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. Cambridge extremal-combinatorics notes use this convention.

The boundary simplicial complex of the cross-polytope has vertices $\pm e_i$, and its faces choose at most one vertex from each opposite pair. Radial projection identifies its geometric realization with the round sphere $S^n$. It is a regular antipodal triangulation of a sphere and is useful as a target for signed vertex labels: a vertex map is a simplicial map precisely when no edge receives opposite labels.