Synthetic weaving in a nutshell

Every mesh has a weave pattern. To find the pattern, connect the midpoints of the edges around each face (this is known as the medial construction.)

A single wire can trace the weave pattern without crossing itself. To find such a circuit, first find a spanning tree in the dual mesh. The circuit tries to avoid crossing edges in the primal mesh, while never crossing the spanning tree.

Helical 'staples' wind on to make the crossings, and overlap & interlock to form a 3-ply wire basket.