Mathematics of Islamic Tiling Patterns

There is a fascinating article about the mathematics of Islamic tiling patterns as discovered by Peter J. Lu of Harvard University while visiting his cousin in the Peace Corps in 2005 in Bukhara, Uzbekistan. We owe a great deal to ancient Islamic mathematicians, not only their own work, but also their translation and preservation of Greek works that otherwise would have been lost.  The discovery is explained by Peter J. Lu in a video lecture...

The original article and interesting supplementary material is freely available... 

 Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture

The Boston Globe has a helpful article on the five basic girih tiles... 

Science Imitates Art? 

 For your own exploration of  Islamic Tiling Patterns, The Boston Globe Science Department provided some PDF files of the five basic girih tiles (shown above) that can be printed and cut out...

Shape 1: Bowties

Shape 2: Decagon 

Shape 3: Rhombus

Shape 4: Pentagon

Shape 5: Hexagon

To create your own complex tiling designs online, Nikolaus Baumgarten provides an excellent and free browser-based 

  

GIRIH TILING EDITOR 

Baumgarten's great editor also has a feature where the tiles can have interlacing strapping lines:

Any tiling you do can be easily translated to this mode.  Here is one of mine:

Another interesting way to experiment with Girih tiling on your computer is using the free SketchUp program.   
 
Google SketchUp is a free, fun, easy-to-use 3D modeling application.  Originally created for architects and designers, SketchUp is also a great tool for teaching geometry.
 

Sketchup Projects Including Girih Tiling

Any image editing software can be used to build fascinating girih tile patterns.  Below is a tiling I build up using the excellent and free open source image software Gimp:

And here is the final result:

If you look carefully at the above tiling, you can see the faint outlines of the five basic girih tiles:

In my tiling, I made them all blue in color to emphasize the overall pattern.  In Islamic tilings these five basic tiles can seldom be seen.  That is why their use was not known until Peter J. Lu noticed them.