Visually intriguing tiling patterns arise from an asymmetric threading technique for constructing space-filling curves in the square. The method uses four square tiles in varying or uniform sizes, each the mirror or negative image of a single directed line segment. On varying size quilt dissections, the simplest pattern, built of just seven threaded tiles, is as fundamental a construction as the classic Peano or Hilbert Curves. For uniform n×n tilings, a computer enumeration finds 0,0,0,6,5,366,0,4110384, . . . of these special threaded tile designs for 1 <= n <= 8. The combinatoric explosion of possible patterns permits one to pick and choose among them using æsthetic criteria.