Symmetry Is the sine qua non of Shape

Abstract

Three-dimensional (3D) shape has been studied for centuries despite the absence of a commonly accepted definition of this property. The absence of a useful definition has been a major obstacle in making progress towards understanding the mechanisms that are responsible for the perception of shape. Today, in the absence of the needed new definition, there is no consensus about whether shapes are, or can be, perceived veridically. This chapter reviews the main definitions of shape in use before our new definition was formulated, calling attention to their shortcomings. It then describes our new definition, which is based on the assumption that 3D shape is based on 3D geometrical self-similarities (3D symmetries) of an object, rather than on similarities of an object with respect to other objects. We explain the new definition by discussing the invariants of three types of symmetry groups in 3D and then derive the invariants of the perspective projection from a 3D space to a 2D image. In our definition, the invariants of 3D symmetries serve as the basis for describing the 3D shapes, and the invariants of perspective projections are essential for recovering 3D shapes from one or more 2D images. We conclude by discussing several implications of this new definition which makes it clear: (i) that the veridicality of shape perception is no longer only an empirical concept—the new definition provides a principled theory of when and how the veridicality of shape can be achieved; (ii) how shape constancy applies to non-rigid objects; (iii) that there are informative, but objective, shape priors that do not have to be learned from objects or updated on the basis of experience: these priors are the object’s symmetries and (iv) that what had loomed as a controversy between view-invariant and view-dependent shape perception has been resolved.