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Abstract
The local constraint of double twist cannot lead to long range order in our physical space ℝ. Cubic blue phases can be interpreted as an array of disclination lines. The symmetry 1432, P4232 and 14,32 of cubic blue phases can be deduced from the symmetry groups of the P, F and G infinite periodic minimal surfaces by removing all the mirror operations incompatible with chirality. Axes of double twist cylinders are shown to be the same as the core of the labyrinths of infinite periodic minimal surface. The blue phase configuration is then driven by boundary conditions on the axes of double twist cylinders and on the minimal surface. We show that the configuration which minimizes the elastic free energy on the minimal surfaces is defined by the asymptotic lines.