Ground states of the antiferromagnetic Ising model on finite triangular lattices of simple shape

The ground state energy of the antiferromagnetic Ising model on finite triangular lattices of some simple shapes is derived using simple arguments. It is shown that the ground state entropy density vanishes for a parallelogram with a free boundary. Numerical calculation of the ground state entropy density for some other simple shapes with free boundaries illustrates the approach to the thermodynamic limit. The results illustrate, by explicit examples, the known intricate relationship between boundary conditions, degeneracy and the ground state entropy in the thermodynamic limit. They are also relevant to some applications of the Ising model in biophysics