  Symmetry reduction by the method of slices quotients the continuous
symmetries of chaotic flows by replacing the original state space by a set of
charts, each covering a neighborhood of a dynamically important class of
solutions, qualitatively captured by a `template'. Together these charts
provide an atlas of the symmetry-reduced `slice' of state space, charting the
regions of the manifold explored by the trajectories of interest. Within the
slice, relative equilibria reduce to equilibria and relative periodic orbits
reduce to periodic orbits. Visualizations of these solutions and their unstable
manifolds reveal their interrelations and the role they play in organizing
turbulence/chaos.
