Retinal neurons and vessels are not fractal but space-filling

Many branched patterns in nature are hypothesized to be fractal, i.e., statistically self-similar across a range of scales. We tested this hypothesis on the two-dimensional arbors of retinal neurons and blood vessels. First, we measured fractalness on synthetic fractal and nonfractal patterns. The synthetic fractal patterns exhibited self-similarity over a decade of scale, but the nonfractal "controls" showed hardly any self-similarity. Neuronal and vascular patterns showed no greater self-similarity than the controls. Second, we manipulated a synthetic fractal pattern to remove its self-similarity and found this to be reflected in a loss of measured fractalness. The same manipulation of the nonfractal control and also of the neural and vascular patterns did not alter their measured fractalness. Third, we "grew" patterns of branched line segments according to a variety of nonfractal algorithms. These patterns were, if anything slightly more fractal than the neural and vascular patterns. We conclude that the biological patterns studied here are not fractal. Finally, we measured extended versions of these patterns: a contiguous array of homotypic neuron arbors and a vascular pattern with a high degree of total detail. These patterns showed a "fractal dimension" of 2, which implies that down to some cut-off scale they fill space completely. Thus, neural and vascular patterns might best be described as quasi-regular lattices.