Simplicial grid refinement: on Freudenthal's algorithm and the optimal number of congruence classes

In the present paper we investigate Freudenthal's simplex refinement algori thm which can be considered to be the canonical generalization of Bank's we ll known red refinement strategy for triangles. Freudenthal's algorithm sub divides any given (n)-simplex into 2(n) subsimplices, in such a way that re cursive application results in a stable hierarchy of consistent triangulati ons. Our investigations concentrate in particular on the number of congruen ce classes generated by recursive refinements, After presentation of the me thod and the basic ideas behind it, we will show that Freudenthal's algorit hm produces at most n!/2 congruence classes for any initial (n)-simplex, no matter how many subsequent refinements are performed. Moreover, we will sh ow that this number is optimal in the sense that recursive application of a ny affine invariant refinement strategy with 2n sons per element results in at least n!/2 congruence classes for almost all (n)-simplices.