48-747 Shape Grammars

SHAPEGRAMMARS

computation

shapegrammarshavesomethingtodowithcomputation

Constructingtheprofileofaclassicaltaperedcolumn

Therearefivesteps.

1.  Determinetheheightandlargestdiameterofthecolumn,d.Thereareclearrulesaboutpreferredproportionsbetweentheheightanddiameterofvarioustypesofclassicalcolumns(doric,ionicetc.)Thesemeasuresarenormallyrelatedtoeachotherasintegralmultiplesofacommonmodule,m.Figureshowstheshaftofacolumnwithdiameter2mandheight12m.Thatis,theproportionbetweendiameterandheightis2:12or1:6.

2.  Atoftheshaft'sheight,drawastraightline,l,acrosstheshaftanddrawasemi‐circle,c,aboutthecenterpointofl,C,withradiusd(1minthefigure).Theshaftwillhavetheuniformdiameterdbelowlinel.

3.  Determinethesmallestdiameteratthetopoftheshaft(1.5minourcase).Drawaperpendicular,l',throughanend‐pointofthediameter.l'intersectscatapointP.ThelinethroughPandCdefinestogetherwithlasegmentofc.

4.  Dividethesegmentintosegmentsofequalsizeanddividetheshaftabovelintothesamenumberofsectionsofequalheight.

5.  Eachofthesesegmentsintersectscatapoint.DrawaperpendicularlinethrougheachofthesepointsandfindtheintersectionpointwiththecorrespondingshaftdivisionasshownintheFigure.Eachintersectionpointisapointoftheprofile.

GiacomoBarozzidaVignola(1507‐73)Anelementarytreatiseonarchitecturecomprisingthecompletestudyoffiveorders,withindicationontheirshadowsandthefirstprinciplesofconstruction

Constructingtheprofileofaclassicalcolumnwithenthasis

Here,too,therearefivesteps.

1.  Determinetheheightanditsdiameter(orradius)whereitiswidestandatthetop.FollowingVignola,thebaseisagainassumedtobe2mwide,andtheheightis16m;thatis,theproportionofthediametertoheightis1:8.Thewidestradiusoccursatofthetotalheightandis1+1/9m.Theradiusatthetopis5/6m.

2.  Drawaline,l,throughthecolumnwhereitiswidest.CallthecenterpointofthecolumnonthatlineQandthepointatdistance1+1/9m.fromQonl,P.

3.  Callthepointatdistance5/6mfromthecenteratthetopandonthesamesideasP,M.DrawacirclecenteredatMwithradius1+1/9m,thatis,thecirclewiththewidestradiusofthecolumn.ThiscircleintersectsthecenterlineofthecolumnatpointR.

4.  DrawalinethroughMandRandfinditsintersection,O,withl.

5.  Drawaseriesofhorizontallinesthatdividetheshaftintoequalsections.AnysuchlineintersectsthecenterlineatapointT.DrawacircleabouteachTwithradiusm.Thepointofintersection,S,betweenthiscircleandthelinethroughOandTisapointontheprofile.

computationis…

creative

descriptive

‘mechanical’

whatisashapegrammar?

IllustrationbyPeterMurray,"theArtchitectureoftheItalianRenaissance",ShockenBooksInc.1963,Pp.96

Shape

Shaperela)on

derivation

howaretheyused?

analyses

originaldesigns

asexplanatorydevices

mughalgardens

par$subdivision

dealingwithcanals

dealingwithborders

another‐ice‐raydesigns

demo

apartmentbuildinginmanhattan

culturalmuseuminLA

developingashapegrammar

stagesinashapegrammardevelopment

shapes

spatialrelations

rules

shapegrammar

designs

shapes

basiccomponentsofgrammarsanddesign

forourpurposes,shapeis…

Shapeisafinitearrangementoflinesofnon‐zerolengthwithrespecttoacoordinatesystem

Shapescanbeformedbyadditionofshapeswhichconsistsoflinesinbothshapes.

Shapescanbeformedbysubtractionofshapeswhichconsistsofthelinesinthefirstshapethatarenotinthesecond.

ShapescanbeformedbycombinationsofthetwounderanEuclideantransformation

Acentralnotioninthisdefinitionofshapesispictorialequivalence

euclideantransformations

translation

rotation

reflection

glidereflection–exampleofacombination

scale

euclideantransformations+scale

spatialrelations

WhentwoormoreshapescombinetheyformaspatialrelationThatis,asetofshapesspecifiesaspatialrelation

shaperules

Shape A,B

Relation A+B

Rule A→A+B

B→A+B

fromrelationtorule

Relation

Rule

toderivation

rule shape possibleresults

TheapplicationofEuclideantransformationdoesnotalterthespatialrelation

labelingruleshelpscontrolruleapplication

labeledrulebasedderivation

Labeledrule

derivation

labeledrulebasedderivation

Labeledrule

derivation

labeledrulebasedderivation

Labeledrule

derivation

labeledrulebasedderivation

Labeledrule

derivation

shapegrammar

rules designs

shapegrammar

shapegrammarembodieschangeimpliesgeneration

vocabulary

shapesmadeupfromthesevocabulary

Inthegeneralcase,shapesareparameterizedschemes

productionrules(orrulesofchange,encapsulateaspatialrelation)

seedshape(wehavetostartsomewhere)

+a“notion”ofruleapplication

formally:ashapegrammaris

shape grammar G = (S, L, I, R)

initial (seed) shape belongs to the universe of labeled shapes made up of shapes in S and labels in L

shape rules

initial or seed shape

label set

vocabulary

R contain rules of the form a → b where a and b belong to the universe of labeled shapes made of shapes in S and labels in L except a cannot be empty

vocabulary

Vocabularyisalimitedsetofshapesnotwoofwhicharesimilar.

Thevocabularyprovidesthebasicbuildingblocksbymeansofwhichshapescanbegeneratedthroughshapearithmeticandgeometric(euclidean)transformations.

universe

IfwearegivenasetofshapesS,thenwecancreateasetUcalledtheuniverseofSinthefollowingmanner:

TheemptyshapeisinU

EveryshapeinSisinU

IfsandtareshapesinU,theirsums+tisinU

IfsandtareshapesinU,theirsumundertransformationsfandg,f(s)+g(t)isinU

UisthusclosedundershapeadditionandtheEuclideantransformations.

Whatcanyousayabouttheuniverseofthesetofshapesconsistingjustoneshape,asinglelineofunitlength,{(0.0),(1,0)}?

shaperuleapplication

Aruleisapplicabletothecurrentshapewhichiseithertheinitialshape

orashapeproducedfromtheinitialshapewheneverthelefthandsideoftherule‘occurs’intheobjectinwhichcaseitisreplacedbytherighthandsideoftheruleunderruleapplication

ruleapplication

Arulea→bisappliesonlyifa‘occurs’inthegivenshapeuundersome‘transformation’TinwhichcaseT(a)issubstitutedbyT(b)inthecurrentshape

Ruleapplication

v←u−T(a)+T(b)providedT(a)≤u

Wedescribethisasu⇒v

difference sum subshape

shape

Implicitinthisdefinitionisthefactthat'parts'ofshapesarerecognizableinarbitraryways

additionandsubtraction

subshapes

Ashapeisasubshapeofanotherifallthelinesinthefirstshapearelinesinthesecond

Asubshapeidentifiesapartofashape

Ashapehasindefinitelymanysubshapes

decomposingashape

reductionrulesforaddandsubtraction

additionandsubtraction

ametaphorforshapegrammars

metaphorforshapegrammars

apencil–

leadatoneendtoaddmarks

aneraserattheothertosubtractmarks

leadedsidehasashape

erasersidehasanothershape,notnecessarilythesame

togethertheyspecifyarule

–thatis,seesomethingtakeitawayanditsplacedo(make/add)somethingelse

derivingdesignsasasequenceofdesignsbasedontheworkofTerryKnighthttp://www.mit.edu/~tknight/IJDC/

language

ThelanguageofagrammarG,isthesetofallshapes(i.e.,withoutnon‐terminals)thatareproducedfromtheinitialshapethroughruleapplication

Language={shape|initial‐shape⇒*shape}

Bylabelingshapeswecan“scaffold”theprocessofgeneratingshapes

ashaperelationandacorrespondingrule

somepossiblelabelingpositionsforasquare

considerthesymmetryofasquare

labeledrules

alabeledruleandsamplederivations

anotherlabeledruleandsamplederivations

anothershaperelationandtwocorrespondingrules

somepossiblelabelingpositions

considerthesymmetryofasquare

andtherectangle

andpossibleshaperules

examplelabeling

labeling8,2

labeling6,3

labeling4,2

labeling6,4

derivationusingrulewith4,4labeling

applicationinarealdesigncontext

inspiration

franklloydwrightTheNaturalHouse

everyhouseworthconsideringasaworkofartmusthaveagrammarofitsown

”grammar”inthissense,meansthesamethinginanyconstruction‐whetheritbeofwordsorofstoneorwood

itistheshaperelationshipbetweenthevariouselementsthatenterintotheconstructionofthething

the“grammar”ofthehouseisthemanifestarticulationofitsparts.thiswillbethe“speech”ituses

tobeachieved,constructionmustbegrammatical

the worlds we study can be understood by capturing underlying relationships