What is the meaning of a basis?

[noamriemer]

Hi!
There is a concept I don't understand and would love to have is cleared...
What is the meaning of a lattice with a basis?
What do I need it for?
Say I have a honeycomb structure. (fig 1) and a basis as mentioned there (did I understand it right? is it the basis? )
why does it become a simple lattice spanned by ax,ay,az
with a two point basis

0, a/2(x+y+z)

or a four -point basis

0, a/2(x+y), a/2(x+z), a/2(y+z) ?

I don't understand the whole thing. For a start, how can in be (mathematically) that 0 is a part of a basis? What is this 0?
Second thing, how can something be spread by either 2 or 4 dimension basis?
It does not fit what I know about a basis...
I can't find answers to these questions... and I guess I miss the whole concept...
Thank you in advance for your help :)

 

[cgk]

A basis in the crystallographic sense not a basis in the sense of a vector space. The two are completely disjunct concepts, which just happen to have the same name by accident.

The unit-cell basis you are describing is effectively just a bunch of points in the unit-cell. The entire lattice is then spanned by translating these points (i.e., all the points separately) by integer-amounts of the lattice vectors.

 

[noamriemer]

Thank you!