Estimating the Number of Similarity Classes for Marked Bisection in General Dimensions

To measure the stability of a marked bisection method, we estimate in general dimensions an upper bound of the number of generated similarity classes. Moreover, to understand the cyclic similarity structure, we estimate the number of uniform refinements required to generate all the similarity classes. We first prove that switching to the newest vertex bisection after n uniform refinements is equivalent to switching after \(n-2\) uniform refinements. Then, we obtain the similarity bound, a bound that we use to derive the number of uniform refinements required to generate all the similarity classes. Although the similarity bound is not tight, the results show that it estimates the magnitude of the expected number of classes. We also show the ratio of the number of similarity classes for marked bisection and newest vertex bisection. Because this ratio grows exponentially with the dimension, we conclude that marked bisection is suitable for low-dimensional applications.