New book chapter contribution to the editorial “Shape Computation: Fifty Years, 1972–2022,” published by Springer Birkhäuser and the Mathematics and the Built Environment book series. The publication is sponsored by the MIT Department of Architecture.

Abstract

This chapter presents a new spatial representation for shapes for applications in architecture, engineering design or related spatial design areas. “Construction lines” and “registration marks” in shape grammars are two geometric concepts that ground the appearance of shapes and enable an algorithmic approach to visual calculating. This chapter studies the two concepts in unity as a new spatial representation called a point-line arrangement. It develops topics such as the comparison and classification of shapes by their arrangements, the finite “geometries” that arrangements give rise to, the algebra of arrangements, and others. In this way, the chapter provides a more holistic overview of the mathematical features of construction lines and registration marks and their use in visual calculating.

Context

The “mathematics of shape grammars” began with the introduction of the formalism in the 1970s (for example, see this publication). Among the various topics of research that have emerged over the past five decades, the following three relate to the work I present in this paper:

(a) The algebras of shapes Ui, defined according to the dimensionality of the basic elements that form a shape.
(b) Structural descriptions defined in terms of embedded parts that are related in special ways, such as in finite topologies or other decompositions. For example, see my publication on shape topologies.
(c) Algorithmic features of visual calculating with shapes and properties that distinguish it from other formal models of computation. For example, see my publication on rule continuity.

The Table 1, featured in the image slideshow, summarizes some of the key properties of shapes related to the topics (a) and (b).

This chapter presents a new research direction for the third topic above i.e., topic (c), and also contributes results to the first, i.e., topic (a).

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