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Microtubules are dynamic polymers that play fundamental roles in all eukaryotes. Despite their importance, how new microtubules form is poorly understood. Textbooks have focused on variations of a nucleation–elongation mechanism in which monomers rapidly equilibrate with an unstable oligomer (nucleus) that limits the rate of polymer formation; once formed, the polymer then elongates efficiently from this nucleus by monomer addition. Such models faithfully describe actin assembly, but they fail to account for how more complex polymers like hollow microtubules assemble. Here, we articulate a new model for microtubule formation that has three key features: (1) microtubules initiate via rectangular, sheet-like structures that grow faster the larger they become; (2) the dominant pathway proceeds via accretion, the stepwise addition of longitudinal or lateral layers; and (3) a “straightening penalty” to account for the energetic cost of tubulin’s curved-to-straight conformational transition. This model can quantitatively fit experimental assembly data, providing new insights into biochemical determinants and assembly pathways for microtubule nucleation.