We show that a planar aperiodic lattice, mimicking the appearance of a sunflower, supports photonic bandgaps for weak dielectric contrast. The pattern's high orientational order and spatially uniform modal pitch yields an isotropic Fourier space. A 2D structure of cylinders (=2) in air possesses a wide 21% TM bandgap, versus 5.6% for a sixfold lattice or 14% for a 12-fold fractal tiling. The isotropic gap frequencies imply flat bands, and thus application in nonlinear optics and low threshold lasers, where a reduced group velocity in all directions may be desired.