Self-consistent field theory predicts that the complex phase behavior of block copolymers does not originate solely from the interface seeking constant mean curvature as once thought, but instead reflects competing minimization of interfacial tension and packing frustration. To test this prediction, we directly measure interfacial curvature distributions from a 3D image reconstruction of the bicontinuous gyroid morphology. Results obtained here reveal that the gyroid interface is not constant mean curvature and confirm the importance of packing frustration in the stabilization of such complex nanostructures. PACS numbers: 68.35.Ct, 42.30.Wb, 47.20.Hw, 83.70.Hq Block copolymers exhibit periodic nanostructures due to immiscibility between the dissimilar (A and B) sequences [1]. Classical block copolymer nanostructures include spheres of A(B) on a body-centered cubic lattice in a B(A) matrix, cylinders of A(B) on a hexagonal lattice in a B(A) matrix, and coalternating lamellae. Of considerable recent interest are several complex (bicontinuous) nanostructures-the perforated lamellar (PL), gyroid (G), and double-diamond (D) morphologies [2][3][4][5][6][7]. These nanostructures may develop if the copolymer composition ( f) falls within a narrow range between the cylindrical and lamellar morphologies, and can be difficult to distinguish experimentally. Block copolymer nanostructures once believed [2] to be D, exemplified by a Schwarz D surface with Pn3m symmetry, have been reclassified [8] on the basis of their small-angle X-ray scattering (SAXS) signatures as G, which is represented by the Schoen G surface with Ia3d symmetry. Identification of complex nanostructures by transmission electron microscopy (TEM) is often inconclusive, since they appear identical along several projection axes.Complex nanostructures also develop in surfactant and lipid systems due to the formation of surfaces with constant mean curvature (CMC) that minimize contact between immiscible moieties [9]. Since block copolymer nanostructures share common topological features with those of other self-organized systems, the concept of CMC minimal surfaces has been used [3] to explain the stability of complex block copolymer nanostructures. On the basis of self-consistent field theory (SCFT), Matsen and Bates [10,11] have recently proposed that the area-averaged mean curvature ͑͗H͒͘ governs the gross morphology (lamellar, bicontinuous, cylindrical, or spherical), whereas the standard deviation of the mean curvature distribution (s H ) determines the delicate stability of the complex nanostructures (G, D, or PL). This additional consideration results from packing frustration [12] and implies that, while a surface strives toward CMC, the mean curvature cannot be constant everywhere along the interface since the microdomain-forming blocks must uniformly fill space in the most entropically favored manner. Thus far, neither ͗H͘ nor s H has been measured experimentally despite their apparent importance.Three-dimensional visualization of bicontinuous morpho...