  In this work we will discuss the facial structure of the cone of nonnegative
ternary quartics with real coefficients. We will establish an equivalence
relation on the set of all faces, which preserves certain properties like
dimension or the number of common zeros. Moreover we will give a complete list
of equivalence classes and we will discuss their dimension and inclusions
between them. Eventually we will be able to present a complete lattice of all
faces of this particular cone.
