3-Periodic weavings: piecewise linear embeddings of symmetric structures.

We provide a systematic account using symmetry and coordinates to explore symmetric (vertex- and edge-transitive) 3-periodic weavings of piecewise linear threads. The dia-w graph, derived from the diamond structure, generates an infinite family of supersymmetric 3-periodic thread weaves - a unique property shared only with the 2-periodic sql-w family of fabric weaves. Additionally, we describe a selection of symmetric 3-periodic, 3-coordinated weavings related to the srs graph and a supersymmetric Borromean weave of hcb graphs.