Structure from Appearance: Topology with Shapes, without Points

Abstract: A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies are built with shapes, which are formalized without points, and with structures defined from their parts. An interpretative, aesthetic dimension is introduced according to which the topological structure of a shape is not inherited from an ambient space but is induced based on how its appearance is interpreted into sets of parts. The proposed approach provides a more natural, spatial framework for studies on the mathematical structure of design objects in art and design. More generally, it shows how mathematical constructs (here, topology) can be built directly in terms of objects of art and design, as opposed to the more common opposite approach where objects of art and design are subjugated to canonical mathematical constructs.