Although Vietoris–Rips complexes are frequently used in topological data analysis to approximate the “shape” of a dataset, their theoretical properties are not fully understood. In the case of the circle, these complexes exhibit a surprising progression of homotopy types (from $S^1$ to $S^3$ to $S^5$, etc.) as the scale increases. However, much less is known about the Vietoris–Rips complexes of higher-dimensional spheres. I will present work that explores Vietoris–Rips complexes of the $n$-sphere $S^n$ and shows how the appearance of nontrivial homotopy groups of $\mathrm{VR}(S^n; t)$ can be controlled by covering properties of $S^n$ and real projective space $\mathbb{R}P^n$. Specifically, if the first nontrivial homotopy group of $\mathrm{VR}(S^n; \pi-t)$ occurs in dimension $k$, then $S^n$ can be covered by $2k+2$ balls of radius $t$, but there is no covering of $\mathbb{R}P^n$ by $k$ balls of radius $t/2$. This is joint work with Henry Adams and Žiga Virk.