= Regular antipodal triangulation of a sphere
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. https://www.dpmms.cam.ac.uk/~par31/notes/extcomb.pdf[Cambridge extremal-combinatorics notes] use this convention.
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