Decompositions of shapes

The work is conducted along the lines of the formal design theory championed by Stiny. Algebras of designs and shape algebras developed as abstractions of (design) practice are central to the theory. The definition of shape algebras as two-sorted is promoted as an elaboration of Stiny's original definition. It allows for the structure and symmetry--the indispensable ingredients of design--to be handled uniformly by a single algebra.; Decompositions as tools for assigning meaning to shapes are central to this work. Their formal properties and possible applications are explored in detail.; The prospect of describing shapes and computations in decompositions is investigated. A number of formal devices is originated to aid in the inquiry. Different algebras of decompositions are developed. Argument lattices and argument decompositions, as a means of explaining computations with shapes are introduced, as well as their augmented and diminished versions. Decompositions structured as algebras and their applications in design are discussed. In particular, lattice decompositions are studied in detail, and so are their more practical instances: hierarchies and topologies.