  The turbulence problem at the level of scaling exponents is hard in part
because of the multifractal scaling of small scales, which demands that each
moment order be treated and understood independently. This conclusion derives
from studies of velocity structure functions, energy dissipation, enstrophy
density (that is, square of vorticity), etc. However, it is likely that there
exist other physically pertinent quantities with uncomplicated structure in the
inertial range, potentially resulting in huge simplifications in the turbulence
theory. We show that velocity circulation around closed loops is such a
quantity. By using a large databases of isotropic turbulence, generated from
numerical simulations of the Navier-Stokes equations over a wide range of
Reynolds numbers, we show that circulation exhibits a bifractal behavior at the
highest Reynolds number considered: space filling for moments up to order $3$
and a mono-fractal with an unchanging dimension of about $2.5$ for higher
orders; this change in character roughly at the third-order moment is
reminiscent of a "phase transition". We explore the possibility that
circulation becomes effectively space filling at much higher Reynolds numbers
even though it may technically be regarded as a bifractal. We confirm that the
circulation properties depend on only the area of the loop, not its shape; and,
for a figure-$8$ loop, the relevant area is the scalar sum of the two segments
of the loop.
