After working out how to use Blender to paint hyperbolic tilings I then thought of using Blender to look at planar symmetric tilings.
A while back I wrote a GIMP plugin which takes a selection from an image (“cell”) and produces a new image according to one of the 17 plane symmetry groups. These are also known as wallpaper groups or plane crystallographic groups. Basically it rotates and or flips copies of the cell, combines them to form a tile and then copies that tile to fill a new image. The GIMP plugin dosen't update in real time. If you want to make changes you need to re-run the plugin to generate a new tiling.
Using UV mapping its really easy to get Blender to draw tilings from any of the 17 plane symmetry groups that update as you paint.
This blend file (sym_tile.blend) has been set up with 17 plane grids each on a separate layer. Each layer implements one of the symmetry groups via a UV map.
Here's a screencast that shows the live drawing in action and explains how I set up the UV maps.
For more on plane symmetry groups see the galleries of images produced with the GIMP plugin, my post on symmetry group notation which also has links to other online resources and my post on quilt design.