  An important idea underlying a plausible dynamical theory of circulation in
three-dimensional turbulence is the so-called Area Rule, according to which the
probability density function (PDF) of the circulation around closed loops
depends only on the minimal area of the loop, not its shape. We assess the
robustness of the Area Rule, for both planar and non-planar loops, using
high-resolution data from Direct Numerical Simulations. For planar loops, the
circulation moments for rectangular shapes match those for the square with only
small differences, these differences being larger when the aspect ratio is
further from unity, and when the moment-order increases. The differences do not
exceed about $5\%$ for any condition examined here. The aspect-ratio dependence
observed for the second-order moment are indistinguishable from results for the
Gaussian Random Field (GRF) with the same two-point correlation function (for
which the results are order-independent by construction). When normalized by
the SD of the PDF, the aspect ratio dependence is even smaller $(< 2\%)$ but
does not vanish unlike for the GRF. We obtain circulation statistics around
minimal area loops in three dimensions and compare them to those of a planar
loop circumscribing equivalent areas, and we find that circulation statistics
match in the two cases only when normalized by an internal variable such as the
SD. This work highlights the hitherto unknown connection between minimal
surfaces and turbulence.
