Lattice points and lattice basis

[Kitten]

Hi!
I'm struggling in identifying the lattice points and atom basis.

As I understand in a cube, there are 8 lattice points, on on each corner of a cube. But in 2d it is any square between 4 points which are the lattice points. Is this correct?

So if the points on the corners are the lattice points. What confuses me is that a primitive cell can only have 1 lattice point and so if it's a square I thought it would have four lattice points not one.

Also I don't understand how to define a basis with each lattice point?

 

[gckaufman]

In a square net, or any net for that matter, each corner holds only one fourth of the total motif. So if your motif is atoms, then each corner of that square net will only have a single atom even though there are four total lattice points. Same thing with a cube. Though there are a total of eight sites where a motif can reside, each of those sites will only contain one eighth of the total motif (in this case atoms). So for a primitive cubic structure, there are eight lattice points but only a single atom!