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| [[File:Fund un prim cell.svg|250px|thumb|right|{{show
| | The Clip Level blade is without doubt one of the most popular styles. The highest aspect of this knife is straight from the handle to the mid blade space. It then curves downward and barely back up to the tip of the knife The curved area on top of the leading edge is called the "clip". Any such knife is widespread in mounted blade knives You can find whittling knives is special kits. These kits may have several knives , each with a unique edge form. Or the package might need a knife deal with and several in another way shaped blade edges which are interchangeable. Completely different edges enable for a lot better maneuverability.<br><br>It isn't that [http://city-wiz.com/node/298767 onerous] actually. Victorinox has been for a few years the main firm available in the market for pocket knives and multi-instruments. The Swiss Army Knives have a high quality design far superior to every factor else available in the market, they're manufactured with wonderful supplies to final, and they are again up by a lifetime warranty. So, the logic choice is to select from them. But, despite the fact that you really can't go unsuitable shopping for a Victorinox, plainly some fashions are better than others. Every knife stroke you make needs to be away from your body. Never stroke in direction of your physique. You recognize you are just asking for it.<br><br>Tow Line - In case somebody has develop into incapacitated and is unable to paddle. This will likely solely be as a result of they are too drained or exhausted to take action, however if in case you have no alternative but to proceed paddling for a bit then setting up a tow may be needed Be sure to have the proper sort of tow line, sea/touring tow strains [http://www.thebestpocketknifereviews.com/best-combat-knife-fightingtactical-pocket-knives-reviews/ Best tactical pocket Knives] are totally different to river tow strains. Put on gloves. Think of it this fashion, when you reduce your arms or fingers, you're carried out. You won't be capable to whittle at all till it heals. Wear them, at least till you've developed some real ability at dealing with the knife<br><br>To additional maintain your pocket knife , it is a good suggestion to apply a drop of oil between the spring and the blade or instrument shank. However not any oil will do. Keep in mind that we will probably be using our Victorinox with food, so the oil should be appropriate for use with foodstuffs. Sharpen your knife recurrently. This is not only a great advice for maintaining the usability of your pocket knife , but a security measure too. A pointy blade is safer as a result of you'll not be making use of an excessive amount of drive when chopping something lessening the chances of slipping.<br><br>It will most probably occur on the finish of the first run throughout the room. When reducing drywall, use a pointy knife. A utility knife or a drywall knife will work. A great pocket knife might do the trick, too. Mark a [http://doc.froza.ru/index.php/The_Best_Tactical_Knives straight] line the place the drywall needs to be minimize. Use the knife and lower [http://Www.thebestpocketknifereviews.com/best-combat-knife-fightingtactical-pocket-knives-reviews/ http://Www.thebestpocketknifereviews.com/] via the paper and a bit of into the gypsum inside. Holding the drywall with the reduce on the alternative aspect from your self, pull gently back on one end of the drywall while tapping firmly behind the cut. The drywall sheet should bend and snap cleanly alongside the line.<br><br>The Butterfly Knife , also known as a gravity knife or balisong is usually illegal. They are designed to be deployed quickly with a flipping motion of the wrist, [https://canimpact.org/members/francdrakefor/activity/696130/ producing] a slashing motion. They are a folding knife that has two handles that [http://Act.jinbo.net/forum2006/index.php/Best_Tactical_Knife_Review rotate round] its tang. They are opened by centrifugal power. Both the switchblade and the butterfly knife have an evil or threatening reputation due to their slashing action. The ideas can break off or the blade chip if a ceramic blade Pocket knife is dropped on the ground and you can not use them to pry with. |
| |Click "show" for description.|A [[truncated trihexagonal tiling|periodic tiling]] with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a [[triangular tiling]] of fundamental units will be generated that is [[#MLD|mutually locally derivable]] from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be [[Translation (geometry)|translated]] to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this.}}|alt=]]
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| In [[geometry]], a [[Tessellation|tiling]] is a partition of the plane (or any other geometric setting) into closed sets (called ''tiles''), without gaps or overlaps (other than the boundaries of the tiles).<ref name="Tilings by Regular Polygons">{{Citation|doi=10.2307/2689529|author=Grünbaum B., Shephard G. C.|title=Tilings by Regular Polygons|journal=Math. Mag.|volume=50|issue=5|year=1977|pages=227–247|url=http://vohweb.chem.ucla.edu/voh/classes%5Cspring10%5CM117%20HNRS%20M180ID22%5CGrunbaumShephardTilingByRegPolygons.pdf|postscript=.}}(archived at [http://www.webcitation.org/5t6mxDTfO WebCite])</ref> A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single [[Fundamental domain|fundamental unit]] or [[primitive cell]] which repeats endlessly and regularly in two independent directions.<ref>Edwards S., [http://www.spsu.edu/math/tile/defs/fundamental.htm ''Fundamental Regions and Primitive cells''] (archived at [http://www.webcitation.org/5smoLyN1Y WebCite])</ref> An example of such a tiling is shown in the diagram to the right (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called [[aperiodic tiling|aperiodic]].<ref>Ollinger N. [http://books.google.de/books?id=EbVrWLNiub4C&pg=PA268&dq=nonperiodic+tiling#v=onepage&q=nonperiodic%20tiling&f=false ''Mathematica in action'' (see page 268)]</ref> The tilings obtained from an aperiodic set of tiles are often called [[aperiodic tiling]]s, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".)
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| The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.
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| ==Explanations==
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| {| class="wikitable" style="width:100%;border:0px;text-align:center;"
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| ! Abbreviation !! Meaning !! Explanation
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| | E<sup>2</sup> || [[Euclidean plane]] || normal flat plane
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| | H<sup>2</sup> || [[Hyperbolic space|hyperbolic plane]] || plane, where the [[parallel postulate]] does not hold
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| | E<sup>3</sup> || [[Three-dimensional space|Euclidean 3 space]] || space defined by three perpendicular coordinate axes
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| |-id="MLD"
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| | MLD || Mutually locally derivable || two tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or inserting an edge)
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| |}
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| ==List==
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| {| class="wikitable" style="width:100%;border:0px;text-align:center;"
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| ! Image !! Name !! Number of tiles !! Space !! Publication Date !! refs !! Comments
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| | [[File:Trilobite and cross.svg|100px|center]] || Trilobite and cross tiles || 2 || E<sup>2</sup> || 1999 || <ref name="Small aperiodic planar set">{{Citation|doi=10.1006/eujc.1998.0281|author=Goodman-Strauss C.|title=A Small Aperiodic Set of Planar Tiles|journal=[[European Journal of Combinatorics]]|volume=20|issue=5|year=1999|pages=375–384}} (preprint available [http://comp.uark.edu/~strauss/papers/newsmall.pdf here])</ref> || Tilings MLD from the [[chair tiling]]s
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| | [[File:Penrose P1.svg|100px|center]] || [[Penrose tiling#The original pentagonal Penrose tiling (P1)|Penrose P1 tiles]] || 6 || E<sup>2</sup> || 1974{{Ref label|note01|Note 1|^}} || <ref>Mikhael J. [http://elib.uni-stuttgart.de/opus/volltexte/2010/5240/pdf/Diss_Mikhael.pdf ''Colloidal Monolayers On Quasiperiodic Laser Fields'' (see page 23)] (archived at [http://www.webcitation.org/5t5ar5MGD WebCite])</ref> || Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
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| | [[File:Kite Dart.svg|100px|center]] || [[Penrose tiling#Kite and dart tiling (P2)|Penrose P2 tiles]] || 2 || E<sup>2</sup> || 1977{{Ref label|note02|Note 2|^}} || <ref>Gardner M. [http://maa.org/pubs/focus/Gardner_PenroseTilings1-1977.pdf ''Penrose tiles to trapdoor ciphers'' (see page 86)] (archived at [http://www.webcitation.org/5t5AJdAUM WebCite])</ref> || Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
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| | [[File:Penrose rhombs.svg|100px|center]] || [[Penrose tiling#Rhombus tiling (P3)|Penrose P3 tiles]] || 2 || E<sup>2</sup> || 1978{{Ref label|note03|Note 3|^}} || <ref name="Pentaplexity">{{Citation|author=Penrose R.|title=Pentaplexity|journal=Math. Intell.|volume=2|year=1979/80|pages=32–37|url=http://www.ma.utexas.edu/users/radin/pentaplexity.html|postscript=.}}(archived at [http://www.webcitation.org/5sxsFR2i9 WebCite])</ref> || Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
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| | [[File:Binary tiles.svg|100px|center]] || [[Penrose tiling#Related tilings|Binary tiles]] || 2 || E<sup>2</sup> || 1988 || <ref name="LanBil1">{{Citation|doi=10.1051/jphys:01988004902024900|author=Lançon F., Billard L.|title=Two-dimensional system with a quasi-crystalline ground state|journal=J. Phys. France|volume=49|issue=2|year=1988|pages=249–256|url=http://hal.archives-ouvertes.fr/docs/00/21/06/91/PDF/ajp-jphys_1988_49_2_249_0.pdf|postscript=.}} (archived at [http://www.webcitation.org/5t6dEDXX5 WebCite])</ref><ref name="LanBil2">{{Citation|doi=10.1051/jp1:1992134|author=Lançon F., Billard L.|title=A simple example of a non-Pisot tiling with five-fold symmetry|journal=J. Phys. I France|volume=2|issue=2|year=1992|pages=207–220|url=http://hal.archives-ouvertes.fr/docs/00/24/64/73/PDF/ajp-jp1v2p207.pdf|postscript=.|bibcode=1992JPhy1...2..207G}}(archived at [http://www.webcitation.org/5t6dOMDQv WebCite])</ref> || Although similar in shape to the P3 tiles, the tilings are not MLD from each other, developed in an attempt to model the atomic arrangement in binary alloys
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| |- id="Robinson tiles"
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| | [[File:Robinson tiles.svg|100px|center]] || [[Aperiodic_tiling#Aperiodic_hierarchical_tilings|Robinson tiles]] || 6 || E<sup>2</sup> || 1971{{Ref label|note04|Note 4|^}} ||<ref name="AHT">{{Citation|author=Goodman-Strauss C.|title=Aperiodic Hierarchical tilings|journal=Proc. of NATO-ASI "Foams, Emulsions, and Cellular Materials" Ser. E|volume=354|year=1999|pages=481–496|url=http://www.webcitation.org/5sxslGkXb|postscript=.}}</ref> || Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
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| | No image || [[Ammann-Beenker tiling|Ammann A1 tiles]] <!-- Is this right? They were discovered by Ammann, but are they the missing tiles from G&S? --> || 6 || E<sup>2</sup> || 1977<ref name=ColMath>{{Cite book|author=Gardner, Martin|title=The Colossal Book of Mathematics|publisher=W. W. Norton & Company|page=76|year=2001}}</ref> || <ref name=TilPat>{{cite book|author=Grünbaum B. and Shephard G. C.|title=Tilings and Patterns}}, according to [http://www.uwgb.edu/dutchs/symmetry/aperiod.htm]; cf [http://www.quadibloc.com/math/til03.htm]</ref> || Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
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| | [[File:Ammann A2.svg|100px|center]] || Ammann A2 tiles || 2 || E<sup>2</sup> || 1986{{Ref label|note05|Note 5|^}} || <ref name="Aperiodic tiles">{{Citation|doi=10.1007/BF02293033|author=Ammann R., Grünbaum B. and Shephard G. C.|title=Aperiodic Tiles|journal=Discrete Comp Geom|volume=8|year=1992|pages=1–25|url=http://www.webcitation.org/5rjblcapr|postscript=.}}</ref>
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| | [[File:Ammann A3.svg|100px|center]] || Ammann A3 tiles || 3 || E<sup>2</sup> || 1986{{Ref label|note05|Note 5|^}} || <ref name="Aperiodic tiles"/>
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| | [[File:Ammann A4.svg|100px|center]] || [[Ammann–Beenker tiling|Ammann A4 tiles]] || 2 || E<sup>2</sup> || 1986{{Ref label|note05|Note 5|^}} || <ref name="Aperiodic tiles"/><ref>Harris E., Frettlöh D. [http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a4 ''Ammann A4'']</ref> || Tilings MLD with Ammann A5.
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| | [[File:Ammann A5.svg|100px|center]] || [[Ammann–Beenker tiling|Ammann A5 tiles]] || 2 || E<sup>2</sup> || 1982{{Ref label|note06|Note 6|^}} || <ref name="RepAmmBee">{{Citation|author= Komatsu K., Nomakuchi K., Sakamoto K., Tokitou T.|title=Representation of Ammann-Beenker tilings by an automaton|journal=Nihonkai Math. J.|volume=15|year=2004|pages=109–118|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nihmj/1273779794|postscript=.}}(archived at [http://www.webcitation.org/5t6emwHVh WebCite])</ref><ref>Harris E., Frettlöh D. [http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_beenker ''Ammann-Beenker'']</ref> || Tilings MLD with Ammann A4.
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| | No image || Penrose Hexagon-Triangle tiles || 2 || E<sup>2</sup> || 1997<ref name="AperiodicPair">{{Citation|author=Penrose R.|title=''Remarks on tiling: Details of a (1+ε+ε<sup>2</sup>) aperiodic set.''|journal=Nato Asi Series C|volume=489 '''The Mathematics of Long-Range Aperiodic Order'''|year=1997|pages=467–497|isbn=978-0-7923-4506-0|url=http://books.google.com/?id=3YGXr_nQFw8C&printsec=frontcover&dq=quasicrystals+and+geometry#v=onepage&q=quasicrystals%20and%20geometry&f=false|postscript=.}}</ref> || <ref name="AperiodicPair"/><ref>Goodman-Strauss C., [http://comp.uark.edu/~strauss/distribution/tilings/penhex.pdf ''An aperiodic pair of tiles'']</ref>
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| | No image || [[Golden triangle (mathematics)|Golden Triangle]] tiles || 10 || E<sup>2</sup> || 2001 <ref>{{cite journal|author=Danzer, Ludwig|author2=van Ophuysen, Gerrit|title=A species of planar triangular tilings with inflation factor <math>\sqrt-\tau</math>|journal=Res. Bull. Panjab Univ. Sci.|volume=50|issue=1-4|pages=137–175|year=2001|mr=1914493}}</ref> || <ref>{{cite journal|author=Gelbrich, G|title=Fractal Penrose tiles II. Tiles with fractal boundary as duals of Penrose triangles|journal=Aequationes Math.|year=1997|volume=54|pages=108–116|mr=MR1466298}}</ref> || date is for discovery of matching rules. Dual to Ammann A2
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| | [[File:Socolar.svg|100px|center]] || Socolar tiles || 3 || E<sup>2</sup> || 1989{{Ref label|note07|Note 7|^}} || <ref>{{cite web |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.8116 |last1 = Gähler | first1 = F. | last2 = Lück | first2 = R. | last3 = Ben-Abraham | first3 = S. I. | last4 = Gummelt | first4 = P. | id = {{citeseerx|10.1.1.26.8116}} | title = Dodecagonal tilings as maximal cluster coverings |deadurl=no |accessdate=2013-09-25}}</ref><ref>[http://www.quadibloc.com/math/dode01.htm ''The Socolar tiling'']</ref> || Tilings MLD from the tilings by the Shield tiles
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| | [[File:Shield.svg|100px|center]] || Shield tiles || 4 || E<sup>2</sup> || 1988{{Ref label|note08|Note 8|^}} || <ref>Gähler F., Frettlöh D. [http://tilings.math.uni-bielefeld.de/substitution_rules/shield ''Shield'']</ref><ref name="Matching rules for quasicrystals: the composition-decomposition method">{{Citation | author=Gähler F.|title=Matching rules for quasicrystals: the composition-decomposition method|journal=J. of Non-crystalline Solids|volume=153&154|year=1993|pages=160–164|url=http://elib.uni-stuttgart.de/opus/volltexte/2009/3989/pdf/gaeh11.pdf|postscript=.}}(archived at [http://www.webcitation.org/5t9d3IP8h WebCite])</ref> || Tilings MLD from the tilings by the Socolar tiles
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| | [[File:Square triangle tiles.svg|100px|center]] || Square triangle tiles || 5 || E<sup>2</sup> || 1986<ref>{{cite journal|author=Stampfli, P|title=A Dodecagonal Quasiperiodic Lattice in Two Dimensions|journal=Helv. Phys. Acta.|volume=59|year=1986|pages=1260–1263}}</ref> || <ref>Hermisson J., Richard C., Baake M. [http://www.tphys.physik.uni-tuebingen.de/baake/ps/torus.ps.gz ''A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes''] (archived at [http://www.webcitation.org/5tCw2DqAi WebCite])</ref>
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| | [[File:Self-replication of sphynx hexidiamonds.svg|100px|center]] || [[Sphinx tiling]] || 91 || E<sup>2</sup> || || <ref>Goodman-Strauss C., [http://mathweb.sc.niigata-u.ac.jp/ant/Sympo/GS_kyoto2.pdf ''Aperiodic tilings'' (see page 74)]</ref>
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| | [[File:Starfish ivyleaf hex.svg|100px|center]] || Starfish, ivy leaf and hex tiles || 3 || E<sup>2</sup> || || <ref name="Quasicrystals and Penrose patterns">{{Citation|author=Lord E. A.|title=Quasicrystals and Penrose patterns|journal=Current Science|volume=61|year=1991|pages=315|url=http://www.webcitation.org/5t3XY5gd3|postscript=.}}</ref><ref name="Aperiodic dense tiling">{{Citation|doi=10.1051/jphys:0198900500101900|author=Olamy Z., Kléman M.|title=A two dimensional aperiodic dense tiling|journal=J. Phys. France|volume=50|year=1989|pages=19–33|url=http://hal.archives-ouvertes.fr/docs/00/21/08/96/PDF/ajp-jphys_1989_50_1_19_0.pdf|postscript=.}} (archived at [http://www.webcitation.org/5tuZzGc6c WebCite])</ref><ref name="Combined">{{Citation|author=Mihalkovič M., Henley C. L., Widom M.|title=Combined energy-diffraction data refinement of decagonal AlNiCo|journal=J. Non-Cryst. Solids|volume=334&335|year=2004|pages=177–183|url=http://people.ccmr.cornell.edu/~clh/PUBS/dfit-04.pdf|postscript=.}} (archived at [http://www.webcitation.org/5u0CZpy7a WebCite])</ref> || Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
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| |- id="Robinson triangle"
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| | [[File:Robinson triangle decompositions.svg|100px|center]] || [[Golden triangle (mathematics)#Golden gnomon|Robinson triangle]] || 4 || E<sup>2</sup> || || <ref name=TilPat/> || Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
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| | [[File:Danzer triangles.svg|100px|center]] || Danzer triangles || 6 || E<sup>2</sup> || 1996<ref>Nischke, K-P and Danzer, L, {{cite journal|title=A construction of inflation rules based on $n$-fold symmetry|journal=Discrete Comput. Geom.|year=1996|volume=15|issue=2|pages=221–236}} 96j:52035</ref> || <ref>Hayashi H., Kawachi Y., Komatsu K., Konda A., Kurozoe M., Nakano F., Odawara N., Onda R., Sugio A., Yamauchi M. [http://www.jaist.ac.jp/~uehara/JCCGG09/short/paper_29.pdf ''Abstract:Notes on vertex atlas of planar Danzer tiling'']</ref>
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| | [[File:pinwheel 1.svg|100px|center]] || [[Pinwheel tiling|Pinwheel tile]]s || || E<sup>2</sup> || 1994<ref name="Pinwheel tilings">{{cite journal |url=http://www.jstor.org/discover/10.2307/2118575?uid=3739256&uid=2&uid=4&sid=21102691007323 |authorlink=Charles Radin|last=Radin|first=C|title=The pinwheel tilings of the plane|journal=[[Annals of Mathematics]](2)|year=1994|volume=139|issue=3|pages=661–702|mr=95d:52021|doi=10.2307/2118575|jstor=2118575|id={{citeseerx|10.1.1.44.9723}} |deadurl=no |accessdate=2013-09-25}}</ref><ref>{{cite journal |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5319 |title=Symmetry Of Tilings Of The Plane|author=Charles Radin|journal=Annals of Mathematics|year=1994|id={{citeseerx|10.1.1.45.5319}} |deadurl=no |accessdate=2013-09-25}}</ref> || <ref>{{Cite journal|author=Radin, C|author2=Wolff, M|title=Space tilings and local isomorphism|journal=Geom. Dedicata|year=1992|volume=42|issue=3|pages=355–360|mr=1164542}}</ref><ref>{{Cite book|author=Radin, C|chapter=Aperiodic tilings, ergodic theory, and rotations|title=The mathematics of long-range aperiodic order|publisher=Kluwer Acad. Publ., Dordrecht|year=1997|mr=1460035}} <!--489, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. --></ref> || Date is for publication of matching rules.
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| | No image || [[Wang tile]]s || 20426 || E<sup>2</sup> || 1966 || <ref>{{cite journal|last=Burger|first=R|title=The Undecidability of the Domino Problem|journal=Memoirs of the American Mathematical Society|year=1966|volume=66|pages=1–72}}</ref>
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| | No image || [[Wang tile]]s || 104 || E<sup>2</sup> || 2008 || <ref>{{cite book|last=Ollinger|first=Nicolas|title=Two-by-two Substitution Systems and the Undecidability of the Domino Problem|year=2008|publisher=Springer|pages=476–485|url=http://hal.inria.fr/docs/00/26/01/12/PDF/sutica.pdf |format=pdf |deadurl=no |accessdate=2013-09-25}}</ref>
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| | No image || Wang tiles || 52 || E<sup>2</sup> || 1971{{Ref label|note04|Note 4|^}} || <ref name="Deterministic Aperiodic Tilesets">{{Cite journal| first1=J. | last1=Kari | author1-link=Jarkko Kari | first2=P. | last2=Papasoglu | title=Deterministic Aperiodic Tile Sets | journal=Geometric and Functional Analysis | volume=9 | year=1999 | pages=353–369 | doi=10.1007/s000390050090}}</ref> || Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
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| | [[File:Wang 32 tiles.svg|100px|center]] || Wang tiles || 32 || E<sup>2</sup> || 1986 || <ref name="Aperiodic sets of square tiles">{{Citation | last1 = Lagae | first1 = A. | authorlink2 = Jarkko Kari | last2 = Kari | first2 = J. | last3 = Dutré | first3 = P. | title = Aperiodic Sets of Square Tiles with Colored Corners | url = http://graphics.cs.kuleuven.be/publications/LKD06ASSTCC/LKD06ASSTCC.pdf | journal = Report CW | volume = 460 | year = 2006 | pages = 12 | id = {{citeseerx|10.1.1.89.1294}} | archiveurl = http://www.webcitation.org/5tBFyK8Ji |archivedate=2010-10-02}}</ref> || Locally derivable from the Penrose tiles.
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| | No image || Wang tiles || 24 || E<sup>2</sup> || 1986 || <ref name="Aperiodic sets of square tiles"/> || Locally derivable from the A2 tiling
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| | [[File:Wang 16 tiles.svg|100px|center]] || Wang tiles || 16 || E<sup>2</sup> || 1986 || <ref>{{Citation | author1-link=Branko Grünbaum | last1=Grünbaum | first1=B. | last2=Shephard | first2=G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1986 | isbn=0-7167-1194-X}}</ref><ref>{{Citation | author1-link=Alessandra Carbone | last1=Carbone | first1=A. | last2=Gromov | first2=M. | last3=Prusinkiewicz | first3=P. | title=Pattern Formation in Biology, Vision and Dynamics | location=Singapore | publisher=World Scientific Publishing Co. Pte. Ltd. | year=2000 | isbn=981-02-3792-8}}</ref> || Derived from tiling A2 and its Ammann bars
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| | [[File:Wang 14 tiles.svg|100px|center]] || Wang tiles || 14 || E<sup>2</sup> || 1996 || <ref>[[Jarkko Kari|Kari J.]] ''A small aperiodic set of Wang tiles". Discrete Mathematics, 160(1-3):259–264''</ref><ref>Lagae A. [http://people.cs.kuleuven.be/~ares.lagae/publications/L07TBMCG/L07TBMCG_thesis_screen.pdf ''Tile Based Methods in Computer Graphics'' Dissertation (see page 149)] (archived at [http://www.webcitation.org/5tHcXltNP WebCite])</ref>
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| | [[File:Wang 13 tiles.svg|100px|center]] || Wang tiles || 13 || E<sup>2</sup> || 1996 || <ref>Culik K., [[Jarkko Kari|Kari J.]] [http://www.springerlink.com/content/tu47v531178v130g/ ''On aperiodic sets of Wang tiles'']</ref><ref>{{cite web |url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.5421&rep=rep1&type=pdf | last = Culik | first = K. | id = {{citeseerx|10.1.1.53.5421}} | title = An aperiodic set of 13 Wang tiles | archiveurl = http://www.webcitation.org/5tBgUuWku | archivedate = 2010-10-02 |deadurl=no |accessdate=2013-09-25}}</ref>
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| | No image || Decagonal Sponge tile || 1 || E<sup>2</sup> || 2002 || <ref>Zhu F. [http://www.cs.williams.edu/~bailey/Zh02.pdf ''The Search for a Universal Tile'']</ref><ref>{{cite web |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.103.3739 | last1 = Bailey | first1 = D. A. | last2 = Zhu | first2 = F. | id = {{citeseerx|10.1.1.103.3739}} | title = A Sponge-Like (Almost) Universal Tile |deadurl=no |accessdate=2013-09-25}}</ref> || Porous tile consisting of non-overlapping point sets
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| | No image || Goodman-Strauss strongly aperiodic tiles || 85 || H<sup>2</sup> || 2005 || <ref>Goodman-Strauss C., [http://comp.uark.edu/~strauss/papers/STAP.pdf ''A hierarchical strongly aperiodic set of tiles in the hyperbolic plane'']</ref>
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| | No image || Goodman-Strauss strongly aperiodic tiles || 26 || H<sup>2</sup> || 2005 || <ref name="Strongly aperiodic">{{Citation|author=Goodman-Strauss C.|title=A strongly aperiodic set of tiles in the hyperbolic plane|journal=Invent. Math.|volume=159|year=2005|pages=130–132|doi=10.1007/s00222-004-0384-1|bibcode=2004InMat.159..119G}}</ref>
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| | [[File:Goodman-Strauss hyperbolic tile.svg|100px|center]] || Böröczky hyperbolic tile || 1 || H<sup>n</sup> || 1974<ref>{{Citation|author=Böröczky, K.|title=Gömbkitöltések állandó görbületü terekben I|journal=Mat. Lapok.|volume=25|pages=265–306|year=1974|postscript=.}}{{Citation|author=Böröczky, K.|title=Gömbkitöltések állandó görbületü terekben II|journal=Mat. Lapok.|volume=26|pages=67–90|year=1974|postscript=.}}</ref> || <ref name="Strongly aperiodic">{{Citation|author=Goodman-Strauss C.|title=A strongly aperiodic set of tiles in the hyperbolic plane|journal=Invent. Math.|volume=159|year=2005|page=120|doi=10.1007/s00222-004-0384-1|bibcode=2004InMat.159..119G}}</ref> <ref>Dolbilin N., Frettlöh D. [http://www.mathematik.uni-bielefeld.de/baake/frettloe/papers/hypart.pdf ''Properties of Böröczky tilings in high dimensional hyperbolic spaces''] (archived at [http://www.webcitation.org/5sxuXAvMx WebCite])</ref> || Only [[weakly aperiodic]]
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| | No image || [[Schmitt tile]] || 1 || E<sup>3</sup> || 1988 || <ref name="Radin 1995 3543–3548">{{Cite web |url=http://www.ma.utexas.edu/users/radin/papers/schmitt.pdf | last=Radin | first=Charles | year=1995 | title=Aperiodic tilings in higher dimensions | journal=[[Proceedings of the American Mathematical Society]] | volume=123 | issue=11 | pages=3543–3548 | format=pdf | doi=10.2307/2161105 | jstor=2161105 | publisher=American Mathematical Society |deadurl=no |accessdate=2013-09-25}}</ref> || [[screw symmetry|Screw-periodic]]
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| | [[File:SCD tile.svg|100px|center]] || [[Schmitt–Conway–Danzer tile]] || 1 || E<sup>3</sup> || || <ref name="Radin 1995 3543–3548"/> || [[screw symmetry|Screw-periodic]] and [[convex polytope|convex]]
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| | [[File:Socolar Taylor 3D.svg|100px|center]] || [[Socolar-Taylor tile]] || 1 || E<sup>3</sup> || 2010 || <ref>Socolar J. E. S. and Taylor J. M. [http://arxiv.org/abs/1003.4279v1 ''An aperiodic hexagonal tile'']</ref><ref>Socolar J. E. S. and Taylor J. M. [http://arxiv.org/abs/1009.1419v1 ''Forcing nonperiodicity with a single tile'']</ref> || Periodic in third dimension
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| | No image || Penrose rhombohedra || 2 || E<sup>3</sup> || 1981<ref name=PentagonalSnowflake>{{Citation|author=Mackay A. L.|title=De Nive Quinquangula: On the pentagonal snowflake|journal=Sov. Phys. Crystallogr.|volume=26(5)|year=1981|pages=517–522|url=http://materials.iisc.ernet.in/~lord/webfiles/Alan/CV073eng.pdf|postscript=.}} (archived at [http://www.webcitation.org/5tHdBODy6 WebCite])</ref> || <ref>Meisterernst G. [http://edoc.ub.uni-muenchen.de/7000/1/Meisterernst_Goetz.pdf ''Experimente zur Wachstumskinetik Dekagonaler Quasikristalle (Experiments on the growth kinetics of decagonal quasicrystals)'' Dissertation (see page 18-19)] (archived at [http://www.webcitation.org/5tK5OeYPC WebCite])</ref><ref name=StructureTransition>{{Citation|author=Jirong S.|title=Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field|journal=Chinese Phys. Lett.|volume=10, No.8|year=1993|pages=449–452|url=http://cpl.iphy.ac.cn/qikan/manage/wenzhang/0100449.pdf|postscript=.}} (archived at [http://www.webcitation.org/5tK650sb9 WebCite])</ref><ref>Inchbald G. [http://www.steelpillow.com/polyhedra/quasicr/quasicr.htm ''A 3-D Quasicrystal Structure'']</ref><ref name=TilingvsClustering>{{Citation|author=Lord E. A., Ranganathan S., Kulkarni U. D.|title=Quasicrystals: tiling versus clustering|journal=Phil. Mag. A|volume=81|year=2001|pages=2645–2651|url=http://materials.iisc.ernet.in/~lord/webfiles/pmq.pdf|postscript=.}} (archived at [http://www.webcitation.org/5tHfLszq0 WebCite])</ref><ref>Rudhart C. P. [http://elib.uni-stuttgart.de/opus/volltexte/2000/624/pdf/rudhart.pdf ''Zur numerischen Simulation des Bruchs von Quasikristallen (On the numeric simulation of cracking in quasicrystals)''] see page 11</ref><ref name=TilingsCoveringsClustersQuasicrystals>{{Citation|author=Lord E. A., Ranganathan S., Kulkarni U. D.|title=Tilings, coverings, clusters and quasicrystals|journal=Current Science|volume=78, No.1|year=2000|pages=64–72|url=http://materials.iisc.ernet.in/~lord/webfiles/eric/pdfs/55.pdf|postscript=.}} (archived at [http://www.webcitation.org/5tuYuxfQL WebCite])</ref><ref name=TheoryofMathingRules>{{Citation|doi=10.1007/BF01218580|author=Katz A.|title=Theory of Matching Rules for the 3-Dimensional Penrose Tilings|journal=Commun. Math. Phys.|volume=118|issue=2|year=1988|pages=263–288|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161989|postscript=.}} (archived at [http://www.webcitation.org/5tuabzi7H WebCite])</ref>
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| | No image || Wang cubes || 21 || E<sup>3</sup> || 1996 || <ref>{{cite web |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.5897 | last1 = Culik | first1 = K. | authorlink2 = Jarkko Kari| last2 = Kari | first2 = J. | id = {{citeseerx|10.1.1.54.5897}} | title = An aperiodic set of Wang cubes |deadurl=no |accessdate=2013-09-25}}</ref>
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| | No image || Wang cubes || 18 || E<sup>3</sup> || 1999 || <ref>{{Citation | author1-link=Gerd Walther | last1=Walther | first1=G. | last2=Selter | first2=C. | title=Mathematikdidaktik als design science | location=Leipzig | publisher=Ernst Klett Grundschulverlag | year=1999 | isbn=3122000601}}</ref>
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| | No image || Wang cubes || 16 || E<sup>3</sup> || || <ref>Lu A., Ebert D. S., Qiao W., Kraus M., Mora B. [http://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1129&context=ecetr ''Interactive volume illustration using Wang cubes'']</ref>
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| | No image || Danzer tetrahedra || 4 || E<sup>3</sup> || 1989<ref>{{Citation|doi=10.1016/0012-365X(89)90282-3|author=Danzer, L.|title=Three-Dimensional Analogs of the Planar Penrose Tilings and Quasicrystals.|journal=Discrete Mathematics|volume=76|year=1989|pages=1–7}}</ref> || <ref>Zerhusen A., [http://www.ms.uky.edu/~lee/zerhusen/quasi.html ''Danzer's three dimensional tiling'']</ref>
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| | [[File:I and L tiles.png|100px|center]] || I and L tiles || 2 || E<sup>n</sup> for all n ≥ 3 || 1999 || <ref name="Aperiodic pair of tiles in En">{{Citation|doi=10.1006/eujc.1998.0282|author=Goodman-Strauss C.|title=''An Aperiodic Pair of Tiles in E<sup>n</sup> for all n ≥ 3''|journal=[[European Journal of Combinatorics]]|volume=20|issue=5|year=1999|pages=385–395}} (preprint available [http://comp.uark.edu/~strauss/papers/Newpaper.pdf here])</ref>
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| ==Notes==
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| First published in
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| :'''1.'''{{note label|note01|Note 1|^}}Penrose, R. (1974), "The role of Aesthetics in Pure and Applied Mathematical Research", ''Bull. Inst. Math. and its Appl.'' '''10''': 266-271
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| :'''2.'''{{note label|note02|Note 2|^}}Gardner, M. (January 1977), "Extraordinary nonperiodic tiling that enriches the theory of tiles", ''Scientific American'' '''236''': 110-121
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| :'''3.'''{{note label|note03|Note 3|^}}Penrose, R. (1978), "Pentaplexity", ''Eureka'' '''39''': 16-22
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| :'''4.'''{{note label|note04|Note 4|^}}Robinson, R. (1971), "Undecidability and nonperiodicity of tilings in the plane", ''Inv. Math.'' '''12''': 177-209
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| :'''5.'''{{note label|note05|Note 5|^}}{{Citation | author1-link=Branko Grünbaum | last1=Grünbaum | first1=B. | last2=Shephard | first2=G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1986 | isbn=0-7167-1194-X}}.
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| :'''6.'''{{note label|note06|Note 6|^}}Beenker, F. P. M.(1982), "Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus", ''Eindhoven University of Technology, TH Report'' '''82-WSK04'''
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| :'''7.'''{{note label|note07|Note 7|^}}Socolar, J. E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", ''Phys. Rev. A'' '''39''': 10519-51
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| :'''8.'''{{note label|note08|Note 8|^}}Gähler, F., [http://elib.uni-stuttgart.de/opus/volltexte/2009/4662/pdf/gaeh7.pdf "Crystallography of dodecagonal quasicrystals"], published in Janot, C.: Quasicrystalline materials : Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988. Singapore : World Scientific, 1988, 272-284
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * Stephens P. W., Goldman A. I. [http://www.pha.jhu.edu/courses/171_405/Quasixtals.pdf ''The Structure of Quasicrystals'']
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| * Levine D., Steinhardt P. J. [http://www.physics.princeton.edu/~steinh/QuasiPartI.pdf ''Quasicrystals I Definition and structure]
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| * [http://tilings.math.uni-bielefeld.de/ ''Tilings Encyclopedia'']
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| [[Category:Aperiodic sets of tiles| ]]
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| [[Category:Tessellation]]
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| [[Category:Mathematics-related lists]]
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