This page lists some wonderful, interactive, visual, easy-to-use software that generates decorative planar patterns or its study. In particular, software that draws 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 (2002-10) 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. @ www.cgl.uwaterloo.ca…
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