  This paper discusses methods for the construction of approximate real space
renormalization transformations in statistical mechanics. In particular, it
compares two methods of transformation: the "potential-moving" approach most
used in the period 1975-1980 and the "rewiring method" as it has been developed
in the last five years. These methods both employ a parameter, called \chi,
that measures the complexity of the localized stochastic variable forming the
basis of the analysis. Both methods are here exemplified by calculations in
terms of fixed points for the smallest possible values of \chi. These
calculations describe three models for two-dimensional systems: The Ising model
solved by Onsager, the tricritical point of that model, and the three-state
Potts model. The older method, often described as lower bound renormalization
theory, provides a heuristic method giving reasonably accurate results for
critical indices at the lowest degree of complexity, i.e. \chi=2. In contrast,
the rewiring method, employing "singular value decomposition", does not perform
as well for low \chi values but offers an error that apparently decreases
slowly toward zero as \chi is increased. It appears likely that no such
improvement occurs in the older approach.
  A detailed comparison of the two methods is performed, with a particular eye
to describing the reasons why they are so different. For example, the old
method quite naturally employed fixed points for its analysis; these are hard
to use in the newer approach. A discussion is given of why the fixed point
approach proves to be hard in this context.
  In the new approach the calculated the thermal critical indices are
satisfactory for the smallest values of \chi but hardly improve as \chi is
increased, while the magnetic critical indices do not agree well with the known
theoretical values.
