Abstract: We tackle one of the most fascinating problems in reaction network theory: the Global Attractor Conjecture, which has been an open question for four decades now. In this paper, we do not quite prove the conjecture, but we prove a significant special case. We obtain a surprisingly crisp combinatorial understanding of the dynamics. On one hand there is this monstrous system of Ordinary Differential Equations with arbitrary degree non-linear terms on the right-hand side which seems hopeless to understand. On the other hand, there is this geometric diagram, much easier to work with. Remarkably, this simple geometric diagram reveals a surprising amount of information about the non-linear Ordinary Differential Equation system (this is joint work with Ezra Miller, and Anne Shiu and was published in SIAM J. Appl. Dyn. Syst., 13(2), 758–797. (40 pages).