On Shapes and Topologies

Graph theoretic representations of shapes and shape computations

Thomas Grasl

Shape gram­mars are an estab­lished method in the field of design com­pu­ta­tion. Since their ini­tial for­mu­la­tion in the 70s the under­ly­ing the­o­ry has been devel­oped inten­sive­ly. Appli­ca­tions have mostly
been restrict­ed to show­case gram­mars remod­el­ling exist­ing bod­ies of work. Dif­fi­cul­ties in automat­ing shape gram­mars have restrict­ed their extra­mur­al impact.

This the­sis attempts to rem­e­dy the sit­u­a­tion in two ways. First, addi­tion­al appli­ca­tions of gram­mars in the field of archi­tec­ture are shown. Notably a method to gen­er­ate, enu­mer­ate and denom­i­nate topolo­gies of a spe­cif­ic typol­o­gy. Sec­ond, it is shown how to map the prob­lem of shape gram­mars onto graph gram­mars. Graph gram­mars are more wide­ly used and have fur­ther devel­oped imple­men­ta­tions. Struc­tural­ly they resem­ble shape gram­mars and they are well suit­ed to car­ry out the under­ly­ing com­pu­ta­tions. Addi­tion­al­ly by abstract­ing shapes to graphs a nov­el solu­tion to imple­ment­ing para­met­ric shape gram­mars is developed.

Chap­ter 2 “Sort Machines” exam­ines pos­si­ble topolo­gies for the U.S. fed­er­al cour­t­house typol­o­gy. Here it is shown how to use graph gram­mars to gen­er­ate pos­si­ble cour­t­house topolo­gies. The exam­i­na­tion is then extend­ed by enu­mer­at­ing these pos­si­bil­i­ties. Final­ly, the struc­ture intro­duced by the gram­mar is used to devel­op a nomen­cla­ture for the topolo­gies, giv­ing each an infor­ma­tive name for dis­cus­sion and comparison.

Chap­ter 3 “Trans­for­ma­tion­al Pal­la­di­ans” describes an imple­men­ta­tion of a spe­cif­ic gram­mar. The exist­ing shape rules are tran­scribed into graph rules. Due to an appro­pri­ate map­ping all oper­a­tions can be car­ried out on the graph. The result is then trans­formed back to shapes.

This all cul­mi­nates in Chap­ter 4 “From Topolo­gies to Shapes” where a gen­er­al solu­tion for map­ping para­met­ric labelled shape gram­mar to graph gram­mars is pre­sent­ed. The nov­el­ty lies in the flex­i­ble descrip­tion of shapes by con­strained topolo­gies. The pre­sent­ed method recog­nis­es emer­gent shapes and can be extend­ed with arbi­trary weights.