Choosing among spatially distributed options is a central challenge for animals, from deciding among alternative potential food sources or refuges, to choosing with whom to associate. Here, using an integrated theoretical and experimental approach (employing immersive Virtual Reality), with both invertebrate and vertebrate models—the fruit fly, desert locust and zebrafish—we consider the recursive interplay between movement and collective vectorial integration in the brain during decision-making regarding options (potential ‘targets’) in space. We reveal that the brain repeatedly breaks multi-choice decisions into a series of abrupt (critical) binary decisions in space-time where organisms switch, spontaneously, from averaging vectorial information among, to suddenly excluding one of, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains. Close to each bifurcation the ‘susceptibility’ of the system exhibits a sharp increase, inevitably causing small differences among the remaining options to become amplified; a property that both comes ‘for free’ and is highly desirable for decision-making. This mechanism facilitates highly effective decision-making, and is shown to be robust both to the number of options available, and to context, such as whether options are static (e.g. refuges) or mobile (e.g. other animals). In addition, we find evidence that the same geometric principles of decision-making occur across scales of biological organisation, from neural dynamics to animal collectives, suggesting they are fundamental features of spatiotemporal computation.