In this paper we consider the interaction of the Lorenz manifold --- the two-dimensional stable manifold of the origin of the Lorenz equations --- with the two-dimensional unstable manifolds of the secondary equilibria or bifurcating periodic orbits of saddle type. We compute these manifolds for varying values of the parameter ρ in the Lorenz equations, which corresponds to the transition from simple to chaotic dynamics with the classic Lorenz butterfly attractor at ρ = 28. Furthermore, we find and continue in ρ the first 512 generic heteroclinic orbits that are given as the intersection curves of these two-dimensional manifolds. The branch of each heteroclinic orbit emerges from the well-known first codimension-one homoclinic explosion point at ρ = 13.9265, has a fold and then ends at another homoclinic explosion point with a specific ρ-value. We describe the combinatorical structure of which heteroclinic orbit ends at which homoclinic explosion point. This is verified with our data for the 512 branches from which we automatically extract (by means of a small computer program) the relevant symbolic information. Our results on the manifold structure are complementary to previous work on the symbolic dynamics of periodic orbits in the Lorenz attractor. We point out the connections and discuss directions for future research.

With three accompanying animations (GIF format) that convey the intricate geometry and interactions of the Lorenz manifold.