I What is the Relationship Between Lattice Points and Basis Balls?

[Eth338]

Hi, take a look at the picture from my textbook, specifically the bottom part:

there are five lattice points, shouldn't that mean that there are also 5 "small basis balls"? Or can they be "shared"? If so, they are not all oriented in the same way - is that not important since there's no special direction?

 

[.Scott]

That small sampling of 5 lattice points is not the full lattice. When you fill it out, all of the Basis parts are there.
For example, the 5-point one could have been shown this way:
XxXx
. XxXx
XxXx

or
XxXx
xXxX
XxXx