Circular Logic

My objective for this chapter is to draw your attention to a few peculiarities inherent in the logic of periodic functions. I find a visual approach the most fruitful for thinking about such matters. As the pictures involved consist mainly of mappings between circles and products of circles, I must first say a few words about the notions of topological spaces and mappings. This chapter thus has four sections:A.Spaces, with emphasis on rings (i.e., closed loops. To avoid the more exact connotations of the word circle I use ring, trusting the reader do not confuse my meaning with algebraic rings.)B.Mappings, with emphasis on the winding number of mappings to a ringC.Phase singularities of maps (Parts I and II), with emphasis on the consequences of a nonzero winding numberD.Technical details on the application of circular logic to biological rhythms