Polyhedral Catastrophe Theory I: Maps of the Line to the Line

Publisher Summary Thorn [1] has .described seven differentiable catastrophes, and these have been elaborated by Zeeman [2] and by both Thorn and Zeeman in this symposium on dynamic systems. In this paper, we present a beginning of the theory of polyhedral catastrophes and, in particular, we examine analogues of each of the four basic differentiable catastrophes which map the line to the line. The unfoldings of these polyhedral singularities are topologically equivalent to those of the differentiable category, so this alternative view of catastrophe theory can give some additional insight into the structure of differentiable unfoldings. The main advantage of the polyhedral theory is that there is a precise algorithm for determining the singular behavior as opposed to the progressively complicated algebraic calculations required in the differentiable theory.