We investigate the approximation error of functions with continuous and piecewise-linear (CPWL) representations. We focus on the CPWL search spaces generated by translates of box splines on two-dimensional regular lattices. We compute the approximation error in terms of the stepsize, length ratio, and angle that define the lattice. Our results show that hexagonal lattices are optimal, in the sense that they minimize the asymptotic approximation error.