We explore periodic, non-intersecting cylinder packing arrangements known as hexastix and polystix. Groups of congruent cylinder arrangements, related through cubic symmetry, and limited to only three or four directions are defined and classified in this paper. The basic techniques and design principles required to reproduce several polystix models are illustrated. Adding a layer of complexity, we look at the different ways the ends of these symmetric stick weaving constructions can be joined to create intricate knots and links. Several original artworks made with pencils or flameworked glass, based on these cylinder packing arrangements, are presented and explained. Building and playing with these structures is a fun way to explore symmetry and promote mathematics as beautiful art.