Shape and shape grammar implementations generally compute with points, lines, planes and volumes. These shape elements are used in describing shapes and defining operations to change shapes (Stiny 1975, Krishnamurti 1992). A critical aspect of these shape computations is that shapes are described by their parts which are also shapes (Stiny 1994). Informally shapes are not divided into atoms. The consequences of this view offers an interesting perspective on the use of boundaries in the representation of shapes. An earlier paper (Earl 1997) outlines some of the formal apparatus required.

This viewpoint has implications for the ways shapes are considered in design. Geometric modelling representations of shape embedded in Euclidean Space, brings an associated topological structure of point sets with familiar notions of boundary and connectivity. However, in general, boundaries arise from topologies and associated closure structures. Closure algebras, which comprise the shape and its subshapes as well as selected closed subshapes, have associated boundaries in a topological sense. Boundaries are emergent shape features dependent on a particular description of a shape by subshapes.

Several types of boundary will be examined. First, descriptive boundaries are used in the representation and visualisation of shapes. General descriptive boundaries can be constructed from the descriptive boundaries of shape elements. Second, closure boundaries are associated with the topological structures of selected subshapes. New concepts of boundary and connectedness emerge from the consideration of closure algebras on shapes. Designs have selected parts connected to the whole design through their closure boundaries. Third, regions and 'boundary' subshapes can be distinguished.

A brief review of subshape descriptions and their structures serves to emphasise the basic differences between geometric models and subshape descriptions of shape. Subshape representations allow flexible interpretations of boundaries. Before parts are identified in descriptions, a shape is formally fragmented, disconnected and without boundaries. Atoms of shape (such as points composing lines) induce uneccessary boundaries. Shape descriptions are free from the constraints of these artificial boundaries .