Various mechanisms govern pattern formation in chemical and biological reaction systems, giving rise to structures with distinct morphologies and physical properties. The self-organization of polymerizing microtubules (MTs) is of particular interest because of its implications for biological function. We report a study of the microscopic structure and properties of the striped patterns that spontaneously form in polymerizing tubulin solutions and propose a mechanism driving this assembly. Microscopic observations reveal that the pattern comprises wave-like MT bundles. The retardance of the solution and the fluorescence intensity of labeled MTs vary periodically in space, suggesting a coincident periodic variation in MT alignment and density. This wave-like structure forms through the development and coordinated buckling of initially aligned MT bundles. Both static magnetic fields and convective flow can induce the initial alignment. The nesting of the buckled MT bundles gives rise to density variations that are in quantitative accord with the data. We further propose that the buckling wavelength is selected by a balance between the bending energy of the bundles and the elastic energy of the MT network surrounding them. These studies reveal a unique physical chemical mechanism by which mechanical buckling couples with protein polymerization to produce macroscopic patterns. Self-organization of this type may be important to the formation of certain biological structures.