Endothelial tip-stalk selection may determine the initial spacing of angiogenic sprouts from a previously uniform layers of cells. The mathematical model presented here predicts the onset conditions and equilibrium spacing of tip-stalk patterns in the presence of elevated VEGF. A linear stability analysis identified the network elements that enabled tissue-scale patterning, while a numerical simulation predicted the final density of tip cells. The assumptions of this model may have selected endothelial tip cells in an unusually high density if they were to become candidates for sprout outgrowth. Including filopodia or cytonemes that extended the effective range of juxtacrine signaling was only mechanism that enabled sparser patterns. This model provides an early experimental target for observing the controlled breakdown of symmetry in a uniform layer of mammalian cells as predicted by Turing, and may yield insight into the self-assembly of blood vessels, in vitro and in vivo.