Spatial change: continuity, reversibility, and emergent shapes

Spatial composition can be viewed as computation involving spatial changes each expressed as <nobr><i>s</i> - f(<i>a</i>)+f(<i>b</i>),</nobr> where <i>s </i>is a shape, and f(<i>a</i>) is a representation of the emergent part (shape) that is altered by replacing it with the shape f(<i>b </i>). We examine this formula in three distinct but related ways. We begin by exploring the conditions under which a sequence of spatial changes is continuous. We next consider the conditions under which such changes are reversible. We conclude with the recognition of emergent shapes, that is, the determination of transformations f that make f(<i>a </i>) a part of <i>s</i>. We enumerate the cases for shape recognition within algebras <i>U_ij</i>, 0 <=<i> i </i> <=<i> j </i> <= 3, and within Cartesian products of these algebras.