  Spin states of maximal projection along some direction in space are called
(spin) coherent, and are, in many aspects, the "most classical" available. For
any spin $s$, the spin coherent states form a 2-sphere in the projective
Hilbert space $\mathbb{P}$ of the system. We address several questions
regarding that sphere, in particular its possible intersections with complex
lines. We also find that, like Dali's iconic clocks, it extends in all possible
directions in $\mathbb{P}$. We give a simple expression for the Majorana
constellation of the linear combination of two coherent states, and use Mason's
theorem to give a lower bound on the number of distinct stars of a linear
combination of two arbitrary spin-$s$ states. Finally, we plot the image of the
spin coherent sphere, assuming light in $\mathbb{P}$ propagates along
Fubini-Study geodesics. We argue that, apart from their intrinsic geometric
interest, such questions translate into statements experimentalists might find
useful.
