  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
$\mathrm{VR}(S^n;\pi-\delta)$ of the $n$-sphere at scale $\pi-\delta$ occurs in
dimension $k$, i.e., suppose that the connectivity is $k-1$. Then
$\mathrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$.
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
$\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$
changes infinitely many times as the scale $r$ increases.
