  Equilibrium solutions are believed to structure the pathways for ergodic
trajectories in a dynamical system. However, equilibria are atypical for
systems with continuous symmetries, i.e. for systems with homogeneous spatial
dimensions, whereas relative equilibria (traveling waves) are generic. In order
to visualize the unstable manifolds of such solutions, a practical symmetry
reduction method is required that converts relative equilibria into equilibria,
and relative periodic orbits into periodic orbits. In this article we extend
the fixed Fourier mode slice approach, previously applied 1-dimensional PDEs,
to a spatially 3-dimensional fluid flow, and show that is substantially more
effective than our previous approach to slicing. Application of this method to
a minimal flow unit pipe leads to the discovery of many relative periodic
orbits that appear to fill out the turbulent regions of state space. We further
demonstrate the value of this approach to symmetry reduction through
projections (projections only possible in the symmetry-reduced space) that
reveal the interrelations between these relative periodic orbits and the ways
in which they shape the geometry of the turbulent attractor.
