Reflections on Bateson's rule: Solving an old riddle about why extra legs are mirror-symmetric

William Bateson was an obsessive observer of animal oddities, and at some point in his herculean survey of museum collections leading up to his monumental 1894 monograph (Materials for the study of variation), he noticed a peculiar trend among the preserved specimens (mainly insects) that possessed extra legs: multiple legs that branched from the same socket tended to be mirror images of their adjacent neighbors. He did not know why. These symmetry relationships have come to be known as Bateson's rule, and they have defied a satisfactory explanation for 125 years. In the past few decades, tantalizing clues have emerged from various lines of investigation, and those lines have converged on a possible solution. An attempt is made here to fit all of those clues together to form a coherent picture of the etiology. Two case studies have proven to be pivotal: a fly mutant whose extra legs are caused by patches of dying cells and a frog syndrome whose extra legs are caused by a parasitic flatworm. The conclusion reached is that the extra legs of insects and vertebrates obey Bateson's rule for the same reason, but that reason has nothing to do with the specific molecules in their signaling pathways. Rather, it is an emergent property of the circuitry of the pathways and their polarized alignments along the limb axes. A parade of theoretical models have tried and failed to crack this mystery in the past, and they are reviewed here as part of the narrative.