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Volume 19, No.3
BooleanHexagonalAutomata
byAngelaCoxeandCliffReiter
[Ed:Clickonthe iguresforlarger,moredetailedversions]
Cellularautomataarelocalrulesthatareusedtoevolvediscretearrangementsofcells.Perhapsthe
mostfamouscellularautomatonisConway’sGameofLife[2].Itisbasedupontheiterationofsimple
rulesonarectangularlatticeofBooleanvaluedcells.Incrediblyrichstructureshavebeendiscovered
andthereisagrowingliteratureofamazingfactsaboutthatautomaton[4].
One‐dimensionalcellularautomataarealsoknowntohaverichstructure,evenwhenconsidering
Booleanautomataonsizethreeneighbourhoods[7].Wolfram’sbook,ANewKindofScience[7],
suggests that simple automata are the principle behind many forces, whether they be artistic,
physical,orsocial,intheworld.Evenwithoutagrandviewofautomata,theyareclearlyrelatedto
importanttechniques,suchaslocal iltersforimageprocessing.
OneintriguingautomatonthatWolframdescribesisde inedonahexagonallattice;itsbehaviorhas
somefeaturessuggestiveofasnow lake.Acelliseitherfrozen(1)ornot(0);itstaysthesameatthe
nextgenerationunlessitiscurrentlyunfrozenandexactlyoneofitsneighboursisfrozen,inwhich
caseitfreezes.Inthisnotewewillinvestigatecellularautomataonhexagonallatticesandtheir
implementationinJ.FirstweconsiderWolfram’ssnow lakeautomaton.
Wolfram’sSnow lakeAutomaton
AhexagonallatticemaybeviewedasacollectionofpointsarrangedlikethecirclesinFigure1.A
hexagonal array of cells may be implemented via a rectangular array where alternate rows are
imaginedtobeoffsetbyhalfaposition.Inarectangulararrangementofcells,thenearestneighbour
neighbourhoodsconsistof3by3blocksofcells.InFigure1,thecelllabelled4couldbethoughtofas
thecentreofa3by3blockofcells,whicharelabelled0‐9.However,inthehexagonalview,thecell
labelled4hassixneighbours,015763,listedclockwiseandshowninbold.Weconsiderthis
hexagontobealeftcon igurationinthe3by3neighbourhood.Onalternaterows,thehexagonal
neighbourhoodswillalternatebetweenleftandrightcon igurations.
Figure1.Ahexagonalarrangementofcellswithonerectangularneighbourhood,withcells
numbered0‐8,andthecorrespondinghexagonalneighbourhood,withcellsinbold.
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Thefollowing12utilitiesimplementthehexagonalautomatonandallowustoviewit.Weexplain
howtheutilitiesworkbelow.
First,padallowsustocreateinitialcon igurations.Thus,thefollowinggivesasimplecon iguration
withasingleinitialfrozenspeck.
Ifweiteratetheautomatontwice,weseethefollowing.However,thesymmetryishardtosee.
Bydoublingthenumberofpixels(imagine2by2blockstobeacell)androtatingalternatepairsof
rowsbyone,thehexagonalsymmetrybecomesreasonablyapparent.However,itisstilleasiertosee
thesymmetryinthe igures.
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Thecentreandleftneighbourhoodisproducedfroma3by3cellvia
;theright‐handversionis
producedby
.Weconsidertheresultinglistof7numberstobethehexagonalneighbourhood.
Thecentreandnumberofneighbours(withvalue1)ofahexagonalneighbourhoodaregivenby
and
.Theheartoftheautomatonissimpletodescribe.Recallthatfrozencorrespondsto1.
Thus,thevalueofacellonlychangesvaluewhenitiscurrently0buthasexactlyoneneighbourwith
value1.Onahexagonalneighbourhood,
gives1ifthecentreis1;
otherwise the result is the result of the test of whether there is exactly one neighbour, which
completely describes the desired automaton. Now these need to be put together on all
neighbourhoods (3 by 3 blocks of cells in the rectangular view). The key idea is all 3 by 3
(halfhexagonal
neighbourhoodsareselected,andthelocalautomatonisapplied.Theadverb
automaton)accomplishesthatoneithertheleftorrightneighbourhoods;theadverbargumentis
accordinglyeither
or
.Noticethatcut 3withaleftargumentof
isperfectfor
gettingthe3by3tesselationsoffsetby1alongoneaxisand2alongtheother.The inalautomaton,
,appliestheperiodicextensionappropriatetoeachaxis,appliesthelocalautomatonand
thentheresultsareintertwined.
Figures2and3showiterations107and127onaninitialcon igurationwithasingleinitialpoint.
Whilethereissomestructuredgrowthatiteration107,thegrowth illsinoniteration127.Thisis
truenearotherpowersoftwo,hencethestructuresweseearetypicalforthisautomaton.
Figure2.Iteration107ofWolfram’ssnow lakeautomaton.
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Figure3.Iteration127ofWolfram'ssnow lakeautomaton.
MakeaMovie
Wecancreateananimationoftheevolutionofthisautomatonusingtheimage3addon[6]. The
followingfunctionmaybeusedtocreate128framesoftheevolution;thesubsequentlinesillustrate
howthatmaybedoneandthenbeassembledintoamovie.Theresultmaybeviewedat[3].
Themovieshowsthatwhiletheautomataexhibitsbeautifulgrowth;italsocon irmsourremarkthat
onlylimitedtypesofbehaviourappear.
Snow lakes
BentleyandHumphreys[1]tookthousandsofphotographsofsnow lakesandicecrystals.Figure4
showsfourexampleswhichgiveaglimpseofthediversityofsnow lakes.
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Figure4.FourofBentleyandHumphreys’snow lakes.
See[5]forawonderfuldiscussionofsnow lakegrowth.Weseethatrealsnow lakeshaveafarmore
diversestructurethanseeninWolfram’sautomaton.Inthenextsectionwebrie lyinvestigateother
automataonahexagonallatticethatmaintainthesymmetryofahexagon.
OtherHexagonalAutomata
Thesymmetriesofthehexagon(whichisthedihedralgroupdenotedD6)arethesix‐foldrotations
alongwithre lectionsthroughoppositevertices.Ourconventionistousealistofthecentreand
then 6 neighbours to represent a hexagonal neighbourhood. Thus, the rotations of the hexagon
correspondtorotationofthelast6elementsandthemirrorsymmetrycorrespondstoreversalof
thoseelements.Thus,wecanapplyallthesymmetriestoalistoflengthsixvia ,whichisde ined
below.
ThenalltheBooleanvaluespossibleonthe6neighboursmaybeclassi iedinto13classesbyusing
“key”wherethe irstelementfromtheorderedapplicationofd6markstheclass.Fouroftheclasses
areshownbelow.
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Thefollowinggivestheclassnumber(from
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)ofeachofthe64classes.
Sinceeachclassappearswithacentrevalueof0and1,thereare26Booleanvaluestobespeci iedin
ordertospecifyanautomatonwiththesymmetryofahexagon.
,de inedbelow,allowsustoassociateanumberwiththose26Booleanvalues.The
Thefunction
adverb
createsafunctionthatlooksupthevaluefromtheruleoftheappropriateclass
,andwecan
fortheinputneighbourhood.Thenweneedonlychangethelocalautomaton,
applyconstructionsasintheearliersection.
Figures5‐7showtheresultofrules25629998,27541687and32824527atiteration127.
Figure5.Automatonwithhexagonalsymmetry(rule25629998).
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Figure6.Automatonwithhexagonalsymmetry(rule27541687).
Figure7.Automatonwithhexagonalsymmetry(rule27541687).
Animationsoftheevolutionoftheseautomataappearon[3].Theseexhibitintricate,fairlysolidand
fernybehaviours.
Withfurtherexperimentation,weseethatnontrivialBooleanautomataofthistypeseemtohavea
hexagonalframeafter127iterations.Thisisnotsurprisinggiventhefactthatinordertobephysical,
weshouldprobablyrequirethat0neighbourhoodsremain0.Moreover,theinitialcon iguration
involvesonlyoneothercon igurationthatcouldallowgrowth,namely,a0cellwithexactlyone
neighbour.Ifgrowthistooccur,thatcon igurationmustbecome1.ThatisexactlytheWolframrule
thatallowschange.Hence,thatruletendstocreateanenvelopeofthebehavioursthatoccur.
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Theautomatawehavestudieddonotexhibitthesamebehavioursseeninordinarysnow lakes.
However, these automata have very rich behaviours. We have also seen that they are readily
implementedinJandeasilypresentedwithananimation.
References
[1]W.A.BentleyandW.J.Humphreys,SnowCrystals,DoverPublications,NewYork,1962.
[2]E.Berlekamp,J.Conway,andR.Guy,WinningWaysForYourMathematicalPlays,Academic
Press,NewYork,1982.
[3] A. Coxe and C. Reiter, Auxiliary Materials for Boolean Hexagonal Automata,
http://www.lafayette.edu/~reiterc/mvp/hx_auto/index.html
[4]A.Hensel,Conway'sGameofLife,http://hensel.lifepatterns.net/
[5]
K.
Libbrecht,
http://www.its.caltech.edu/~atomic/snowcrystals/physics/physics.htm
Snow
Crystals,
[6]C.ReiterandZ.Reiter,Image3Addon,www.jsoftware.com,toappear.
[7]S.Wolfram,ANewKindofScience,WolframMedia,Champaign,2002.
AngelaM.Coxe
LafayetteCollegeBox8916
Easton,PA18042USA
[email protected]
CliffReiter
DepartmentofMathematics
LafayetteCollege
Easton,PA18042USA
[email protected]
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