328:consists of infinitely many copies of
320:represents an interval of length one,
2643:
2493:
2393:
2289:
2251:
2125:from the original on 23 February 2007
1989:
1811:
1784:Discrete & Computational Geometry
1513:
1394:
1151:
1149:
1147:
736:The chair substitution tiling system.
569:such that the substitution tiling is
1899:
1123:
1031:
775:have found several subsequent sets.
550:substitution and expansion rules and
1773:Nischke, K.-P.; Danzer, L. (1996).
1281:
1131:"The Nobel Prize in Chemistry 2011"
841:. This method was later adapted by
816:
785:Conway-pinwheel substitution tiling
582:and periodic ones. Sometimes these
13:
2644:
2035:Block, J.; Weinberger, S. (1992).
1866:
1202:
1144:
993:10.1038/scientificamerican0177-110
14:
3361:
2226:
2054:10.1090/s0894-0347-1992-1145337-x
1611:European Journal of Combinatorics
1013:Penrose Tiles to Trapdoor Ciphers
76:In March 2023, four researchers,
2380:
2373:
2277:
1865:See, for example, the survey by
1678:(corrected paperback ed.).
709:
526:
2198:
2179:
2136:
2110:
2061:
2028:
1983:
1929:Goodman-Strauss, Chaim (2005).
1922:
1893:
1859:
1842:
1805:
1766:
1739:
1696:
1642:; Geoffrey C. Shephard (1986).
1604:Goodman-Strauss, Chaim (1999).
1460:Goodman-Strauss, Chaim (1998).
1421:
1388:
1339:
1318:
1236:
935:List of aperiodic sets of tiles
898:Confusion regarding terminology
444:tried to determine whether the
312:of the line that looks like ...
1648:. W.H. Freeman & Company.
1520:Journal d'Analyse Mathématique
1184:
1168:
999:
961:
948: – Mosaic tile decoration
278:{\displaystyle 1/\varepsilon }
1:
2606:Regular Division of the Plane
2394:
1428:Conover, Emily (2023-03-24).
1179:Mathematics Genealogy Project
1164:. Cambridge University Press.
955:
428:(i.e. each patch occurs in a
2290:
1916:10.1016/0012-365X(95)00120-L
1877:10.1007/978-94-015-8784-6_13
1405:10.1007/978-94-015-8784-6_16
584:geometric matching condition
298:{\displaystyle \varepsilon }
250:{\displaystyle \varepsilon }
7:
2514:Architectonic and catoptric
2412:Aperiodic set of prototiles
1116:10.1103/PhysRevLett.53.1951
922:
553:the cut-and-project method.
114:Definition and illustration
10:
3366:
1680:Cambridge University Press
1675:Quasicrystals and geometry
868:
864:
783:found rules enforcing the
713:
435:
92:, i.e., a solution to the
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2726:
2695:
2657:
2653:
2639:
2500:
2494:
2489:
2402:
2389:
2371:
2298:
2285:
1968:10.1007/s00222-004-0384-1
1725:10.1103/PhysRevB.39.10519
1557:Treviño, Rodrigo (2023).
749:Trilobite and Cross tiles
340:are the tilings with one
2188:Acta Phys. Superficierum
2165:10.1103/PhysRevE.47.R788
2068:Socolar, Joshua (1990).
1993:Inventiones Mathematicae
1936:Inventiones Mathematicae
1296:Shephard, Geoffrey Colin
1246:Inventiones Mathematicae
1214:Shephard, Geoffrey Colin
1017:. W H Freeman & Co.
164:contains all translates
1852:, 39–52, 53–66 (1981).
1346:Lagarias, J.C. (1996).
1095:Physical Review Letters
80:, Joseph Samuel Myers,
2205:Kaplan, Craig (2009).
1625:10.1006/eujc.1998.0281
813:
795:Cut-and-project method
752:
737:
659:
647:
516:
487:cut-and-project method
403:
299:
279:
251:
231:
158:
43:
32:
29:translational symmetry
2233:The Geometry Junkyard
2014:10.1007/s002220050153
1815:Annals of Mathematics
1467:Annals of Mathematics
859:alternating condition
810:
773:Chaim Goodman-Strauss
746:
735:
657:
645:
589:Sometimes additional
476:Chaim Goodman-Strauss
404:
300:
280:
252:
232:
159:
86:Chaim Goodman-Strauss
38:
22:
2117:Steinhardt, Paul J.
1903:Discrete Mathematics
1645:Tilings and Patterns
1301:Tilings and Patterns
1219:Tilings and Patterns
571:sibling-edge-to-edge
356:
289:
261:
241:
186:
133:
2157:1993PhRvE..47..788E
2086:1990CMaPh.129..599S
2006:1997InMat.128..603M
1949:2004InMat.159..119G
1717:1989PhRvB..3910519S
1364:1996CMaPh.179..365L
1259:1971InMat..12..177R
1107:1984PhRvL..53.1951S
985:1977SciAm.236a.110G
973:Scientific American
716:Substitution tiling
602:Penrose tiling (P1)
521:Socolar–Taylor tile
460:Raphael M. Robinson
2095:10.1007/BF02097107
1798:10.1007/BF02717732
1670:Senechal, Marjorie
1533:10.1007/BF02793412
1373:10.1007/BF02102593
1267:10.1007/BF01418780
855:amenable manifolds
827:Sturmian sequences
814:
753:
738:
660:
648:
646:The Robinson Tiles
399:
295:
275:
247:
227:
154:
90:aperiodic monotile
50:is a non-periodic
44:
33:
3350:Aperiodic tilings
3337:
3336:
3333:
3332:
3329:
3328:
2635:
2634:
2526:Computer graphics
2485:
2484:
2369:
2368:
2238:Aperiodic Tilings
2145:Physical Review E
1886:978-90-481-4832-5
1754:-aperiodic set".
1689:978-0-521-57541-6
1655:978-0-7167-1194-0
1414:978-90-481-4832-5
1101:(20): 1951–1953.
1024:978-0-7167-1987-8
787:system. In 1998,
3357:
2655:
2654:
2641:
2640:
2593:Conway criterion
2520:Circle Limit III
2491:
2490:
2424:Einstein problem
2391:
2390:
2384:
2377:
2313:Schwarz triangle
2287:
2286:
2272:
2265:
2258:
2249:
2248:
2221:
2220:
2202:
2196:
2195:
2183:
2177:
2176:
2151:(2): R788–R791.
2140:
2134:
2133:
2131:
2130:
2114:
2108:
2107:
2097:
2065:
2059:
2058:
2056:
2032:
2026:
2025:
1987:
1981:
1980:
1970:
1960:
1926:
1920:
1919:
1910:(1–3): 259–264.
1897:
1891:
1890:
1863:
1857:
1846:
1840:
1839:
1809:
1803:
1802:
1800:
1770:
1764:
1763:
1743:
1737:
1736:
1711:(15): 10519–51.
1700:
1694:
1693:
1666:
1660:
1659:
1640:Grünbaum, Branko
1636:
1630:
1629:
1627:
1601:
1592:
1591:
1580:10.1090/noti2759
1563:
1554:
1545:
1544:
1511:
1502:
1501:
1483:
1457:
1446:
1445:
1443:
1442:
1425:
1419:
1418:
1392:
1386:
1385:
1375:
1343:
1337:
1336:
1334:
1332:
1326:"Bromley Tilers"
1322:
1316:
1315:
1292:Grünbaum, Branko
1288:
1279:
1278:
1240:
1234:
1233:
1210:Grünbaum, Branko
1206:
1200:
1199:
1188:
1182:
1172:
1166:
1165:
1153:
1142:
1141:
1139:
1138:
1133:. Nobelprize.org
1127:
1121:
1120:
1118:
1086:
1077:
1076:
1074:
1062:
1053:
1052:
1050:
1038:
1029:
1028:
1016:
1003:
997:
996:
965:
951:
907:non-periodic. A
831:Beatty sequences
817:Other techniques
812:distinguishable.
567:hereditary edges
511:
408:
406:
405:
400:
392:
391:
332:and one copy of
304:
302:
301:
296:
284:
282:
281:
276:
271:
256:
254:
253:
248:
236:
234:
233:
228:
223:
222:
217:
169:
163:
161:
160:
155:
153:
152:
147:
94:einstein problem
48:aperiodic tiling
3365:
3364:
3360:
3359:
3358:
3356:
3355:
3354:
3340:
3339:
3338:
3325:
2802:
2795:
2728:
2722:
2691:
2649:
2631:
2496:
2481:
2398:
2385:
2379:
2378:
2365:
2356:Wallpaper group
2294:
2281:
2276:
2229:
2224:
2217:
2203:
2199:
2184:
2180:
2141:
2137:
2128:
2126:
2115:
2111:
2066:
2062:
2033:
2029:
1988:
1984:
1958:10.1.1.477.1974
1927:
1923:
1898:
1894:
1887:
1864:
1860:
1847:
1843:
1828:10.2307/2118575
1810:
1806:
1779:-fold symmetry"
1771:
1767:
1744:
1740:
1701:
1697:
1690:
1667:
1663:
1656:
1637:
1633:
1602:
1595:
1561:
1555:
1548:
1512:
1505:
1481:10.1.1.173.8436
1458:
1449:
1440:
1438:
1426:
1422:
1415:
1393:
1389:
1344:
1340:
1330:
1328:
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1312:
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1230:
1207:
1203:
1189:
1185:
1173:
1169:
1154:
1145:
1136:
1134:
1129:
1128:
1124:
1087:
1080:
1063:
1056:
1039:
1032:
1025:
1007:Gardner, Martin
1004:
1000:
969:Gardner, Martin
966:
962:
958:
949:
925:
900:
873:
867:
843:Goodman-Strauss
819:
797:
789:Goodman-Strauss
722:
714:Main articles:
712:
672:Euclidean group
579:
559:Penrose tilings
547:matching rules,
529:
507:
472:Craig S. Kaplan
438:
430:uniformly dense
387:
383:
357:
354:
353:
290:
287:
286:
267:
262:
259:
258:
242:
239:
238:
218:
213:
212:
187:
184:
183:
165:
148:
143:
142:
134:
131:
130:
116:
82:Craig S. Kaplan
71:Penrose tilings
17:
12:
11:
5:
3363:
3353:
3352:
3335:
3334:
3331:
3330:
3327:
3326:
3324:
3323:
3318:
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3308:
3303:
3298:
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3253:
3248:
3243:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3203:
3198:
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3188:
3183:
3178:
3173:
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3158:
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3133:
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3013:
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3003:
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2943:
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2933:
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2893:
2888:
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2878:
2873:
2868:
2863:
2858:
2853:
2848:
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2838:
2833:
2828:
2823:
2818:
2813:
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2797:
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2773:
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2715:
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2705:
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2689:
2684:
2679:
2674:
2669:
2663:
2661:
2651:
2650:
2637:
2636:
2633:
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2630:
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2624:
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2600:
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2578:
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2528:
2523:
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2511:
2501:
2498:
2497:
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2483:
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2480:
2479:
2474:
2469:
2468:
2467:
2453:
2448:
2443:
2438:
2433:
2432:
2431:
2429:Socolar–Taylor
2421:
2420:
2419:
2409:
2407:Ammann–Beenker
2403:
2400:
2399:
2387:
2386:
2372:
2370:
2367:
2366:
2364:
2363:
2358:
2353:
2352:
2351:
2346:
2341:
2330:Uniform tiling
2327:
2326:
2325:
2315:
2310:
2305:
2299:
2296:
2295:
2283:
2282:
2275:
2274:
2267:
2260:
2252:
2246:
2245:
2240:
2235:
2228:
2227:External links
2225:
2223:
2222:
2215:
2197:
2178:
2135:
2109:
2080:(3): 599–619.
2060:
2047:(4): 907–918.
2027:
2000:(3): 603–611.
1982:
1943:(1): 119–132.
1921:
1892:
1885:
1858:
1841:
1822:(3): 661–702.
1804:
1791:(2): 221–236.
1765:
1738:
1695:
1688:
1661:
1654:
1631:
1618:(5): 375–384.
1593:
1546:
1527:(1): 139–186.
1503:
1490:10.2307/120988
1474:(1): 181–223.
1447:
1420:
1413:
1387:
1358:(2): 356–376.
1338:
1317:
1310:
1280:
1253:(3): 177–209.
1235:
1228:
1201:
1192:Berger, Robert
1183:
1167:
1143:
1122:
1078:
1054:
1030:
1023:
998:
979:(1): 111–119.
959:
957:
954:
953:
952:
943:
937:
932:
924:
921:
917:Penrose tiling
899:
896:
869:Main article:
866:
863:
818:
815:
796:
793:
761:Joshua Socolar
711:
708:
698:from Σ for an
696:infinite words
591:matching rules
578:
575:
555:
554:
551:
548:
538:domino problem
534:undecidability
528:
525:
491:N.G. de Bruijn
446:domino problem
437:
434:
398:
395:
390:
386:
382:
379:
376:
373:
370:
367:
364:
361:
352:is centred at
344:somewhere and
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25:Penrose tiling
15:
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1175:Robert Berger
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910:
905:
895:
893:
888:
887:Faraday waves
883:
882:Robert Ammann
878:
877:Dan Shechtman
872:
862:
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840:
836:
832:
828:
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790:
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782:
781:Charles Radin
778:
774:
770:
769:Ludwig Danzer
766:
765:Roger Penrose
762:
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745:
741:
734:
730:
727:
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717:
710:Substitutions
707:
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689:
686:(also called
685:
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527:Constructions
524:
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518:
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469:
468:Robert Ammann
465:
464:Roger Penrose
461:
457:
452:
451:Robert Berger
447:
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431:
427:
422:
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416:
413:s only. Thus
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105:Dan Shechtman
102:
101:quasicrystals
97:
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2612:Regular grid
2604:
2518:
2451:Quaquaversal
2395:
2349:Kisrhombille
2279:Tessellation
2206:
2200:
2191:
2187:
2181:
2148:
2144:
2138:
2127:. Retrieved
2112:
2077:
2073:
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2044:
2040:
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1997:
1991:
1985:
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1907:
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1705:Phys. Rev. B
1704:
1698:
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1644:
1634:
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1609:
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1565:
1524:
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1465:
1439:. Retrieved
1435:Science News
1433:
1423:
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1390:
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1351:
1341:
1329:. Retrieved
1320:
1300:
1250:
1244:
1238:
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1170:
1161:
1135:. Retrieved
1125:
1098:
1094:
1012:
1001:
976:
972:
963:
940:Quasicrystal
914:
909:non-periodic
908:
903:
901:
892:interference
874:
871:Quasicrystal
858:
847:Shahar Mozes
838:
834:
820:
798:
777:Shahar Mozes
754:
739:
725:
723:
691:
687:
683:
676:translations
665:
661:
649:
637:
634:
630:hierarchical
629:
611:
607:
599:
590:
588:
583:
580:
570:
566:
563:
556:
542:
530:
501:
499:
483:substitution
480:
456:Hans Läuchli
439:
423:
418:
414:
410:
349:
345:
341:
337:
333:
329:
325:
321:
317:
314:aaaaaabaaaaa
313:
309:
307:
179:
175:
171:
166:
129:of a tiling
124:
117:
109:
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47:
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2647:vertex type
2505:Anisohedral
2460:Self-tiling
2303:Pythagorean
1574:(8): 1183.
1198:(66): 1–72.
1156:Baake, M.;
929:Girih tiles
894:phenomena.
823:Jarkko Kari
704:uncountably
120:graph paper
78:David Smith
66:tilings.
41:David Smith
2551:Pentagonal
2129:2007-03-26
1762:: 467–497.
1441:2023-03-25
1158:Grimm, Uwe
1137:2011-10-06
1072:2305.17743
1048:2303.10798
956:References
851:Lie groups
668:isometries
595:Wang tiles
495:Meyer sets
426:repetitive
316:... where
56:prototiles
2659:Spherical
2627:Voderberg
2588:Prototile
2555:Problems
2531:Honeycomb
2509:Isohedral
2396:Aperiodic
2334:honeycomb
2318:Rectangle
2308:Rhombille
2104:123629334
2022:189819776
1953:CiteSeerX
1672:(1996) .
1588:260818304
1541:121775031
1476:CiteSeerX
1382:122753893
904:aperiodic
902:The term
802:de Bruijn
688:corridors
680:rotations
517:ein Stein
509:‹See Tfd›
485:and by a
462:in 1971.
397:…
378:…
293:ε
273:ε
245:ε
210:∈
140:⊂
60:aperiodic
3344:Category
2741:V3.4.3.4
2576:Squaring
2571:Heesch's
2536:Isotoxal
2456:Rep-tile
2446:Pinwheel
2339:Coloring
2292:Periodic
2123:Archived
1298:(1987).
1275:14259496
1216:(1987).
1160:(2013).
1009:(1988).
923:See also
720:L-system
700:alphabet
622:Robinson
503:einstein
442:Hao Wang
64:periodic
3201:6.4.8.4
3156:5.4.6.4
3116:4.12.16
3106:4.10.12
3076:V4.8.10
3051:V4.6.16
3041:V4.6.14
2941:3.6.4.6
2936:3.4.∞.4
2931:3.4.8.4
2926:3.4.7.4
2921:3.4.6.4
2871:3.∞.3.∞
2866:3.4.3.4
2861:3.8.3.8
2856:3.7.3.7
2851:3.6.3.8
2846:3.6.3.6
2841:3.5.3.6
2836:3.5.3.5
2831:3.4.3.∞
2826:3.4.3.8
2821:3.4.3.7
2816:3.4.3.6
2811:3.4.3.5
2766:3.4.6.4
2736:3.4.3.4
2729:regular
2696:Regular
2622:Voronoi
2546:Packing
2477:Truchet
2472:Socolar
2441:Penrose
2436:Gilbert
2361:Wythoff
2173:9960162
2153:Bibcode
2082:Bibcode
2002:Bibcode
1977:5348203
1945:Bibcode
1856:, I, II
1836:2118575
1733:9947860
1713:Bibcode
1360:Bibcode
1331:26 July
1255:Bibcode
1177:at the
1103:Bibcode
981:Bibcode
946:Zellige
865:Physics
757:Amman's
726:enforce
674:, e.g.
670:of the
618:Läuchli
536:of the
436:History
3091:4.8.16
3086:4.8.14
3081:4.8.12
3071:4.8.10
3046:4.6.16
3036:4.6.14
3031:4.6.12
2801:Hyper-
2786:4.6.12
2559:Domino
2465:Sphinx
2344:Convex
2323:Domino
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626:Ammann
513:German
474:, and
419:aaaaaa
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52:tiling
3206:(6.8)
3161:(5.6)
3096:4.8.∞
3066:(4.8)
3061:(4.7)
3056:4.6.∞
3026:(4.6)
3021:(4.5)
2991:4.∞.4
2986:4.8.4
2981:4.7.4
2976:4.6.4
2971:4.5.4
2951:(3.8)
2946:(3.7)
2916:(3.4)
2911:(3.4)
2803:bolic
2771:(3.6)
2727:Semi-
2598:Girih
2495:Other
2100:S2CID
2018:S2CID
1973:S2CID
1832:JSTOR
1584:S2CID
1562:(PDF)
1537:S2CID
1494:JSTOR
1378:S2CID
1271:S2CID
1067:arXiv
1043:arXiv
614:Knuth
421:....
58:) is
3291:8.16
3286:8.12
3256:7.14
3226:6.16
3221:6.12
3216:6.10
3176:5.12
3171:5.10
3126:4.16
3121:4.14
3111:4.12
3101:4.10
2961:3.16
2956:3.14
2776:3.12
2761:V3.6
2687:V4.n
2677:V3.n
2564:Wang
2541:List
2507:and
2458:and
2417:List
2332:and
2211:ISBN
2169:PMID
1881:ISBN
1729:PMID
1684:ISBN
1650:ISBN
1409:ISBN
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692:arms
678:and
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532:the
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