0 Decomposition of the Height Function of Scherk ’ s First Surface Randall

As x → ±∞, z = ±y tan( 1 2 α)+(n+ 1 2 )π sec( 1 2 α), respectively for integers n. Thus Scherk’s first surface connects two infinite sets of parallel planes at x = ±∞, equally spaced by π and rotated by an angle α with respect to each other. It has been used as a model for grain boundaries in diblock copolymers and smectic liquid crystals [2,3] where the multi-valued height function represents the peak of the one-dimensional density modulation of these materials. In the limit that α → 0, the solution z = h[x, y;α] in (2) is the height function for another well-known minimal surface, the helicoid: