Geometry of Arrangements that Determine Shapes

Abstract

Shape grammars compute over shapes which are defined in the universe $U^*$. Shapes in the universe $U^*$ are analogous to line drawings that can be physically realized in the plane. Any shape is embedded or contained in an arrangement of points and lines in the plane called, respectively, registration marks and construction lines, that satisfy special incidence laws. In this expository article, arrangements that contain shapes are studied as incidence structures and the finite geometries they give rise to are characterized. In particular, arrangements that contain shapes are distinguished into those that give rise to finite near-linear and linear spaces, and those that do not give rise to any proper form of geometry (in the strict mathematical sense). Arrangements that constitute finite geometries (near-linear and linear spaces) give an alternative characterization of determinate rules in shape grammars. This paper contributes to the body of work related to the mathematics of shapes in the area of shape grammar theory.

Figures (18)

• Figure 1: Adjacent co-linear and overlapping co-linear line segments are not permitted to form shapes.
• Figure 2: (a) The union of two shapes may not be a shape, since the result may contain adjacent co-linear or overlapping co-linear line segments. (b) The sum of two shapes is always a shape. The red cross-hair indicates the origin of the plane.
• Figure 3: Some shapes and their underlying arrangements.
• Figure 4: Some sets of points and/or lines in the plane which are not arrangements for shapes.
• Figure 5: Configurations of points and lines like Pappus's (left) and Desargues's (right) are not arrangements that determine shapes.