Spatial Relations and Grammars

Dear Sir, In Stiny (1980b), I showed that spatial relations provide a very natural basis for the definition of languages of designs by shape grammars. Spatial relations can also be employed to define languages of designs in another way. This new method employs a simple variant of the shape grammar formalism. By means of the definitions and notation in Stiny (1980a), a set grammar G is defined for a set of shapes S and a set of symbols L. Rules have the form A -> B, where A and B are subsets of (S, L) and (S, Z,)*, respectively. A rule A -> B applies to a set of labelled shapes C under a transformation r when T(A) is a subset of C to produce a new set of labelled shapes given by [C~ r{A)] + T(B), where the operations + and are set union and difference. Rules thus incorporate the constructive mechanism used by Stiny (1980b, page 426) in his 'kindergarten' procedure for Froebel's building gifts and by Knight (1981) in her 'shape equivalence rules' which apply to change spatial relations. A set of labelled shapes is generated when rules are applied recursively to a subset I of (S, L) called the initial set. Designs are formed by taking the shape union of elements in sets so constructed. Such designs are in the language defined by G whenever they have no symbols associated with them. Of course, these definitions can be extended in the usual way to define parametric set grammars. This new formalism allows for designs to be formed from sets of labelled shapes which are constructed by recursively adding elements to or subtracting elements from sets according to given spatial relations. This process involves two basic types of rules: