Fundamental Regions and Primitive cells

Any periodic tiling has no end of fundamental regions and primitive cells.  The difference between
the two is that a fundamental region generates the tiling when all symmetries are applied,
but only translations are used for a primitive cell to generate. If we start with a periodic tiling:

Pick any point, and translate that point by all of the translations of the symmetry group of the tiling.  This gives a lattice
for the tiling.  The lattice of a tiling is not unique.  Different choices for an initial  point can give
different spatial locations for the lattice. Also, there is no end of ways to get two sets of parallel lines that go through
all the lattice points of any given tiling.  That is, the shape of the lines joining the lattice points for a given tiling is not unique.
The same lattice points can seem to have different characters depending upon which lines are chosen to join the points.
Pictured here are two different sets of lines for the same lattice.

Every parallelogram arising from the parallel lines of a lattice is a primitive cell.  
To get a fundamental region, cut away parts that are repeated.  You can then cut 
and paste pieces of a fundamental region to get other fundamental regions for the tiling.  

Tiling Definitions  Centered Cells