  In experiments and numerical simulations we measured angles between the
symmetry axes of small spheroids advected in turbulence ("passive directors").
Since turbulent strains tend to align nearby spheroids, one might think that
their relative angles are quite small. We show that this intuition fails in
general because angles between the symmetry axes of nearby particles are
anomalously large. We identify two mechanisms that cause this phenomenon.
First, the dynamics evolves to a fractal attractor despite the fact that the
fluid velocity is spatially smooth at small scales. Second, this fractal forms
steps akin to scar lines observed in the director patterns for random or
chaotic two-dimensional maps.
