Weaving typically involves forming a surface by interlacing fibers into a mechanically stable arrangement, effectively making a two-dimensional object out of one-dimensional objects. Moorish Fretwork involves interweaving solid helical elements into mechanically stable two-dimensional arrangements by exploiting the helices’ screw symmetry. A three-dimensional extension of this idea was demonstrated by Alexandru Usineviciu at Bridges in 2015 [15]. Here we expand the idea further by considering cases informed by invariant cylindrical rod packing, and discuss interweaving geodesics of the gyroid. Simulations and/or physical models of nineteen triply periodic arrangements of interwoven helices are shown, with physical models demonstrated for eight.