Over a thousand years ago, artisans in the Islamic world began to develop a system for constructing intricate geometric art based on radially symmetric starlike figures. As the centuries progressed, they raised this practice into a high art form, adorning architectural surfaces with colourful symmetric patterns (like the one on the top of this page) of limitless variety. The genre's masterpiece is surely the Alhambra palace in Granada, Spain. The era of building great bejewelled palaces is behind us, and with its passing went the craftsmen who designed these beautiful motifs. The techniques were closely guarded secrets that have not been handed down to the present day. Thus, we are forced to re-engineer the original design techniques from what clues survive. Many different systems have been hypothesized in modern times. What's weird is that most of them work, even though they're all so different. In truth, we can't know for sure how the Islamic artisans figured out these designs. But we can invent systems to create designs similar to theirs. And revel in the exploration.
Taprats is a Java applet that implements two such design technique for Islamic star patterns. The first technique is based largely on the work of Hankin  in the early part of the twentieth century and on a more recent paper by A.J. Lee . The technique was further refined by Craig S Kaplan . In a nutshell, we start with a tiling of plane made up at least in part of regular polygons. The polygons are filled with radially symmetric motifs like those found in the Islamic tradition. The tiles forming the gaps between the regular polygons are then filled in by finding natural extensions of the lines meeting their boundaries. The result is a network of lines that has nice graph-theoretic properties. The graph structure enables it to be coloured in various ways, or even rendered as a weave, or interlacing, as were many of the original designs.
The second technique is based on work by Peter J. Lu and Paul J. Steinhardt of Harvard University . It centers on the realization that many classical Islamic tilings are based on a small set of tiles with equilateral sides, now dubbed Girih tiles. When each tile is decorated with a particular set of lines, they can be assembled to produce many of the intricate designs found in mediaval Islamic architecture.
Taprats has a library of built-in tilings that can be used to construct many famous Islamic designs. Even better, the construction of these designs is parameterized in certain ways, so you can use Taprats as a vehicle for exploration of the vast space of Islamic designs.
The research that went into this applet appeared in print in the proceedings of the third annual Bridges conference at Southwestern College in Kansas. You can read the paper online  as part of the Bridges issue of the journal Visual Mathematics. You can also view a PDF of the paper  as it appeared in the Bridges proceedings.
You can find the Java source code for this applet on Sourceforge.
Original author: Craig S. Kaplan
Additional work by: Pierre Baillargeon
email: pierrebai at hotmail.com
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see www.gnu.org.
-  E.H. Hankin, Memoirs of the Archaeological Societry of India, volume 15. Government of India, 1925.
-  A.J. Lee. Islamic Star Patterns. Muqarnas, 4:182-197, 1995.
-  http://vismath4.tripod.com/kaplan/index.html
-  http://www.cgl.uwaterloo.ca/~csk/washington/tile/papers/kaplan_bridges2000.pdf
-  http://www.peterlu.org//content/decagonal-and-quasicrystalline-tilings-medieval-islamic-architecture