specified in Equations (2.12) and (2.13) to get To gain a better understanding of how the two sets of bounds compare, we compute these bounds explicitly for the case of a 2-current in a regular tetrahedral complex (thus, p = 3 and d = 2). Notice that this instance is close to a best case for Sullivan’s bounds, as less regular complexes affect them more severely. With this point in mind, we present in Table 2.1 our bounds and Sullivan’s bounds on both a regular tetrahedral complex and one on which we stretch the regular tetrahedra by a factor of 10 in a direction normal to one of their faces (i.e., turn them into skinny, spike-like simplices). Table 2.1: Comparison of our bounds with those obtained by Sullivan for a 2-current in a (1) 3-complex of congruent regular tetrahedra and (2) a 3-complex of congruent stretched tetrahedra which are created by taking regular tetrahedra and multiplying their height by a factor of 10. 45 Quantity Sullivan’s bound Our bound Regular tetrahedra M ( P ) (1 . 2 × 10 $^{5}$) M( T ) (1 . 6 × 10 $^{3}$) M( T ) + (2 . 5 × 10 $^{6}$) Δ M( ∂T ) M ( ∂P ) (8 . 7 × 10 $^{3}$) M( ∂T ) (2 . 5 × 10 $^{6}$) M( ∂T ) F$_{λ}$ ( T, P ) (2 . 4 × 10 $^{5}$) Δ M( T ) + (1 . 7 × 10 $^{3}$) Δ λ M( ∂T ) (1 . 6 × 10 $^{3}$) Δ M( T ) + (2 . 5 × 10 $^{6}$) Δ λ M( ∂T ) Stretched tetrahedra M ( P ) (5 . 5 × 10 $^{9}$) M( T ) (3 . 7 × 10 $^{4}$) M( T ) + (1 . 4 × 10 $^{9}$) Δ M( ∂T ) M ( ∂P ) (1 . 1 × 10 $^{7}$) M( ∂T ) (1 . 4 × 10 $^{9}$) M( ∂T ) F$_{λ}$ ( T, P ) (1 . 1 × 10 $^{10}$) Δ M( T ) + (2 . 3 × 10 $^{7}$) Δ λ M( ∂T ) (3 . 7 × 10 $^{4}$) Δ M( T ) + (1 . 4 × 10 $^{9}$) Δ λ M( ∂T ) F$_{λ}$ ( T, P ) ≤ Δ (4 ϑ$_{K}$ ) p − d $^{(}$M( T ) + λ (1 + 4 ϑ$_{K}$ ) M( ∂T ) $^{)}$.