Symmetry of the continuum percolation threshold in systems of two different size objects

We study the continuum percolation in systems composed of overlapping objects of two different sizes. We show that when treated as a function of the volumetric fraction f as opposed to the concentration x, the percolation threshold exhibits the symmetry ηc(f,r)=ηc(1−f,r) where r is the ratio of the volumes of the objects. Knowledge of this symmetry has the following benefits: (i) the position of the maximum of the percolation threshold is then known to be at exactly f=1/2 for any r and (ii) full knowledge of the percolation threshold is obtained by performing simulations only for f∈[0,12] or f∈[12,1], whichever is computationally easier.