  We provide an existence proof for two 1-parameter families of embedded triply
periodic minimal surfaces of genus three, namely the tG family with tetragonal
symmetry that contains the gyroid, and the rGL family with rhombohedral
symmetry that contains the gyroid and the Lidinoid, both discovered numerically
in the 1990s. The existence was previously proved within a neighborhood of the
gyroid and the Lidinoid, using Weierstrass data defined on branched rectangular
tori. Our main contribution is to extend the technique to branched tori that
are not necessarily rectangular.
