Visual Structures of Embedded Shapes

Abstract

Shape computations recognise parts and create new shapes through transformations. These elementary computations can be more than they seem, inducing complicated structures as a result of recognising and transforming parts. This paper introduces, what is perhaps in principle, the simplest case where the structure results from seeing embedded parts. It focusses on lines because, despite their visual simplicity, if a symbolic representation for shapes is assumed, lines embedded in lines can give rise to more complicated structures than might be intuitively expected. With reference to the combinatorial structure of words the paper presents a thorough examination of these structures. It is shown that in the case of a line embedded in a line, the resulting structure is palindromic with parts defined by line segments of two different lengths. This result highlights the disparity between visual and symbolic computation when dealing with shapes—computations that are visually elementary are often symbolically complicated.