Standard simplicial decomposition of a sphere ($F^n=\partial \mathrm{conv}\{\pm e_1,\dots ,\pm e_{n+1}\}$)

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.