  Riemann's non-differentiable function is a celebrated example of a continuous
but almost nowhere differentiable function. There is strong numeric evidence
that one of its complex versions represents a geometric trajectory in
experiments related to the binormal flow or the vortex filament equation. In
this setting, we analyse certain geometric properties of its image in
$\mathbb{C}$. The objective of this note is to assert that the Hausdorff
dimension of its image is no larger than 4/3 and that it has nowhere a tangent.
