We survey what is known and unknown about Vietoris–Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris–Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let \(S^n\) be the n-sphere with the geodesic metric, and of diameter \(\pi \) , and let \(\delta > 0\) . Suppose that the first nontrivial homotopy group of the Vietoris–Rips complex \(\textrm{VR}(S^n;\pi -\delta )\) of the n-sphere at scale \(\pi -\delta \) occurs in dimension k, i.e., suppose that the connectivity is \(k-1\) . Then \(\textrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \textrm{cov}_{\mathbb {R}\textrm{P}^n}(k)\) . In other words, there exist \(2k+2\) balls of radius \(\delta \) that cover \(S^n\) , and no set of k balls of radius \(\frac{\delta }{2}\) cover the projective space \(\mathbb {R}\textrm{P}^n\) . As a corollary, the homotopy type of \(\textrm{VR}(S^n;r)\) changes infinitely many times as the scale r increases.
