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48-747 Shape Grammars
FormingNewLanguagesfromOld
SpatialMetathesis
RECAPgrammarparadigm
Vocabulary!
GRAMMAR!
Spatial Relations!
Rules!
LANGUAGE of designs!
Initial shape!x→ s+t(additive)s+t→x(subtractive)
augmentedby
theuseoflabels Rules!
LANGUAGE!
Encapsulates“theme” Exploredfordesign“styles”Perhapsmotivatedbyrevivalorrestoration
GothicClassicalIslamicEasternstyles
Languagesgivebirthtonewlanguages
metathesis
transpositionofletters,words,sounds,syllables
bird←brid
evelate←elevate(spoonerism)
whynottranspositionofshapesorimagesbyshapereplacement?
metathesis
considertheletterequivalencep↔t
pot
tot
top
pop
byanalogythisleadstothenotionofshapeequivalencea↔b
shapeequivalencerule
Isoftheforma↔bwhereneitheranorbisempty
ApplytheruletoaspatialrelationR,asetofshapes,toproduceanewspatialrelationNprovidedRcontainsashapesandthereisageometricaltransformationfsuchthateithers=f(a)ors=f(b)
N=S–f(a)+f(b)ifs=f(a)
N=S–f(b)+f(a)ifs=f(b)
exampledesigns
ifwecanhaveshapeequivalenceruleswhynotshapeequivalenceschemas?
shapeequivalenceschema
shapeequivalenceschema
Isaschemaoftheforma↔bwhereneitheranorbisempty,aandbhaveopenterms
ApplytheschematoaspatialrelationR,asetofshapes,toproduceanewspatialrelationNprovidedRcontainsashapes,thereisanassignmentgtoallopenvariablesinaandb,andthereisageometricaltransformationfsuchthateithers=f(a)ors=f(b)
N=S–f(g[a])+f(b)ifs=f(g[a])
N=S–f(b)+f(a)ifs=f(g[b])
WhatwehaveseensofarisaFLIP‐FLOPbetweenshapes/schemaswiththeimplicitPROVISOthatnonewshapesareintroducedintotherelation
whataboutintroducingnewshapesintotheequivalencerule
transitionfromRomanesquetoGothicarches?
anymorevariations?
Wehavea↔b
Wecanconstructclassesofspatialrelationsbylookingatf[h(a)]andg[j(b)]sothat
N=S–f(a)+g(b)
N=S–f(h(b))+g(j(a))
tomakethistransitionideaworkonemustconsiderheuristicsinhowtheshapeequivalencerulesareapplied.
TransformationofGrammars
thebasicidea
Vocabulary1 Vocabulary2
SpatialRelation1 SpatialRelation2
Rules1 Rules2
Language2Language1
metatheticalchangerules
isomorphism
derivationalstructure derivationalstructure
GRAMMAR1 GRAMMAR2Transformation
tocomparelanguages
weneedtoensurethatgrammarsarespecifiedinannormalizedfashion–i.e.,inthesamesortofwayeverytime
hence,grammarsinnormalform
Vocabulary
PurelyAdditiverules
PurelySubtractiverules
Labelsarespatial
–how
–where
Statesarenonspatial–when
nonspatialorstatelabels
spatial–whereandhowlabels
stateandspatiallabels
grammarinnormalform
recursivestructureR(G)
Isabasicpropertyofgrammars
Expressesarelationshiponrules,theinitialshapeandselectedtypicalderivationsofdesignsinthegrammar
R(G)={(rulex,ruley)…}where(rulex,ruley)isamemberofR(G)whenever
• Rulexisadditiveoristheinitialshape
• Ruleyispurelyadditiveorpurelysubtractiveandruleyisappliedtothatpartofthedesignthatincludesasubshapeofalabeledshapewhichwasaddedbyapreviousapplicationofrulex
i.e.,rulexmakesruleypossible
transformationofgrammars
Comprisestwoindependentstages
DefiningshapechangerulesspecifyingtransformationTAbetweenaninitialandfinalsetofrelations
DefiningstatechangerulesspecifyingtransformationTB
TAandTBarecombinedtoproduceacompletetransformationTofG.
derivingafinalsetofrelations
transformation
PrairietoUsonian
theprairiehousegrammarinformalform
initialsetofspatialrelations
changerules
derivingasetoffinalspatialrelations
transformation!