  An exact formula is derived, as an integral, for the mean square winding
angle of Brownian motion (that is, diffusion) after time t, around an
infinitely long impenetrable cylinder of radius a, having started at radius
R(>a) from the axis. Strikingly, for the simpler problem with a=0, the mean
square winding angle around a straight line, is long known to be instantly
infinite however far away the starting point lies. the fractally small, fast,
random walk steps of mathematical Brownian motion allow unbounded windings
around the zero thickness of the straight line. A remedy if it is required, is
to accord the line non-zero thickness, an impenetrable cylinder, as analysed
here. The problem straight away reduces to a 2D one of winding around a disc in
a plane since the axial component of the 3D Brownian motion is independent of
the others. After deriving the exact mean square winding angle, the integral is
evaluated in the limit of a narrow cylinder a<<R, highlighting the limits of
short and long diffusion times addressed by previous approximate treatments.
