The Vietoris–Rips complex is a construction central to applied topology, including applications to geometric group theory, topological data analysis, and more. However, even for simple spaces such as spheres, their homotopy types are yet to be characterized. In this talk, I will present connectivity bounds for the Vietoris–Rips complexes of spheres $S^n$ in terms of covering properties of $\mathbb{R}P^n$. We leverage the connection to neighborhoods of $S^n$ in the tight span $E(S^n)$ (a.k.a hyperconvex hull) and tools from equivariant topology. These techniques generalize to the study of Vietoris–Rips complexes of antipodal metric spaces. This is joint work with Florian Frick.