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!