Based on prior work, we know that at an early stage the visual system has three different systems that independently extract visual motion information from visual inputs. At later stages, these systems combine their outputs. Here, we consider a much studied (>650 publications) class of visual stimuli, plaids, which are combinations of two sine waves. Currently, there is no quantitative theory to explain the perceived motion of plaids. With an important methodological improvement, we obtained a large set of data exploring the various dimensions in which plaids differ. We found that only two of the three motion systems are active in plaid processing. With this clue, we were able to formulate a pure theory for the outputs of the first-order and third-order motion systems and how they combine. With zero parameters estimated from the data, the theory captures the essence of the full range of our plaid data and supports some surprising, counter-intuitive conjectures about how vision works.