  Lebesgue's universal covering problem is re-examined using computational
methods. This leads to conjectures about the nature of the solution which if
correct could provide a blueprint for a complete solution. Empirical lower
bounds for the minimal area are computed using different hypotheses based on
the conjectures. A new upper bound of 0.844113 for the area of the minimal
cover is derived improving previous results. This method for determining the
bound is suggested by the conjectures and computational observations but is
proved independently of them. The key innovation is to modify previous best
results by removing corners from a regular hexagon at a small slant angle to
the edges of the dodecahedron used before. Simulations indicate that the
minimum area for a convex universal cover is likely to be around 0.84408.
