Special project: Tiling with groups

A good reference for this section is Thurston's article [Th].

Let be any oriented path on a rectangular lattice. Let be any elements of some group . We associate to a word'' in as follows. For each move by one unit in , write . For each move by one unit in , write . For each move by one unit in , write . For each move by one unit in , write .

Example 4.9.1   Let be the closed path pictured below, where denotes the initial (and terminal) point.

The word associated to is

Exercise 4.9.2   Let be the word associated to a path . Let denote the path associated to the path with every edge having the opposite orientation. Show .

Exercise 4.9.3   Conside the tile in Example 4.9.1. Find the words associated to each of its rotations and reversals.

Exercise 4.9.4   Show that the square grid can be tiled using the tile in Example 4.9.1 along with its rotations and reversals.

Exercise 4.9.5   Using group theory, show that the square grid cannot be tiled using the tile in Example 4.9.1 along with its rotations and reversals.

Exercise 4.9.6   Show that the square with missing upper left-hand and lower right-hand corners

cannot be tiled by the tile

along with rotations by multiples of and reversals.

david joyner 2008-04-20