chapter2 Lattices Networks and Their EvolutionWord文件下载.docx

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Configurationsofelementsintwo-dimensional（2D）andthree-dimensional（3D）spacesareclassifiedintothefollowingthreetypes.

（1）Tessellation:

Tofillthespacewithoutoverlappingoropening.Itisequivalenttodividethespace.

Examples:

ropeoff,houseplan,crystalstructure,borderofcountries,tiling,etc.

（2）Packing:

Tofillthespacewithoutoverlappingbutallowingopening.

packingofobjectsinacontainer,pilingupofobjects,etc.

（3）Covering:

Tofillthespacewithoutopeningbutallowingoverlapping.

effectivedistributionsofradiowavetowersorwatchspots,spraypainting,etc.

PropertiesofConfigurations

Thereareseveraltypesofconfigurationswithrespecttotranslationsymmetry.Notethatthereareseveraltypesofaperiodicones.

（1）Periodic（=translationsymmetry）:

Theunitofperiodicityiscalledaunitstructure.

crystalstructures,etc

（2）Aperiodic（=withouttranslationsymmetry）

（3）Nearlyperiodic（quasi-periodic）

daylychangeoftheairtemperature,soundwavefromaninstrument,etc.

（4）Random（=irregular）

Examples:

configurationofgasmolecules,soundwaveofnoises,etc

（5）Quasi-crystal:

deterministicbutaperiodic（seeSec.5）

quasicrystal,Penrosetiling,etc.

Nearlyperiodictilingperiodicpackingrandomcovering

Fig.2-1Thethreetypesofconfigurationsofelementsin2D

2.1.2Tilingbycongruentregularpolygons

Tilingbycongruentregularpolygonsispossibleonlyforregulartriangle,squareandregularhexagon.Itsreasonisunderstoodbyobservingthenumberofpolygonsgatheringatavertex.Atthevertex360degisdividedintoequalangles,whichshouldbeanangleofregularpolygons.360/2=180isimpossible.360/3=120,360/4=90and360/6=60arepossible.Otherdivisions,suchas360/5=72and360/7=51.4,areimpossible.

Fig.2-2Atthevertex360degisdividedintoequalangles

However,ifweaddothertypesofpolygons,tilingsofregularpentagons,regularheptagons,regularoctagons,etc.becomepossible.

Fig.2-3Left:

Combinationofregularpentagonsandrhombi.Thicklineindicatesaunitstructureofperiodicity.Middle:

Combinationofregularheptagonsandpentagons.Right:

Combinationofregularoctagonsandsquares

2.1.3Randomtiling-Collinslattice

Regulartrianglesandsquareswithequallengthsofedgescanbetiledrandomly.ThistilingiscalledCollinslattice.TwoexamplesofCollinslatticeareshownbelow.

Fig.2-4Collinslattice

Notethattherearefourtypesofarrangementsaroundavertex,asshownbelow.

Fig.2-5FourtypesofarrangementsinCollinslattice

2.1.4Voronoitessellation

Letapointdistributionbegivenon2Dplane.Ifweconnectneighboringthreepoints,theplaneisdividedbytriangles（solidlinesintherightfigure）.ThisnetworkoftrianglesiscalledaDeraunayNetwork.Next,ifwedrawperpendicularbisectorlinesforalledgesofDeraunaynetwork（dashedlinesintherightfigure）,weobtaindivisionoftheplanebypolygonsofvariousshapes.ThisdivisioniscalledaVoronoitessellationoraVoronoinetwork.ThepolygonsarecalledVoronoipolygons.Bythewaythethreeperpendicularbisectorlinesforeachtriangleareprovedtomeetatonepoint.

Asimpleexampleisthepointdistributionwithregulartriangleconfiguration,whichleadstoaVoronoitessellationcomposedofequalregularhexagons（seetherightfigure）.

AnimportantpropertyoftheVoronoitessellationisthatanypointontheVoronoinetworkisseparatedequallyfromthetwopointsonbothsidesofthenetworkline（seetherightfigure）Thispropertyseemstogiveamechnismhowbordersofcountriesaredetermined.ThefigurebelowshowsaVoronoitessellationofEuropebasedontheprincipalcities（accordingtoamapin1985）.TherealbordersofcountriesaredeviatedfromtheVoronoitessellation.But,roughlyspeaking,theyarenearroeachother.

Fig.2-6DeraunayNetworkandVoronoitessellation

Fig.2-7ExamplesoftheVoronoitessellation

Similartessellationispossibleinthreedimensionalspace,ifpointdistributionisgivenin3Dspace.Inthiscasewemakeperpendicularbisectorplanesinsteadofperpendicularbisectorlines.TheresultingpolyhedraarecalledVoronoiplyhedra.

2.1.5Quasicrystal

Molecularconfiguration,whichisdeterminedexactlybyapplyingageneratingrulebutdoesnothaveperiodicity,iscalledaquasicrystal.Itissynthesizedfromalloys,orothermaterials.Structureofquasicrystalincludesapartwithregularpentagon,ascanbe