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LindenmayerSystems:DetailsOverview

Asabiologist,AristidLindenmayerstudiedgrowthpatternsvarioustypesofalgae.In1968hedevelopedLindenmayersystems(orLsystems)asamathematicalformalismfordescribingthegrowthofsimplemulticellularorganisms.Morerecently,Lsystemshavefoundseveralapplicationsincomputergraphics.Twoprincipalareasincludegenerationoffractalsandrealisticmodelingofplants.

Basically,aLindenmayersystembeginswithastringofsymbolscalledtheaxiom,andappliestotheaxiomasetofproductionruleswhichareusedtorewritetheaxiom.

Forexample,lettheaxiomofaLindenmayersystembethestringofthreesymbols'peg',andletthesetofproductionrulesbethesinglerule:'e=i'.Thisissummarizedinthefollowingtable:

axiom: pegrules: e=i

Whentheruleisappliedtotheaxiom,each'e'symbolsintheaxiomisreplacedwithan'i'symbol.Thus,theaxiom'peg'becomes'pig'.Thisnewstring'pig'iscalledthefirstgenerationoftheLsystemsinceitistheresultofapplyingtherulestotheaxiomonetime.

ThesecondgenerationofthisLsystemistheresultofapplyingtherulestothefirstgeneration.Sincethereareno'e'symbolsinthefirstgeneration,thereisnothingfortheruletoreplace.Thusthesecondgenerationisalsothestring'pig'.ThisratherboringLsystemissummarizedbelow:

BoringLSystemaxiom: pegrules: e=igeneration1: piggeneration2: piggeneration3: pig

Lsystemsaremoreinterestingwhentherulesarerecursive.Arecursiveruleisarulethatreplacesasymbolwithacopyofitselfplussomethingextra.Anexampleofarecursiveruleis'e=eie'.ThetablebelowshowsthefirstthreegenerationsofanLsystemwiththisrecursiverule:

RecursiveLSystemaxiom: pegrules: e=eiegeneration1: peieggeneration2: peieieieggeneration3: peieieieieieieieg

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Inthefirstgeneration,thesingle'e'symbolintheaxiomwasreplacedwith'eie'creatingthestring'peieg'.SincetheruleinthisLsystemisrecursive,thefirstgenerationstringhastwonew'e'symbols,eachofwhichgetreplacedwith'eie'inthesecondgeneration...andsoitgoes.

ThenextsectionsummarizesthedefinitionofanLsystemmoreformally.

DefinitionofanLSystem

AnLsystemisaformalgrammarconsistingof4parts:

1. Asetofvariables:symbolsthatcanbereplacedbyproductionrules(seebelow).IntheFractalGrowersoftware,variablescanbeanyofthe26lowercaseEnglishlettersathroughz

2. Asetofconstants:symbolsthatdonotgetreplaced.IntheFractalGrowersoftware,theconstantsareanyofthefollowingsymbols:!,[,],+,.

3. Asingleaxiomwhichisastringcomposedofsomenumberofvariablesand/orconstants.Theaxiomistheinitialstateofthesystem.

4. Asetofproductionrulesdefiningthewayvariablescanbereplacedwithcombinationsofconstantsandothervariables.Aproductionconsistsoftwostringsthepredecessorandthesuccessor.

GraphicInterpretationofStrings

RecursiveLsystems,liketheonedescribedabove,oftenproduceintricatelycomplexpatternsthatareselfsimilaracrossmultiplescales.Thesepatternsarealmostimpossibleforahumantoperceivedirectlyfromlongstringsofsymbols.Aswithmanytypesofdata,agraphicalrepresentationofthestringscanexposethesepatterns.

ThemostcommongraphicalinterpretationappliedtoLSystemsisbasedonturtlegraphicswhichisthebestknownfutureoftheLogoprogramminglanguage.Logowasinventedin1967byWallyFeurzeigandSeymorurPapertasakidfriendlywayofteachingcomputerscience.Theturtleinturtlegraphicsisanonscreencursor(oronfloorrobot)whichcanbegivendrawing(ormovement)instructionssuchasmoveforwardbyaspecifieddistanceorturnrightorleftbyaspecifiedangle.Traditionallythecursorinturtlegraphicsisrepresentedpictoriallyasaturtleicon.

WhenapplyingturtlegraphicstoLsystems,astateoftheturtleisdefinedasaquadruple(x,y,a,c).TheCartesiancoordinates(x,y)representtheturtle'sposition.Theanglea,calledtheheading,isinterpretedasthedirectioninwhichtheturtleisfacing.Thecolorcisinterpretedasthecolorpenthattheturtlecurrentlyhaspressedtothefloorsothatanymovementoftheturtlewillcreatealineofthatcolor.Giventhestepsizedandtheangleincrementb,theturtlecanrespondtosymbolsinanLsystemstringaccordingtothefollowinginstructions:

f

Moveforward(inthedirectionofthecurrentheading)adistancedwhiledrawingalineofcolorc.Thestateoftheturtlechangesto(x',y',a,c),where

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x'=x+dcos(a)andy'=y+dsin(a).

h Sameasf.

g Sameasfexceptnolineisdrawn.

!

Likethefandhsymbols,the!symbolmovestheturtleforward.Unlikethefandhsymbols,thedistancethatthe!symbolmovestheturtleisafunctionofthesymbol'sage.

The!symbolisaconstant,soonceitisaddedtoastring,itremainsinthestringforallfuturegenerations.EveryNEW!symbolmovestheturtlethesamedistanceasdoeseveryf,g,andhsymbol.Everygenerationthata!symbolages,itslengthisdividedbythevalueofShrinkage.Thus,aShrinkagevalue>1makesolder!symbolsshorter,andaShrinkagevalue

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thesesymbolsareignoredbytheturtle,theyarestillusefulintheLsystemgrammarforbuildingstringstructurescontainingsymbolsnotignoredbytheturtle.

Noteoncolor:thevariables'a'through'e'onlyeffectthecolorwhenthe'colorfandhsegmentsbyimbeddedvariables'radiobuttonisselectedonthecolortaboftheLSystemcontrolpanel.Ifthisradiobuttonisnotselected,thentheturtleignoresthe'a'through'e'symbols,andthecolorofeachlinesegmentisdeterminedbythelocationofeachforhinthestring:fandhsymbolsatthebeginningofthestringdrawredsegments,inthemiddleofthestringdrawgreensegments,andattheendofthestringdrawpurplesegments.

Noteonsteplength:InLsystemsthatdonotusethe'!'symbol,thesteplengthhasnoeffectontheappearanceofdrawing.Thisisbecause,all'f'and'h'symbolsaredrawnwiththesamesteplength,thenthecompletedrawingisscaledsoastofillthedrawingwindow.Alargestepsizewilljustcausethesoftwaretoscaledownthefinaldrawingmore.However,whenthe'!'isused,thenthesteplengthhasagreateffectonthedrawingsince'f'and'h'symbolsalwaysmoveforwardadistanceequaltothesteplengthwhereas'!'symbolsmoveforwardadistanceequaltothesteplengthraisedtothepowerequaltothesymbol'sage.

Example:DragonCurve

Manyfractals(oratleasttheirfiniteapproximations)canbethoughtofassequencesoflinesegments.

Thedragoncurveisonesuchfractal,anditsfiniteapproximationcanbecreatedwithanLSystem:DragonCurveLSystemvariables: fhconstants: +axiom: f

rules: f=fhh=f+hangleincrement: 90degreesgeneration1: fhgeneration2: fhf+hgeneration3: fhf+hfh+f+hgeneration4: fhf+hfh+f+hfhf+h+fh+f+h

ThisLSystemhastworules:oneforreplacing'f'symbolsandoneforreplacing'h'symbols.Therefore,both'f'and'g'arevariablesoftheLSystem.The'+'and''symbolsareconsideredconstantsbecausetherearenorulesinthesystemforreplacingthesetwosymbols.Theaxiomofthesystemisasingle'f'symbol.Therefore,whencreatingthefirstgeneration,theruleforreplacing'h'symbolsisnotused.

Theredandgreencolorsinthetablehelpmakethesubstitutionprocesseasiertoperceive.Ateachgeneration,thenewsymbolsthatresultfromreplacingan'f'symbolareshowningreen,andthenewsymbolsthatresultfromreplacingan'h'symbolareshowninred.Eachsymbolthatismorethenonegenerationoldisshowninblack.Onlyconstantscanbemorethenonegenerationold,

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becausevariablesarealwaysreplacedateachgeneration.Onlysomeofthe'+'and''symbolsareshowninblackbecauseonlythesearemorethenonegenerationold.

Let'stakeastepbystepexaminationofthesecondgenerationofthisLSystem.Thegeneration2stringis:'fhf+h'.Assumingthattheinitialheadingoftheturtleisupwardonthescreen,thenthefirst'f'willdrawalinestraightup.The''causestheturtletochangeitsheadingby90degreestobepointeddirectlytotheleft.The'h'drawsalinedirectlytotheleftofthescreen.Thesecond''symbolagainturnstheturtle90degreestoitsleftwhichisdirectlydownonthescreen.Thefsymboldrawsaunitlengthlineinthedowndirection.Thelastturnsymbolinthestringisa'+'whichturnstheturtletoitsrightby90degrees.However,sincecurrentheadingoftheturtleispointeddownward,itsrightisthescreen'sleft,soagaintheturtle'sheadingistotheleftofthescreen.Finally,thehsymboldrawsaunitlengthlinetothescreen'sleft.Theresultingimageisshownisfigure1withlinesegmentcolorsmatchingthe'f'and'h'colorsinthedescriptionabove.

figure1

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Copyright:JoelCastellanos,19962007

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