  The Gram spectrahedron of a real form $f\in\mathbb{R}[\underline{x}]_{2d}$
parametrizes all sum of squares representations of $f$. It is a compact,
convex, semi-algebraic set, and we study its facial structure in the case of
ternary quartics, i.e. $f\in\mathbb{R}[x,y,z]_4$. We show that the Gram
spectrahedron of every smooth ternary quartic has faces of dimension 2, and
generically none of dimension 1. We complete the proof that the so called
Steiner graph of every smooth quartic is isomorphic to $K_4\coprod K_4$.
Moreover, we show that the Gram spectrahedron of a generic psd ternary quartic
contains points of all ranks in the Pataki interval.
