Since at least the work of Gestalt psychologists and modernists like Josef Albers, designers have been mesmerized by the perceptual ambiguities of overlapping shapes. These overlaps and occlusions can suggest intricate surface patterns or partial projections of deep spaces.

Here we present an experiment in architectural surface which takes its departure from a defined, if arbitrary, boundary or chain of boundaries. It asks a simple question: what are the spaces which can be defined with a single continuous curve, or a braid of such curves? Each of these surfaces is a spanning surface: a surface whose boundary is exactly some specified curve. Such surfaces are conditioned by their boundary but also must resolve certain conditions on their interior consistently, which induces individual tectonic responses which reciprocally determine final form.

We have developed software tools to expand these overlapping shapes into intricate 3d surfaces and spaces, creating complex forms from simple closed curves and chains. These forms have the remarkable property that from specific views, they appear to collapse into 2d patterns, while from other perspectives they appear as fully-formed volumes.