One type of non-periodic pattern is a Penrose Tiling . A Java applet that lets you play with Penrose Tiles was written by Shuxiang Zeng . A collection of programs to help you experiment with tilings can be found at The Geometry Center at the University of Minnesota. Jim Millar has a nice collection of fractal tiles, constructed using one of them. Many beautiful images of fractal patterns with wallpaper symmetry are offered by Michael Field .
Origami provides a hands-on way to construct patterns.
Helena Verrill gives instructions for making origami quilts and many more mathematical objects. Key Curriculum Press offers some books about mathematics and quilt-making . Another site for quiltmakers is the MathQuilt site. Symmetry is basic to the study of crystals. You can see moving models of each of the 230 types of crystals.