Fig. 7

An orientation map on the sphere sampled from a random vector field which has \(\mathit{SO}(3)\)-shift-twist symmetry. We plotted the restriction to a hemisphere of the random map exploring \(\mathcal{H}^{\mathrm{exact}}_{10}\); beware that the color coding has a different meaning than in Figs. 1, 2, 5, and 6. Here, the sample map is a vector field on the sphere, and there is no complex number; to visualize the direction of the emerging vector at each point, we apply the orthogonal projection from the drawn hemisphere to the “paper” plane, thus getting a vector field on the unit ball of the Euclidean plane, and plot the resulting orientation map using the same color code as in Figs. 1, 2, 5, and 6