This page lists some wonderful, interactive, visual, easy-to-use software that generates decorative planar patterns or its study. ⁖ periodic tilings, penrose tilings, Arabic weavings, or drawing programs that draw based on mathematical analysis of symmetry.
Kali is a freeware by Jeff Weeks. It is a interactive program for drawing figures of plane symmetry. The program is so appealing that it is suitable for grade-level kids too. A user selected symmetry pattern automatically appears based on whatever you draw on the screen. Mac ＆ Windows versions at: http://www.geometrygames.org/Kali/index.html MacWin
Escher Web Sketch. This is a Java program that lets user interactively draw symmetrical patterns. By Wes Hardaker and Gervais Chapuis, at http://escher.epfl.ch/escher/ Java
Taprats is a Java applet that generates Islamic patterns. Excellent. By Craig S Kaplan, at http://sourceforge.net/projects/taprats/ Java
Also check out his thesis work. Computer Graphics and Geometric Ornamental Design By Craig S Kaplan. @ http://www.cgl.uwaterloo.ca/~csk/phd/
Tyler is a Java applet that draw tilings. It is written by Melinda Green and Don Hatch, at http://www.superliminal.com/geometry/tyler/gallery/ Java
Bob is a Windows program that does Penrose tilings, by Stephen Collins at http://www.stephencollins.net/Penrose/Default.aspx Win