A hyperbolic isometry acts on the compactified hyperbolic plane with North-South dynamics and a single invariant axis. The same is true for a pseudo-Anosov mapping class acting on a Teichmuller space and other hyperbolic-like settings. However, while a fully irreducible free group outer automorphism acts on compactified Outer space with North-South dynamics, there can be many axes for a single fully irreducible φ ∈ Out(F_r). With this in mind, Handel and Mosher define the axis bundle for a fully irreducible φ ∈ Out(F_r). And then Handel-Mosher and Bridson-Vogtmann ask about the geometry of the axis bundle. In joint work with Chi Cheuk Tsang, we show that the axis bundle of a nongeometric fully irreducible outer automorphism admits a canonical “cubist” decomposition into branched cubes that fit together with special combinatorics. From this structure, we locate a canonical finite collection of periodic fold lines in each axis bundle. This can be considered as an analogue of results of Hamenstadt and Agol from the surface setting, which state that the set of trivalent train tracks carrying the unstable lamination of a pseudo-Anosov map can be given the structure of a CAT(0) cube complex, and that there is a canonical periodic fold line in this cube complex. Our “cubist” decomposition also gives a “hands on” solution to the fully irreducible conjugacy problem in Out(F_r).