Wallpaper Groups
We can also use non-regular polygons, such as rectangles and parallelograms (a rectangle kicked on the head) but mathematics tells use that we will always find that the patterns fall into a small number of categories, just 17, that share the same set of symmetries (in terms of translations, rotations and reflections). These are called the "Wallpaper Groups", because every wallpaper pattern you have ever seen or can ever construct would fall into one of these categories. See this Wikipedia article for a complete catalog of all possibilities.
If you think about filling three dimensional space with regular shapes you get to the "crystallographic groups" that enumerate all the possible ways of making natural crystals. This is a concept of huge importance for the science of materials.