It has been suggested that as much as 90% of the discharge from the Antarctic Ice Sheet is drained through a small number of fast-moving ice streams and outlet glaciers fed by relatively stable and inactive catchment areas. Here, evidence obtained from balance velocity estimates suggests that each major drainage basin is fed by complex systems of tributaries that penetrate up to 1000 kilometers from the grounding line into the interior of the ice sheet. This finding has important consequences for the modeled or estimated dynamic response time of past and present ice sheets to climate forcing.