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
To get a fundamental region, cut away parts that are repeated. You can
and paste pieces of a fundamental region to get other fundamental regions for
Tiling Definitions Centered Cells