It is worth noting that for small dimensions of the lattice, the energy differs depending on whether we use periodic boundary conditions or free ends. This means also that the partition functions will be different, as discussed below. In the thermodynamic limit we have \( N\rightarrow \infty \), and the final results do not depend on the kind of boundary conditions we choose.
For a one-dimensional lattice with periodic boundary conditions, each spin sees two neighbors. For a two-dimensional lattice each spin sees four neighboring spins. How many neighbors does a spin see in three dimensions?