48-747 shape grammars -
TRANSCRIPT
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