= Standard simplicial decomposition of a sphere
{title2=$F^n=\partial\operatorname{conv}\{\pm e_1,\ldots,\pm e_{n+1}\}$}
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.
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