This paper is the third and final one in a sequence of three. All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically. The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precursor. In the present paper we consider the more complicated case of two-dimensional nets, where each "child" proposition is probabilistically justified by two "parent" propositions. Surprisingly, it turns out that probabilistic justification in two dimensions takes on the form of Mandelbrot's iteration. Like so many patterns in nature, probabilistic reasoning might in the end be fractal in character.