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FOREWORD BY #ECIL "ALMOND AFTERWORD BY 3ANFORD +WINTER 0RINCETON !RCHITECTURAL 0RESS .EW 9ORK
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)NTRODUCTION "ENJAMIN !RANDA #HRIS ,ASCH
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3PIRALING PRODUCES A SHAPE UNLIKE ANY OTHER BECAUSE IT IS SELDOM EXPERIENCED AS GEOMETRY BUT RATHER AS ENERGY
3PIRALING
8IFUIFSTUBST TUPSNDMPVET PSQFUBMTPGBnPXFS UIFTQJSBM JTPOMZEFUFDUBCMFCZPCTFSWJOHUIFUIJOHTDBVHIUJOJUTXBLF %SPQMFUTPGGFSSPMJRVJEQMBDFEJOBQPMBSJ[FETPMVUJPO BCPWF SFWFBMUIBUNBHOFUJDFOFSHZOBUVSBMMZEJTUSJCVUFTJUTFMGJOB TQJSBMNBOJGFTUBUJPOɨFGPSNJTBMTPJOIFSFOUJOQMBOUHSPXUI QBUUFSOT XIJDIBMMPXUIFNBYJNVNOVNCFSPGQFUBMTUPHSPX JOUIFMFBTUBNPVOUPGTQBDFɨFTQJSBMJTOPUTPNVDIBTIBQF BTUIFFWJEFODFPGBTIBQFJOGPSNBUJPO
3PIRALING
2ECIPE FOR 3PIRALING 0ICK AN ANGLE D )MAGINE A CIRCLE 0LOT ONE POINT ON THIS CIRCLE AT D DEGREES FROM THE ORIGIN
0 LOT ANOTHER POINT AT D DEGREES FROM THE LAST POINT ON A CONCENTRIC CIRCLE THAT IS SLIGHTLY BIGGER THAN THE CIRCLE BEFORE IT 2EPEAT STEP oq BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
4HE COORDINATES OF THE KTH POINT IN A SPIRAL LATTICE WITH DIVERGENCE D AND EXPANSION PARAMETER ' ARE GIVEN BY 'K COSKD 'K SINKD
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o )N A SPIRAL LATTICE THE EYE TENDS TO CONNECT NEAREST POINTS INTO SPIRALS 4HESE SPIRALS WITHIN THE SPIRAL ARE CALLED PARASTICHIES )N PLANTS THE NUMBER OF THESE VISIBLE SPIRALS ARE MOST OFTEN TWO SUCCESSIVE ELEMENTS OF THE &IBONACCI SEQUENCE ONE IN WHICH EACH NUMBER IS THE SUM OF THE PREVIOUS TWO
p 4HIS MODEL IS TYPICALLY USED TO DESCRIBE 0HYLLOTAXIS 'REEK PHYLLO LEAF TAXIS ARRANGEMENT A NATURALLY OCCURRING PLANT GROWTH PATTERN THAT GOVERNS THE ARRANGEMENT OF LEAVES FLOWER PETALS PINE CONES ETC
3PIRALING
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
3PIRALING
3PIRALING
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3PIRALING
3PIRALING
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3PIRALING
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3PIRALING
0ACKING PRODUCES STABILITY THROUGH ADJACENCY
0ACKING
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0ACKING
2ECIPE FOR 0ACKING #REATE A SHAPE OF A RANDOM SIZE 0ICK A RANDOM POINT A )F THE SHAPE IS INSIDE ANOTHER SHAPE OR OVERLAPS ANOTHER SHAPE o THROW IT AWAY AND GO BACK TO STEP q B )F NOT PLACE IT 'O TO STEP BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
3PHERES CIRCLES ARE NATURALLY STABLE SHAPES 7HEN PACKED TOGETHER THEY
CREATE A VERY STRONG CONSTRUCTION OWING TO THE SPHERES INHERENT STABILITY AND TENDENCY FOR A COLLECTION OF SPHERES TO PRODUCE MULTIPLE POINTS OF TANGENCY
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o )F THE DISTANCE BETWEEN THE CIRCLES IS LESS THAN THE SUM OF THEIR RADII THEN THEY OVERLAP
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0ACKING
%XPERIMENTS 4HREE $IMENSIONAL 3LICING #FZPOETFSWJOHBTBEZOBNJDTZTUFNGPSQSPEVDJOHGMBUPSHBOJ[BUJPOT QBDLJOHBMTPPGGFSTSJDIUISFFEJNFOTJPOBMBOENBUFSJBMTUSBUFHJFTXIFO DPNCJOFEXJUIPUIFSPQFSBUJPOT&OUFSUIFDVUCZTMJDJOHBXBZBUUIFQBDLJOH TUVEJFT UIFBMHPSJUINJDMPHJDJTSFMFBTFEJOUPOFXBSDIJUFDUVSBMQPTTJCJMJUJFT
0ACKING
0ACKING
02/*%#4
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0ACKING
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6IEW 7
(SPVOEGMPPSTIBQFBOEPSJFOUGPSWJFXT
4FDPOEGMPPSNBYJNJ[FTPMBSHBJO CFESPPNCBSEFTJHO 5IFIJMMCPSFUIFSBNQBOEBMMPXFEGPSBUXPTUPSZCVJMEJOH5IF TFDPOEGMPPSJTDPODFJWFEPGBTBiCFESPPNCBSwXIJDISFDFJWFTTVO GSPNUIFTPVUIFSOXBMMPGXJOEPXTBOEJTJTPMBUFEGSPNUIFHSPVOE NBLJOHJUBOJEFBMUIFSNBMNBTT"MMCFESPPNTBOEBNFOJUJFTBSF BMPOHUIFMPOHFBTUXFTUBYJTPGUIFCVJMEJOHJOBOUJDJQBUJPOPGQBTTJWF TPMBSIFBUJOH
(SPVOEGMPPSQMBO
0ACKING
02/*%#4 ,/' #!").
4ILING 5IFOPSUIGBDBEFJTMFGUNPSFPQBRVFUPNJOJNJ[FIFBUMPTT 4FMGTJNJMBSQBUUFSOTPGTFFNJOHMZJOGJOJUFWBSJFUZDBOCF QSPEVDFECZNJSSPSJOHUJMFTBOEKPJOJOHDPEFEFEHFT
0ACKING
'RADIENT 5IFTPVUIGBDJOHGBDBEFPQFOTVQUPXBSEUIFXFTUFSOFEHFPG UIFCVJMEJOH XIFSFUIFDPNNPOTQBDFTBSFHBUIFSFEUPUBLF BEWBOUBHFPGUIFWJFX
'RADIENT
0ACKING
7EAVING PRODUCES STRENGTH BY COMBINING TWO WEAK SYSTEMS IN A RECIPROCAL PATTERN
7EAVING
8FBWJOHJTUIFTZOUIFTJTPGUXPEJGGFSFOUTZTUFNT JOUFSMPDLJOH JOPSEFSUPHJWFTFMGTVQQPSUJOHGPSNUPUIFJSDPNCJOFEXIPMF 5SBEJUJPOBMMZSFGFSSFEUPBTBiXBSQwBOEBiXFGUwQBUUFSO OFJUIFSDPVMETVQQPSUUIFNTFMWFTBMPOF CVUUPHFUIFSUIFZ CFDPNFTUSPOH5IFFOEMFTTWBSJFUZPGXFBWJOHTFFOJOCBTLFU OFU SPQF BOEUFYUJMFEFTJHOQSPWFTUIBUQSPDFEVSBMUFDIOJRVFT BOEDVMUVSBMQSBDUJDFTBSFOPUNVUVBMMZFYDMVTJWF.PTU TVSQSJTJOHBCPVUBXPWFODPOTUSVDUJPOJTUIBUJUJTBDUVBMMZ IBSEFSUPVOSBWFMUIBOUPXFBWFJOUIFGJSTUQMBDF
7EAVING
2ECIPE FOR 7EAVING 4UBSUESBXJOHBTJODVSWFBMJOFUIBUHPFTBSPVOEBDJSDMFBUB TUFBEZSBUF TQSFBEPVUPWFSUJNF -PPQUIFDVSWFCZBEEJOHBUFSNBNBUIFNBUJDBMGVODUJPO MJLFDPT UIBUTQFFETVQBOETMPXTEPXOUIFMJOFBTJUHPFT BSPVOEUIFDJSDMFo q "EENPSFUFSNTUPDSFBUFNPSFMPPQT PWFSMBQT BOETRVJHHMFT .JSSPSUIFDVSWFGPSBEFOTFS JOUFSMPDLJOHGJHVSFf
[ W\ 6LQ IUHT W ZKHUHW WLPH
[ W&RV IUHT W \ 6LQ IUHT W
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
4HE COMPONENTS OF THE EQUATION ARE SCALE FREQUENCY AND AMPLITUDE 4HESE MATHEMATICAL ATTRIBUTES REPLACE THE TRADITIONAL KNOTMAKING PROCEDURES OF TRANSLATION TURNING AND REFLECTION %ITHER ONE OF THESE SETS OF ATTRIBUTES CAN PRODUCE AN ENDLESS VARIETY OF FORMS THAT ARE TRACEABLE BACK TO SIMPLE RULES
7KHFXUYHDERYHPLUURUHGDURXQGWKH ;D[LV [ W&RV IUHT W \ 6LQ IUHT W [ W&RV IUHT W \ 6LQ IUHT W
o !DDING A COS TERM IN THE X PORTION OF THE EQUATION AFFECTS THE HORIZONTAL EXPANSION OF THE POINTS THAT MAKE UP THE CURVE 4HE X TERM PUSHES OR PULLS THESE POINTS ALONG THE hTIMEv LINE T UNTIL THE CURVE BEGINS TO LOOP BACK ON ITSELF
q !DDING A Z TERM GIVES A THREE DIMENSIONAL ASPECT TO THE CURVE e -ANY TRADITIONAL WEAVING PATTERNS MAKE USE OF SYMMETRY BECAUSE IT PROVIDES GUARANTEED
POINTS OF OVERLAP THAT HELP TO STRUCTURE THE WEAVE
7EAVING
%XPERIMENTS #ROSSING 0ATTERNS 5IFTFTLFUDIFTVTFBQBSBNFUSJDFRVBUJPOUPPSHBOJ[FBTFSJFTPGTJOFBOE DPTJOFDVSWFTJOTQBDF5IFXFBWFJTBDSPTTJOHQBUUFSO BiTPGUwTUSVDUVSFPG MPPQTBOELOPUTXIFSFJOUIFTIBQFPGUIFDPOTUSVDUJPOJTEFUFSNJOFEMFTTCZ UIFQSPQFSUJFTPGUIFNBUFSJBMTUIFNTFMWFTUIBOCZUIFQBUUFSOUISPVHIXIJDI UXPTFUTPGNBUFSJBMTJOUFSBDU
7EAVING
7EAVING
02/*%#4
GJH
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
3VMFTJOHMFTFFE
3VMFTJOHMFTFFE
3VMFSBOEPNTFFE
3VMFSBOEPNTFFE
3VMFTJOHMFTFFE 722/,1*
7EAVING
"LENDING IS A FUNDAMENTAL TECHNIQUE IN THE ACT OF NEGOTIATION
"LENDING
#MFOEJOHIPMETUIBUHJWFOBOZUXPPSJHJOBMTUBUFTTVDI BTTQFFE UJNF TIBQF PSDPOUPVSBOJOGJOJUFOVNCFSPG TVCTUBUFTBSFBWBJMBCMFUPFTUBCMJTIBDPOOFDUJPOCFUXFFOUIFN 5IFDVMUJWBUJPOPGMBOEGPSBHSJDVMUVSFEJTQMBZTBCMFOEFE QBUUFSOOFHPUJBUJOHTIBQFBOEDPOUPVS UIFNBQQJOHPGUSBGGJD DPOHFTUJPOPWFSUJNFTIPXTUIBUTQFFETBSFCMFOEFEUISPVHI DSJUJDBMUISFTIPMETPOUIFSPBE5IFJSVOEFSMZJOHNBUIFNBUJDBM MPHJDJTFODPVSBHJOHCFDBVTFJUEJDUBUFTUIBUBDPNNPOBMJUZ CFUXFFOBOZUXPTUBUFTDBOCFFBTJMZEFUFSNJOFE
"LENDING
2ECIPE FOR "LENDING 0ICK TWO SHAPES A BEGINNING STATE AND AN ENDING STATE
"REAK THE SHAPES DOWN INTO AN EQUAL NUMBER OF CONSTITUENT POINTSo IN SPACE $IVIDE UP THE SPACE BETWEEN THE TWO SHAPES IN AN INTERESTING WAYq )NTERPOLATE BETWEEN A POINT ON ONE SHAPE AND THE CORRESPONDING POINT ON THE OTHER SHAPE BY TAKING THE DISTANCE BETWEEN THEM AND MULTIPLYING BY THE CURRENT RATIOe EACH DIVISION IS A RATIO OF THE WHOLE ETC 2EPEAT FOR EACH POINT BY EACH DIVISION BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
$PQZJOHJTBTVCTFUPGCMFOEJOH JUJTUIFNPSQIJOHPGPCKFDUTXJUIPVUDIBOHJOHJUTTUBUF o *OUIJTJOTUBODF QPJOUTBSFUIFDPOTUJUVFOUQBSUTPGPVSRVBMJUZTIBQF5IFTBNFQSJODJQMFIPMETGPS CMFOEJOHDPMPST SHC
QPTJUJPOT Y Z
TDBMF
PSBOZPUIFSBUUSJCVUF
q5IJTJTSFBMMZUIFIFBSUPGUIFBMHPSJUIN*UEFUFSNJOFTUIFTNPPUIOFTTPGUIFUSBOTJUJPOCFUXFFO TUBUFTBOEBMTPUIFiDVSWFwPGUIFDIBOHFJUTGSFRVFODZQSPGJMF5IFTQBDFDBOCFEJWJEFEJOUP FRVBMQBSUT PSBGVODUJPODBOCFBQQMJFEUPBEKVTUUIBUSBUJP"OFYQPOFOUJBMGVODUJPO O NBLFT UIFTQBDFCFUXFFOFBDIEJWJTJPOHFUQSPHSFTTJWFMZCJHHFSXIJMFBSPPUGVODUJPO ǔO XJMMNBLFUIF EJWJTJPOTQSPHSFTTJWFMZTNBMMFS
5IJTDSFBUFTBGJFMEPGQPJOUTCFUXFFOUIFUXPJOJUJBMTIBQFTUIBUDBOCFVTFEUPNPEJGZUIFJOJUJBMHF
f
PNFUSZPSDBOCFUBLFOBTBGSBNFXPSLXJUIXIJDIUPXFBWFBOFOUJSFMZOFXTZTUFNJOUPUIFGJSTUPOF "LENDING
%XPERIMENTS !GGREGATE 3URFACES 5IFTFFYQFSJNFOUTXFSFQSPEVDFECZNPEJGZJOHBOEFYUFOEJOHUIFCMFOEJOH BMHPSJUINBOEJUTWBSJBCMFT QVSTVJOHBOJOUFSFTUJOUIFEFTDSJQUJPOPG TIBQFTJOTQBDFBOEUIFSFMBUJPOTIJQTCFUXFFOUIFN4JODFJUJTOFDFTTBSZUP GJSTUCSFBLBTIBQFEPXOJOPSEFSUPGVMMZVOEFSTUBOEJUTRVBMJUJFT JUJTBO JOWFTUJHBUJPOJOUPEFDPNQPTJUJPO OPUBTBEFTUSVDUJWFBDU CVUBDSFBUJWFPOF
"LENDING
"LENDING
02/*%#4
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
"LENDING
(OW TO (IDE A ,OVELY 6IEW &TUBCMJTIWFDUPSTPOTJUFGPSSFGMFDUJOHUIFWJFXBOETVSSPVOEJOHGPSFTU $SFBUFCMFOE &YUSVEFCMFOEUPDSFBUFBTUSVDUVSBMXBMMPGSFGMFDUJWFTUBJOMFTTTUFFM & YUSBDUIPMFQBUUFSO DBNPVGMBHF GPSWJFXJOHUISPVHIXBMM 1PVSGPVOEBUJPOGPSXBMMBTTFNCMZXIJDIBMTPTFSWFTBTBCFODI 4FMGTUSVDUVSBMDBNPVGMBHFXBMMBDUTBTGPSNXPSLGPSCFODI %JWJEFJOUPPWFSMBQQJOHTFDUJPOTXIJDIBSFUBQFEUPHFUIFSPOTJUF 'MBUUFOGPSMBTFSDVUUJOHBOETIJQQJOH
"LENDING
"LENDING
"LENDING
"LENDING
"LENDING
&OLLOWING THE RULE OF SELF SIMILARITY CRACKING GIVES A SENSE OF THE LARGER WHOLE
#RACKING
#ZSFDBMMJOHJUTTPVSDFTIBQFSFDVSTJWFMZ DSBDLJOHHFOFSBUFT BHFPNFUSZPGTFMGTJNJMBSJUZ'PSJOTUBODF BSJWFSEFMUBIBT WBSJPVTMZTDBMFEiUSJBOHMFTwUIBUBSFFBDIiDSBDLFEwCZUIF JUFSBUJWFBOEBHHSFHBUFQSPDFTTPGGMVWJBMFSPTJPO5IFDSBDL QBUUFSOTJOESZNVEPSQBJOUTIPXBTJNJMBSSFDVSTJPOPG TIBQFTJOBUMFBTUUXPWJTJCMFTDBMFT8IFUIFSJUJTUIFEZOBNJDT PGXBUFSDIBOOFMJOHUISPVHITFEJNFOUUPQSPEVDFBEFMUBPS IFBUBOEESZOFTTDBVTJOHQBJOUUPQFFM DSBDLJOHJTBEJTUJODU BDUJPOQFSGPSNFEUISPVHINBUFSJBMT
#RACKING
2ECIPE FOR #RACKING #HOOSE A SHAPE TO BE CRACKED &IND ITS CENTROID #REATE SUBSIDIARY SHAPES BY CONNECTING THE CENTROID TO EACH END OF ONE EDGE OF THE PARENT SHAPE 2EPEAT STEPS AND FOR EACH NEW SHAPE o #ONTINUE UNTIL A LIMIT IS REACHED #HOOSE AN ITERATION OF THE ALGORITHM WHOSE SUBSIDIARY SHAPES WILL BE LEFT WHOLEq BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
5IJTBMHPSJUINQSPEVDFTBDPOTUSVDUJPOJOXIJDIFBDIFEHFJTTIBSFECZFYBDUMZUXP TIBQFTBOEFBDIFEHFJTDPOUJOVPVTDPOOFDUFEUPBOFEHFXIJDIJTDPOOFDUFEUPBO FEHF BOETPPOOPNBUUFSIPXEFOTFUIFNFTICFDPNFT
o *GPOFXFSFUPMPDBMJ[FUIFDSBDLJOHDSBDLNPSFJOPOFQBSUPGUIFTUSVDUVSFUIBO
BOPUIFSPOFDPVMEDSFBUFQBUDIFTNBEFVQPGBIJHIFSOVNCFSPGTIPSUFSNFNCFST
q &BDIJUFSBUJPODPOUBJOTBOFYQPOFOUJBMMZHSFBUFSOVNCFSPGTIBQFTUIBOUIFPOFCFGPSF JU&BDIJUFSBUJPOUBLFTBOFYQPOFOUJBMMZMPOHFSUJNFUPQSPDFTT
#RACKING
%XPERIMENTS 2ECURSIVE 3KETCHES 5IFDSBDLJOHBMHPSJUINVTFEJOUIFTFTLFUDIFTJTPOFUIBUUBLFTBTIBQFBOE EJWJEFTJUBDDPSEJOHUPBWBMVFTFUCZUIFVTFS*OFBDIJOTUBODF UIFTIBQFJT CFJOHSFDVSSFEUPBOPUIFSTFUPGTFMGTJNJMBSTIBQFT3FDVSSJOHJTBCPVUBSFUVSO UPPSJHJOTXJUIJOFBDIOFXDSBDLUIFTVCEJWJTJPOFYQSFTTFTTPNFUIJOHVOJRVF BCPVUUIFJOJUJBMTUBUFGSPNXIJDIJUCFHBO*UJTBOFNCFEEFEMPHJDJNQPTTJCMF UPFTDBQFBOEGBTDJOBUJOHUPXBUDIQSPMJGFSBUF
#RACKING
G
H
I
J
#RACKING
02/*%#4
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
#RACKING
4FDUJPOT
5IFUPQBOECPUUPNTUSBUBBSFDSBDLFEJOBTJNJMBSGBTIJPO CVUXJUIEJGGFSFOUTUBSUJOHQPJOUT SFTVMUJOHJOBOPGGTFUXIFSF UIFZNFFU
4JUF
#RACKING
02/*%#4 $%,4! ! 2)6%2 -53%5-
/JHIUUJNFWJFXUPXBSEFOUSBODF
#RACKING
*OUFSJPS
$ELTA &ORMATION
REM
3PPG GPSNBUJPO
"VEJUPSJVN PSHBOJ[JOHIVC
C UT OVE
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(SPVOE GPSNBUJPO
0VUEPPSHBMMFSJFT SJWFSFDPMPHZ FYIJCJUT
C UT OVE REM
$SBDL .BKPSQSPHSBNNBUJDEJWJTJPOT
$SBDL .BKPSTUSVDUVSBMFYUSVTJPOT
$SBDL .JOPSTUSVDUVSBMXFCCJOH
$SBDL 5FSUJBSZTUSVDUVSBMSFJOGPSDFNFOU
$VU3FNPWF $SBDLJOHBMXBZTUFOETUPXBSEWFSUJDFT BOETJODFBEFMUBIBTPOMZPOFNPVUI UIFFMJNJOBUJPOPGUIFPUIFSUXP WFSUJDFTXBTOFDFTTBSZ
#RACKING
&LOCKING FINDS ORDER THROUGH ENTROPY
&LOCKING
5IFSFJTOPCFUUFSXBZUPJMMVTUSBUFIPXDPNQMFYJUZDBO FSVQUGSPNNBOZMPDBMJOUFSBDUJOHEFDJTJPOTUIBOGMPDLJOH 5IFJOEJWJEVBMFMFNFOUTNJHIUCFQMBOUTBOEBOJNBMTJOBO FDPTZTUFN WFIJDMFTJOUSBGGJD PSQFPQMFJODSPXET*GPOF DPOTJEFSTUIFDJUZBEZOBNJDJOXIJDITZTUFNTPGGMPXTVDI BTUSBGGJD DJSDVMBUJPOPGHPPET PSDSPXECFIBWJPSDBOCF BQQMJFEUPBOZVSCBOUSBOTGPSNBUJPO UIFOGMPDLJOHQSPWJEFT BWJUBMNPEFMPGDPNQMFYDPPSEJOBUJPOUIBUEFTDSJCFTUIFTF NBUFSJBMTIJGUT
&LOCKING
2ECIPE FOR &LOCKING &OR EACH AGENT FOR EACH INCREMENT OF TIME A !VOID CROWDING LOCAL FLOCKMATES 3TEER TO KEEP A MINIMUM DISTANCE BETWEEN EACH AGENT AND THE ONES AROUND ITo B ! LIGN TOWARDS THE AVERAGE HEADING OF LOCAL FLOCKMATES C #OHERE TO THE FLOCK MOVE TOWARD THE CENTER MASSq OF LOCAL FLOCKMATES
B
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
#RAIG 2EYNOLDS FIRST COMPILED THE CLASSIC FLOCKING ALGORITHM IN IN A PROJECT SIMULATING THE WAY THAT BIRDS AND OTHER FLOCKING HERDING AND SCHOOLING ANIMALS BEHAVE (E CALLED HIS COMPUTER SIMULATED AGENTS "OIDSA CONTRACTION OF BIRDS AND DROIDS &LOCKING CONTINUES TO BE AN EVOCATIVE EXAMPLE OF EMERGENCE WHERE COMPLEX GLOBAL BEHAVIOR CAN ARISE FROM THE INTERACTION OF SIMPLE LOCAL RULES
C
o )N FLOCKING MODELS A BOID REACTS ONLY TO FLOCKMATES WITHIN A CERTAIN NEIGHBORHOOD AROUND ITSELF THERE IS NO GLOBAL STEERING INTELLIGENCE 4HE NEIGHBORHOOD IS DEFINED BY A DISTANCE FROM THE CENTER OF THE BOID AND AN ANGLE AROUND IT MEASURED FROM ITS DIRECTION OF TRAVEL
q 4HE hCENTER MASSv IS THE AVERAGE POSITION OF ALL THE AGENTS
D &LOCKING
%XPERIMENTS 4RAFFIC &LOCKS 0GBMMNPEFTPGIVNBOUSBOTQPSUBUJPO WFIJDVMBSUSBGGJDJTNPTUQPJTFEXJUIUIF QPUFOUJBMUPGMPDL5IFXPSEiUSBGGJDwJTVTVBMMZUBLFOUPNFBOiUPPNBOZDBST wCVU JUDBOBMTPJNQMZUIFTJNQMFGMPXPGDBSTBMPOHBSPBEXBZ5IFDPOHFTUJPOUIBUJU DPNNPOMZEFTDSJCFTJTCVUPOFTJEFPGUIFFRVJMJCSJVN UIBUPGTUBHOBUJPO)PXFWFS XJUIBEKVTUNFOUTUPUIFSPBEXBZ OPMBOFT BOEWFIJDVMBSUFDIOPMPHZ DPMMJTJPO BWFSTJPO
USBGGJDKBNTDBOCFSFWFSTFEJOUPQSPEVDUJWFFOUJUJFT GMPDLTUIBUZJFME GPSNBUJPOTPGPSEFSUFUIFSJOHBUUIFFEHFPGDPOUSPM4UVEJFTJOUSBGGJDGMPDLJOHQPJOU UPXBSENBTTJWFTDBMFE GJSTUDPNFGJSTUTFSWFPSHBOJ[BUJPOTXIFSFEFTJSFTGPSNPWJOH TIPQQJOH BOEQMBZJOHNFSHFPOBTJOHMF TFMGPSHBOJ[FEQMBOFPGNVUVBMDPPQFSBUJPO
&LOCKING
&LOCKING
02/*%#4
4HE "ROOKLYN 0IGEON 0ROJECT 5IF#SPPLMZO1JHFPO1SPKFDUJTBOFYQFSJNFOUJOEFWFMPQJOHBTBUFMMJUFUIBUSFDPSETUIF DJUZBTTFFOCZBGMPDLPGCJSET6TJOHUSBJOFEQJHFPOTBOEXPSLJOHXJUITFBTPOFECJSE GMZFST UIFQSPKFDUUFBNFRVJQTQJHFPOTUIBUGMZJOSFHVMBSTQJSBMQBUUFSOTPWFSTXBUDIFT PG#SPPLMZOXJUIXJSFMFTTWJEFPDBNFSBTBOENJDSPQIPOFT)BSOFTTFEUPUIFTFDVTUPN DBNFSBTBOETNBMMCBUUFSZQBDLT UIFCJSETCFDPNFTBUFMMJUFTDBSSZJOHiFBSUITFOTJOHw FRVJQNFOUUIBUGFFETJNBHFTBOETPVOETPGUIFDJUZCBDLUPBHSPVOEMPDBUJPO5IFJS GMJHIUQBUITDBQUVSFVODPOWFOUJPOBMQPSUSBJUTCPUIPGUIFDJUZCFMPXBOEPGGMPDL NPUJPOT5IJTVOJRVFXBZUPTFF#SPPLMZODPOUSBTUTEJSFDUMZXJUIUIFXBZUIFDJUZJT JODSFBTJOHMZSFDPSEFEBOESFQSFTFOUFEUPEBZ5IFBEWFOUPGHFPHSBQIJDJOGPSNBUJPO UFDIOPMPHJFTBOEUIFSJTFPGOFUXPSLQSPUPDPMTIBWFQMBDFEWJSUVBMMZBMMVSCBOJNBHJOH BOESFNPUFTFOTJOHTZTUFNTiPOUIFHSJEw6TJOHBGMPDLPGCJSETBTPOFDPNQPOFOUPG BOJNBHJOHBQQBSBUVT UIJTQSPKFDUBUUFNQUTUPDPOGSPOUUIFMJNJUTPGUIJTHSJECZ DSFBUJOHBOFRVBMMZSJDIEJTDMPTVSFPGUIFDJUZTFFJOHUIFDJUZBTBGMPDLXPVME
&LOCKING
4BUFMMJUFWJFXUISFFTFDPOEGMJHIU
4BUFMMJUFEPDLJOHJOUPUIF%$1 EBUBDPMMFDUJPO QMBUGPSN
4BUFMMJUFDPNQPOFOUT
&LOCKING
02/*%#4 4(% "2//+,9. 0)'%/. 02/*%#4
4UJMMTGSPNTPSUJF .BZ
&LOCKING
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