the same over the whole range of loads, the predicted coefficients of friction should remain valid anyway. Nevertheless, the existence of a “hard” sliding direction at $30^\circ$, as predicted by our model, is supported by the shearing calculations.

![Plot of the mean friction force $F^{(1)}$ versus the path length $l$ for a load $L=\unit[183]{pN}$. Both periodic paths $0^\circ$ (red circles) and $30^\circ$ (blue crosses) are shown alongside three quasi aperiodic ones (green diamonds for $-10^\circ$, green triangles for $3^\circ$ and green squares for $10^\circ$). Dashed lines are plotted to guide the eye.[]{data-label="fig:5"}](Fig5.pdf){width="1.0\linewidth"}

In Fig. \[fig:5\] the reason why long paths are required ($\sim~\unit[1]{\mu m}$) becomes apparent. While the friction force for the $0^\circ$ path (red circles) shows essentially no dependence on the length of the path, for the $30^\circ$ one (blue crosses) the friction force oscillates in the beginning before converging to a constant value. This is due to the fact that the summation in equations (\[eq:F1\]) and (\[eq:F2\]) terminates at the last minimum encountered. For $0^\circ$ this happens always after a full period, while along the $30^\circ$ direction one has two minima per period (fcc and the hcp position), so that the total friction force average depends on the 