  This is the first in a series of papers in which the gradient flows of
fundamental curvature invariants are used to formulate a visualization of
curvature. We start with the construction of strict Newtonian analogues (not
limits) of solutions to Einstein's equations based on the topology of the
associated gradient flows. We do not start with any easy case. Rather, we start
with the Curzon - Chazy solution, which, as history shows, is one of the most
difficult exact solutions to Einstein's equations to interpret physically. We
show that the entire field of the Curzon - Chazy solution, up to a region very
"close" to the the intrinsic singularity, strictly represents that of a
Newtonian ring, as has long been suspected. In this regard, we consider our
approach very successful. As regrades the local structure of the singularity of
the Curzon - Chazy solution within a fully general relativistic analysis,
however, whereas we make some advances, the full structure of this singularity
remains incompletely resolved.
