For the regular tetrahedral complex and the M( P ) bound, our coefficient of M( T ) is more than 74 times better, but we do have a second term that can be quite large, but diminishes in importance if the complex is subdivided appropriately (see Remark 2.5.10). In the stretched complex, our coefficient on M( T ) is 1 . 5 × 10 5 times better, indicating that our bound is better behaved for irregular complexes. Our bound on M( ∂P ) is about 290 times worse than Sullivan’s for the regular tetrahedra, and about 120 times worse for the stretched complex. For the flat norm bound in the regular complex, we are about 148 times better on the M( T ) term and about 145 times worse on the M( ∂T ) term. On the stretched complex, our M( T ) coefficient is about 3 × 10 5 times better, and our M( ∂T ) coefficient is about 60 times worse. We also note that in the case of the flat norm with scale, our larger M( ∂T ) coefficient becomes less important for small λ . Remark 2.5.12. For the important case where ∂T is empty, i.e., when T is a cycle, we have M( ∂T ) = 0 , and hence our bounds are uniformly better than Sullivan’s. As compared to Sullivan, we are able to take advantage of our simplicial setting to get better bounds on the mass expansion of T . While our mass expansion bounds involving ∂T are currently inferior to Sullivan’s, we suspect our arguments can be tightened and modified to obtain bounds that are better in all cases. More importantly, our bounds are less sensitive to simplicial irregularity. Given the challenges inherent in creating meshes without slivers even in three dimensions [11], bounds that behave well in their presence are highly desirable. We illustrate computations of the multiscale simplicial flat norm by describing the flat norm decompositions of a 2-manifold with boundary embedded in R 3 (see Figure 2.6). 46 2.6 Computational Results