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The mesh of a simplicial complex is the largest diameter of one of its realized simplices, or the supremum when $K$ is not finite:If $\dim K=n$, thenIterating gives

A triangulation of $S^k$ is regular in this antipodal-combinatorics convention when the antipodal map preserves its simplices and its nested coordinate equators $S^0\subset S^1\subset \cdots \subset S^{k-1}$ are subcomplexes. The two closed hemispheres at each stage are then triangulated balls. Barycentric subdivision preserves these conditions while making the mesh of a simplicial complex arbitrarily small. This convention is distinct from other meanings of regular triangulation in polyhedral geometry. Cambridge extremal-combinatorics notes use this convention.

For a finite simplicial complex $K$, a simplicial complex $L$, and a continuous map $f:|K|\to |L|$, some iterated barycentric subdivision $K^{(r)}$ admits a simplicial approximation to $f$. The proof applies the Lebesgue number lemma to the inverse images of target open stars and uses that the mesh of a simplicial complex tends to zero under repeated barycentric subdivision.