  We use the consistency approach to classify discrete integrable 3D equations
of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$
and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes
the most prominent representatives of this class, the discrete KP equation and
its Schwarzian (multi-ratio) version, as well as three further equations. The
combinatorics and geometry of the octahedron type equations are explained. In
particular, the consistency on the 4-dimensional Delaunay cells has its origin
in the classical Desargues theorem of projective geometry. The main technical
tool used for the classification is the so called tripodal form of the
octahedron type equations.
