Any orientable, chess-colorable triangulation can be realized in corrugated plain weaving

A small portion of a chess-colorable triangulation and its realization as a corrugated plain weave

Triaxial plain weaving is not necessarily sparse. If the underlying surface triangulation is chess colorable (i.e., face 2-colorable, or equivalently, its dual is bipartite,) and orientable, a corrugated version of the surface can be woven with weavers that are straight, but corrugated with 90-degree folds.

The basket surface gains stiffness from this corrugation. The corrugated weavers shadow the paths of the straight weavers in the plain, open triaxial weave of the surface.

The tetrahelix is chess-colorable and orientable, so, barring interferences, it has such a corrugated plain weave.