Ogled vira Seznam neperiodičnih množic tlakovanj

[[Slika:Fund un prim cell.svg|350px|thumb|right|[[Prisekano trišestkotno periodično tlakovanje]], ki ima prikazani osnovni celici (trikotnik) in primitivno celico (šestkotnik). Tlakovanje celotne ravnine se dobi s kopiranjem trikotnih delov površine. Da bi to lahko naredili, moramo osnovni trikotnik zavrteti za 180º, da bi pokrili ravnino od roba do roba. Torej bo [[trikotno tlakovanje]] z osnovnimi enotami nastalo iz dveh lokalnih medsebojno izpeljivih tlakovanj, ki sta prikazani z isto barvo. Druga oblika, ki je narisana v tlakovanju, je beli šestkotnik, ki predstavlja osnovno celico. Kopije pripadajočih obarvanih ploščic  se lahko prenesejo s [[translacija|translacijo]] tako, da tvorijo neskončno tlakovanje ravnine. Teh delov tlakovanja ni potrebno zavrteti, da bi to dosegli.]]

'''Seznam neperiodičnih množic tlakovanj''' je v [[geometrija|geometriji]] skupina oblik imenovanih [[protoploščica|ploščice]], ki ravnino   pokrijejo brez lukenj ali prekrivanj <ref name="Tilings by Regular Polygons">{{Citation|doi=10.2307/2689529|author1=Grünbaum B.|author2=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=.|accessdate=2012-06-22|archive-date=2011-09-01|archive-url=https://web.archive.org/web/20110901174726/http://vohweb.chem.ucla.edu/voh/classes%5Cspring10%5CM117%20HNRS%20M180ID22%5CGrunbaumShephardTilingByRegPolygons.pdf|url-status=dead}}(arhivirano pri )</ref>.  

Takšno tlakovanje je sestavljeno iz [[osnovna domena|osnovnih enot]] ali [[primitivna celica|primitivnih celic]]. Imenuje se periodično <ref>Edwards S., [http://www.spsu.edu/math/tile/defs/fundamental.htm ''Fundamental Regions and Primitive cells''](arhivirano pri)</ref>.
 
Zgled takšnega tlakovanja je prikazan desni.
Vsako periodično tlakovanje ima primitivno celici, ki ga generira. Tlakovanje, ki ne more biti iz ene primitivne celice, se imenuje neperiodično. Kadar dana množica ploščic dovoljuje samo neperiodično tlakovanje, je takšna množica aperiodična <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'' (gel stran 268)]</ref>

Prva preglednica pojasnjuje okrajšave uporabljene v drugi preglednici.  Druga preglednica vsebuje vse znane aperiodične množice ploščic in daje nekaj dodatnih podatkov o vsaki množici.
  
{| class="wikitable" style="width:100%;border:0px;text-align:center;"
|-
! okrajšava !! pomen !! pojasnilo
|-
| E<sup>2</sup> || [[evklidska ravnina]] || običajna ravnina
|-
| H<sup>2</sup> || [[hiperbolični prostor|hiperbolična ravnina]] || ravnina na kateri ne velja [[aksiom o vzporednici]] 
|-
| E<sup>3</sup> || [[trirazsežni prostor|evklidski trirazsežni prostor]] || prostor, ki je določen s tremi pravokotnimi osmi
|-id="MLD"
| MLD || vzajemno lokalno izpeljivo || dve tlakovanji sta vzajemno izpeljivi, če se eno tlakovanje dobi  iz drugega s preprostim lokalnim pravilom (brisanje  ali dodajanje roba)
|}

==Pregled ==

{| class="wikitable" style="width:100%;border:0px;text-align:center;"
|-
! slika !! ime !! število  ploščic !! prostor !! datum objave !! reference !! opombe
|-
| [[Slika:Trilobite and cross.svg|100px|center]] || trilobitne in križne ploščice || 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}} (prepis dosegljiv na [http://comp.uark.edu/~strauss/papers/newsmall.pdf here])</ref> || Tlakovanja MLD iz [[sedežno tlakovanje|sedežnih tlakovanj]]
|-
| [[Slika:Penrose P1.svg|100px|center]] || [[Penroseovo tlakovanje#Izvirno pentagonalno Penroseovo tlakovanje  (P1)|Penroseove ploščice P1]] || 6 || E<sup>2</sup> || 1974 || <ref>Mikhael J. [http://elib.uni-stuttgart.de/opus/volltexte/2010/5240/pdf/Diss_Mikhael.pdf ''Colloidal Monolayers On Quasiperiodic Laser Fields'' (glej stran 23)](arhivirano na)</ref> || Tlakovanja MLD iz  tlakovanj od P2 in P3, Robinsonovi trikotniki in "morska zvezda, bršljanov list, heks"
|-
| [[Slika:Kite Dart.svg|100px|center]] || [[Penroseovo tlakovanje#Kite and dart tiling (P2)|Penroseove ploščice P2]] || 2 || E<sup>2</sup> || 1977 || <ref>Gardner M. [https://web.archive.org/web/20110617054708/http://maa.org/pubs/focus/Gardner_PenroseTilings1-1977.pdf ''Penrose tiles to trapdoor ciphers'' (see page 86)](arhivirano pri )</ref> || Tlakovanja MLD iz tlakovanj P1 in P3, Robinsonovih trikotnikov in "morske zvezde, bršljanovega lista in heksa"
|-
| [[Slika:Penrose rhombs.svg|100px|center]] || [[Penroseovo tlakovanje#Rombsko tlakovanje (P3)|Penroseove ploščice P3]] || 2 || E<sup>2</sup> || 1978 || <ref name="Pentaplexity">{{Citation|author=Penrose R.|title=Pentaplexity|journal=Math. Intell.|volume=2|year=1979–1980|pages=32–37|url=http://www.ma.utexas.edu/users/radin/pentaplexity.html|postscript=.|accessdate=2012-06-22|archive-date=2011-06-07|archive-url=https://web.archive.org/web/20110607155418/http://www.ma.utexas.edu/users/radin/pentaplexity.html|url-status=bot: unknown}}(arhivirano pri )</ref> || Tlakovanja MLD iz tlakovanj P1 in P2 ter Robinsonovega trikotnika in "morske zvezde, bršljanovega lista in heksa"
|-
| [[Slika:Binary tiles.svg|100px|center]] || [[Penroseovo tlakovanje#Druga tlakovanja in islamska umetnost|binarne ploščice]] || 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=.|accessdate=2012-06-22|archive-date=2011-05-09|archive-url=https://web.archive.org/web/20110509163641/http://hal.archives-ouvertes.fr/docs/00/21/06/91/PDF/ajp-jphys_1988_49_2_249_0.pdf|url-status=bot: unknown}}(arhivirano pri )</ref><ref name="LanBil2">{{Citation|doi=10.1051/jp1:1992134|author1=Lançon F.|author2=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|accessdate=2012-06-22|archive-date=2012-03-07|archive-url=https://web.archive.org/web/20120307232407/http://hal.archives-ouvertes.fr/docs/00/24/64/73/PDF/ajp-jp1v2p207.pdf|url-status=bot: unknown}}(arhivirano pri )</ref> || Čeprav so po obliki podobne ploščicam P3, tlakovanja niso MLD tlakovanja, ki jih lahko izpeljemo drug iz drugega v modelu razporeditve atomov v   dvojnih (binarnih) zlitinah
|- id="Robinsonove ploščice"
| [[Slika:Robinson tiles.svg|100px|center]] || Robinsonove ploščice || 6 || E<sup>2</sup> || 1971 ||<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.google.de/url?sa=t|postscript=.|accessdate=2012-06-22|archive-date=2012-11-10|archive-url=https://web.archive.org/web/20121110042549/http://www.webcitation.org/5sxslGkXb|url-status=dead}}</ref> || Ploščice dajejo občutek neperiodičnosti s tvorbo neskončne hierarhije kvadratnih mrež
|-
| ni slike || [[Ammann-Beenkerjevo tlakovanje|Ammannove A1 ploščice]] || 6 || E<sup>2</sup> || 1977<ref name=ColMath>{{navedi knjigo|author=Gardner, Martin|title=The Colossal Book of Mathematics|url=https://archive.org/details/colossalbookmath00gard|publisher=W. W. Norton & Company|page=[https://archive.org/details/colossalbookmath00gard/page/n92 76]|year=2001}}</ref> || <ref name=TilPat>{{navedi knjigo|author1=Grünbaum B.|author2=Shephard G. C.|title=Tilings and Patterns}}, according to [http://www.uwgb.edu/dutchs/symmetry/aperiod.htm] {{Webarchive|url=https://web.archive.org/web/20060830155826/http://www.uwgb.edu/dutchs/symmetry/aperiod.htm |date=2006-08-30 }}; cf [http://www.quadibloc.com/math/til03.htm]</ref> || Ploščice dajejo  občutek neperiodičnosti s tvorbo neskončnega hierarhičnega binarnega drevesa.
|-
| [[Slika:Ammann A2.svg|100px|center]] || Ammannove A2 ploščice || 2 || E<sup>2</sup> || 1986 || <ref name="Aperiodic tiles">{{Citation|doi=10.1007/BF02293033|author1=Ammann R.|author2=Grünbaum B.|author3=Shephard G. C.|title=Aperiodic Tiles|journal=Discrete Comp Geom|volume=8|year=1992|pages=1–25|url=http://sites.google.com/site/aperiodictiles/home|postscript=.|accessdate=2012-06-22|archive-date=2012-11-09|archive-url=https://web.archive.org/web/20121109080755/http://www.webcitation.org/5rjblcapr|url-status=dead}}</ref>
|-
| [[Slika:Ammann A3.svg|100px|center]] ||  Ammannove ploščice A3 || 3 || E<sup>2</sup> || 1986 || <ref name="Aperiodic tiles"/>
|-
| [[Slika:Ammann A4.svg|100px|center]] || Ammannove ploščice A4  || 2 || E<sup>2</sup> || 1986 || <ref name="Aperiodic tiles"/><ref>Harris E., Frettlöh D. [http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a4 ''Ammann A4''] {{Webarchive|url=https://web.archive.org/web/20160409220840/http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a4 |date=2016-04-09 }}</ref> || Tlakovanja MLD z Ammannom A5.
|-
| [[Slika:Ammann A5.svg|100px|center]] ||  Ammannove ploščice A5|| 2 || E<sup>2</sup> || 1982 || <ref name="RepAmmBee">{{Citation|author1= Komatsu K.|author2= Nomakuchi K.|author3= Sakamoto K.|author4= 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=.}}(arhivirano pri [http://www.webcitation.org/5t6emwHVh WebCite])</ref><ref>Harris E., Frettlöh D. [http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_beenker ''Ammann-Beenker''] {{Webarchive|url=https://web.archive.org/web/20081005110309/http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_beenker |date=2008-10-05 }}</ref> || Tlakovanja MLD z Ammannom A4.
|-
| ni slike || Penroseove šestkotno-trikotne ploščice || 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>
|-
| ni slike || ploščice [[Zlati  trikotnik (matematika)|zlatega trikotnika]] || 10 || E<sup>2</sup> || 2001 <ref>{{navedi revijo|author1=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> || Datum odkritja primerjalnih pravil. Dualne z Ammannom A2
|-
| [[Slika:Socolar.svg|100px|center]] || sokolar ploščice || 3 || E<sup>2</sup> || 1989 || <ref>Gähler F., Lück R., Ben-Abraham S. I., Gummelt P.[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.8116&rep=rep1&type=pdf ''Dodecagonal tilings as maximal cluster coverings''] (arhivirano pri [https://web.archive.org/web/20130601142008/http://www.google.de/url?sa=t WebCite])</ref><ref>[http://www.quadibloc.com/math/dode01.htm ''The Socolar tiling'']</ref> || Tlakovanje MLD z uporabo zaščitnih ploščic
|-
| [[Slika:Shield.svg|100px|center]] || zaščitne ploščice || 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''] {{Webarchive|url=https://web.archive.org/web/20160303222354/http://tilings.math.uni-bielefeld.de/substitution_rules/shield |date=2016-03-03 }}</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=.|accessdate=2012-06-22|archive-date=2011-07-17|archive-url=https://web.archive.org/web/20110717000428/http://elib.uni-stuttgart.de/opus/volltexte/2009/3989/pdf/gaeh11.pdf|url-status=bot: unknown}}(arhivirano pri )</ref> || Tlakovanja MLD iz Sokolar ploščic
|-
| [[Slika:Square triangle tiles.svg|100px|center]] || kvadratne trikotne ploščice || 5 || E<sup>2</sup> || 1986<ref>{{navedi revijo|author=Stampfli, P|title=A Dodecagonal Quasiperiodic Lattice in Two Dimensions|url=https://archive.org/details/sim_helvetica-physica-acta_1986_59_6-7/page/1260|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''] {{Webarchive|url=https://web.archive.org/web/20160304025952/http://www.tphys.physik.uni-tuebingen.de/baake/ps/torus.ps.gz |date=2016-03-04 }} (arhivirano pri [https://web.archive.org/web/20130601142008/http://www.google.de/url?sa=t WebCite])</ref>
|-
| [[Slika:Self-replication of sphynx hexidiamonds.svg|100px|center]] || [[sfingino tlakovanje]] || 91 || E<sup>2</sup> || || <ref>Goodman-Strauss C., [http://mathweb.sc.niigata-u.ac.jp/ant/Sympo/GS_kyoto2.pdf ''Aperiodic tilings'' (glej stran 74)] {{Webarchive|url=https://web.archive.org/web/20120313051452/http://mathweb.sc.niigata-u.ac.jp/ant/Sympo/GS_kyoto2.pdf |date=2012-03-13 }}</ref>
|-
| [[Slika:Starfish ivyleaf hex.svg|100px|center]] || ploščice morska zvezda, bršljanov list in heks|| 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://materials.iisc.ernet.in/~lord/webfiles/eric/pdfs/49.pdf|postscript=.|accessdate=2012-06-22|archive-date=2012-11-10|archive-url=https://web.archive.org/web/20121110050516/http://www.webcitation.org/5t3XY5gd3|url-status=dead}}</ref><ref name="Aperiodic dense tiling">{{Citation|doi=10.1051/jphys:0198900500101900|author1=Olamy Z.|author2=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=.|accessdate=2012-06-22|archive-date=2011-05-09|archive-url=https://web.archive.org/web/20110509165854/http://hal.archives-ouvertes.fr/docs/00/21/08/96/PDF/ajp-jphys_1989_50_1_19_0.pdf|url-status=bot: unknown}}(arhivirano pri )</ref><ref name="Combined">{{Citation|author1=Mihalkovič M.|author2=Henley C. L.|author3=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=.}} (arhivirano pri [https://web.archive.org/web/20130601142008/http://www.google.de/url?sa=t WebCite])</ref> || Tlakovanje je MLD za  Penroseovo tlakovanje P1, P2, P3 in Robinsonove trikotnike
|- id="Robinsonov trokotnik"
| [[Slika:Robinson triangle decompositions.svg|100px|center]] || [[zlati trikotnik (matematika)#Zlati gnomon|Robinsonov trikotnik]] || 4 || E<sup>2</sup> || || <ref name=TilPat/> || Tlakovanje je MLD za  Penroseovo tlakovanje P1, P2, P3 ter "morsko zvezdo, bršljanov list in heks".
|-
| [[Slika:Danzer triangles.svg|100px|center]] || Danzerjevi trikotniki || 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>
|-
| [[Slika:pinwheel 1.svg|100px|center]] || [[tlakovanje vetrnice|tlakovanja vetrnic]] ||  || E<sup>2</sup> || 1994<ref name="Pinwheel tilings">{{cite journal|author=Radin, 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|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.9723&rank=1|accessdate=2010-04-03}}</ref><ref>{{cite journal|title=Symmetry Of Tilings Of The Plane|author=Charles Radin|journal=Annals of Math|year=1994|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5319&rank=6|accessdate=2010-04-03}}</ref> || <ref>{{Cite journal|author1=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>{{navedi knjigo|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}}</ref> || Datum velja za objavo pravil ujemanja.
|-
| ni slike || [[Wangova ploščica|Wangove ploščice]] || 20426 || E<sup>2</sup> || 1966 || <ref>{{navedi revijo|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> 
|-
| ni slike || [[Wangova ploščica|Wangove ploščice]] || 104 || E<sup>2</sup> || 2008 || <ref>{{navedi knjigo|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}}</ref>
|-
| ni slike || Wangove ploščice || 52 || E<sup>2</sup> || 1971 || <ref name="Deterministic Aperiodic Tilesets">{{navedi revijo| 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}}</ref> || Ploščice dajejo občutek aperiodičnosti s kreiranjem neskončne hierahije kvadratnih  mrež
|-
| [[Slika:Wang 32 tiles.svg|100px|center]] || Wangove ploščice || 32 || E<sup>2</sup> || 1986 || <ref name="Aperiodic sets of square tiles">{{Citation|author1=Lagae A.|author2=[[Jarkko Kari|Kari J.]]|author3=Dutré P.|title=Aperiodic Sets of Square Tiles with Colored Corners|journal=Report CW|volume=460|year=2006|pages=12|url=http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=9F2387ADDE287538436CC970669B4ABC?doi=10.1.1.89.1294&rep=rep1&type=pdf|postscript=.|accessdate=2012-06-22|archive-date=2012-10-09|archive-url=https://web.archive.org/web/20121009020610/http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=9F2387ADDE287538436CC970669B4ABC?doi=10.1.1.89.1294&rep=rep1&type=pdf|url-status=bot: unknown}}(arhivirano pri )</ref> || Lokalno izpeljane iz  Penroseovih ploščic.
|-
| ni slike || Wangove ploščice || 24 || E<sup>2</sup> || 1986 || <ref name="Aperiodic sets of square tiles"/> || Lokalno izpeljano iz tlakovanja A2
|-
| [[Slika:Wang 16 tiles.svg|100px|center]] || Wangove ploščice || 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> ||  Izpeljano iz  tlakovanja A2 in njegovih Ammannovih drogov
|-
| [[Slika:Wang 14 tiles.svg|100px|center]] || Wangove ploščice || 14 || E<sup>2</sup> || 1996 || <ref>[[Jarkko Kari|Kari J.]] ''manjša neperiodična množica Wangovih ploščic". 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)](arhivirano pri)</ref>
|-
| [[Slika:Wang 13 tiles.svg|100px|center]] || Wangove ploščice || 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'']{{Slepa povezava|date=junij 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>Culik K. [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5421 ''An aperiodic set of 13 Wang tiles''] (arhivirano pri [https://web.archive.org/web/20220630122119/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.5421 WebCite])</ref>
|-
| ni slike || dekagonalne  spužvine ploščice || 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>Bailey D. A., Zhu F. [http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=935FA62ED40593EF5057844A22F336E7?doi=10.1.1.103.3739&rep=rep1&type=pdf ''A Sponge-Like (Almost) Universal Tile'']</ref> || Porozne ploščice, ki so sestavljene iz neprekrivajočih se množice točk
|-
| ni slike || Goodman-Straussove strogo aperiodične ploščice || 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>
|-
| ni slike || Goodman-Straussove strogo aperiodične ploščice || 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>
|-
| [[Slika:Goodman-Strauss hyperbolic tile.svg|100px|center]] || Böröczkijeva hiperbolična ploščica || 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"/> <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''] (arhivirano pri [https://web.archive.org/web/20170811183350/http://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0319058-6/S0002-9939-1972-0319058-6.pdf WebCite])</ref> || Samo [[slabo aperiodične]]
|-
| ni slike || [[Schmittova ploščica]] || 1 || E<sup>3</sup> || 1988 || <ref name="Radin 1995 3543–3548">{{Citation | 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 | doi=10.2307/2161105 | jstor=2161105 | publisher=American Mathematical Society | postscript=.}}</ref> || [[vijačna simetrija|Vijačno periodične]]
|-
| [[Slika:SCD tile.svg|100px|center]] || [[Schmitt–Conway–Danzerjeve ploščice]] || 1 || E<sup>3</sup> || || <ref name="Radin 1995 3543–3548"/> || [[vijačna simetrija|Vijačno periodične]] in [[konveksni politop|konveksni]]
|-
| [[Slika:Socolar Taylor 3D.svg|100px|center]] || Socolar Taylorjeva ploščica || 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> || Periodično v tretji razsežnosti
|-
| ni slike || Penroseov romboeder || 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/webSlikas/Alan/CV073eng.pdf|postscript=.}}{{Slepa povezava|date=junij 2023 |bot=InternetArchiveBot |fix-attempted=yes }} (arhivirano pri [https://web.archive.org/web/20110721160040/http://materials.iisc.ernet.in/~lord/webfiles/Alan/CV073eng.pdf 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 (glej stran 18-19)](arhivirano pri)</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=.|accessdate=2012-06-22|archive-date=2011-07-07|archive-url=https://web.archive.org/web/20110707013425/http://cpl.iphy.ac.cn/qikan/manage/wenzhang/0100449.pdf|url-status=bot: unknown}}(arhivirano pri )</ref><ref>Inchbald G. [http://www.steelpillow.com/polyhedra/quasicr/quasicr.htm ''A 3-D Quasicrystal Structure'']</ref><ref name=TilingvsClustering>{{Citation|author1=Lord E. A.|author2=Ranganathan S.|author3=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/webSlikas/pmq.pdf|postscript=.}}{{Slepa povezava|date=junij 2023 |bot=InternetArchiveBot |fix-attempted=yes }} (arhivirano pri [https://web.archive.org/web/20110721155854/http://materials.iisc.ernet.in/~lord/webfiles/pmq.pdf 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|author1=Lord E. A.|author2=Ranganathan S.|author3=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/webSlikas/eric/pdfs/55.pdf|postscript=.}}{{Slepa povezava|date=junij 2023 |bot=InternetArchiveBot |fix-attempted=yes }} (arhivirano pri [https://web.archive.org/web/20110721161356/http://materials.iisc.ernet.in/~lord/webfiles/eric/pdfs/55.pdf 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=.}} (arhivirano pri [http://www.webcitation.org/5tuabzi7H WebCite])</ref>
|-
| ni slike || Wangove kocke || 21 || E<sup>3</sup> || 1996 || <ref>Culik K., [[Jarkko Kari|Kari J.]] [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.5897 ''An aperiodic set of Wang cubes'']</ref>
|-
| ni slike || Wangove kocke || 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>
|-
| ni slike || Wangove kocke || 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>
|-
| ni slike || Danzerjevi tetraedri || 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>
|-
| [[Slika:I and L tiles.png|100px|center]] || I in L ploščice || 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}} (prepis dosegljiv na [http://comp.uark.edu/~strauss/papers/Newpaper.pdf here])</ref>
|}

== Opombe in sklici ==
{{opombe}}

== Zunanje povezave ==
* [http://www.physics.princeton.edu/~steinh/QuasiPartI.pdf Kvazikristali, definicije in struktura] {{ikona en}}

[[Kategorija:Množica neperiodičnih tlakovanj]]
[[Kategorija:Teselacije]]
[[Kategorija:Matematični seznami]]