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LETTERSDisordered,quasicrystallineandcrystallinephasesofdenselypackedtetrahedraAmirHaji-Akbari1*,MichaelEngel1*,AaronS.Keys1,XiaoyuZheng3,RolfeG.Petschek5,PeterPalffy-Muhoray4&SharonC.Glotzer1,2Allhard,convexshapesareconjecturedbyUlamtopackmoredenselythanspheres1,whichhaveamaximumpackingfractionofw5p/!182009|doi:10.1038/nature08641773MacmillanPublishersLimited.Allrightsreserved?2009Figure2ashowstheequationofstatew(P*)obtainedfromsimula-tionsofasmallsystemwith512tetrahedraandalargersystemwith4,096tetrahedra.Here,P*5Ps3/kBTisthereducedpressureandstheedgelengthofatetrahedron.Forthesmallsystem,theequilib-riumpackingfractionexhibitsanS-shapedtransitionatP*558andw50.47fromasimplefluidtoamorecomplexfluid,discussedbelow.Athigherpressurethesystemjams(SupplementaryFigs1and2)and,whencompressedtonearlyinfinitepressure,attainsamaximumpackingfractionofw50.7858.Thelargesystemunder-goesafirst-ordertransitiononcompressionofthefluidphaseandformsaquasicrystal.InFig.2b,weanalysethesystemforthepresenceoflocallydensemotifsintroducedinFig.1A.Weseethatthefractionoftetrahedrabelongingtoatleastonepentagonaldipyramidincreaseswellbeforejammingorcrystallization.Withincreasingpressure,interpenetratingpentagonaldipyramidsformicosahedraandfinallymergeintoapercolatingpentagonaldipyramidnetwork(Fig.2c,d)asthefractionoftetrahedrainpentagonaldipyramidsapproachesunity.Forthelargesystem,thefractionoftetrahedrainicosahedrasuddenlydropsatP*562,whencrystallizationoccurs.Comparisonwiththeglassshowsthatfarfewericosahedraremaininthequasicrystal.Figure2canddsuggestsapercolationtransi-tionofthepentagonaldipyramidnetworkinbothsystemsatP*p55862,beforebothjammingandcrystallization.Wedonotobservetetrahedraticliquidcrystalphases,whichhavebeensug-gestedbytheory24.Structuralchangesinthefluidarerevealedbytheunusualbeha-viourofitsradialdistributionfunctiong(r),asshowninFig.2e.Wefindthatthefirstpeaknear
