Why do not these lattices qualify as Bravais lattices?

Question

I have trouble understanding why one cannot add more Bravais lattice to the already established ones. To me it seems like I can easily come up with more Bravais lattices that fulfills the definition below.

"[Bravais lattice] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by: $\boldsymbol {R}=n_1\boldsymbol {a_1}+n_2\boldsymbol {a_2}+n_3\boldsymbol {a_3}"$ - https://en.wikipedia.org/wiki/Bravais_lattice

Let's look at the 2-dimensional case but my question can easily be extended to 3D.

For two examples of lattices that is not included in the already established ones are the ones above. These two unit cells will create an infinite array by discrete translations. They also fulfill the following "A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice will appear exactly the same from each of the discrete lattice points when looking in that chosen direction" - https://en.wikipedia.org/wiki/Bravais_lattice

So how come these are not included? I can think of more examples but there seems to be something that I fundamentally do no comprehend.

Thanks in advance.

Answer 1

Your examples are included in the Bravais lattice scheme. Your first example is, as was pointed out in the comments, a square lattice that has been rotated by 45 degrees.

Your second example is a little more subtle. Notice that there are 3 different "types" of point in the cell you have drawn, the corner points and the left and right interior points. The translations relating these points generate points that are not in the lattice; the applying the translation that takes you from the left to the right inner point to the corner point will leave you in empty space. So not all of these points are actually lattice points. What you have is known as an orthorombic lattice with a basis. The corner points generate an orthorombic lattice and on top of this underlying structure there is some extra stuff drawn on, which will also repeat with the lattice. The key point is that the lattice is defined by the translational symmetry of the system.