Geometry of Arrangements that Determine Shapes

TL;DR

This paper explores the geometric arrangements of points and lines that define shapes in shape grammars, characterizing the finite geometries they form and their role in determining shape rules.

Contribution

It provides a mathematical characterization of arrangements that contain shapes as finite geometries, linking them to shape grammar rules.

Findings

Arrangements form finite near-linear and linear spaces.

Some arrangements do not form proper geometries.

Finite geometries offer an alternative view of shape grammar rules.

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.