A new treatment of the general reptation algorithm, introducing the notion of a transition state, is presented for close-packed lattices. Dynamic and static properties of short N-bead polymer systems on the two-dimensional triangular lattice are obtained from Monte Carlo simulations using this algorithm for systems ranging from dilute to concentrated conditions. Extrapolation of the simulation results to long chain lengths is found to be consistent with simple asymptotic power law scaling relations for [R-g(2)], the mean-square dimensions of the polymer; their relaxation time, tau(R); and the center of mass self-diffusion coefficient, D-c.m.. Empirical formulas of the form [R-SAW(2)]/[R-2]=alpha X+exp(-beta X) are proposed for the mean-square dimensions, ([R-2]), Of athermal polymers as a function of the scaled density X=(N-1)(d nu-1)rho; where rho is the polymer segment density; nu is the power law exponent for the mean dimensions, [R-SAW(2)], of dilute chains of the same length (N-1); and d is the dimensionality of the system. The fitted constants, alpha and beta, are independent of polymer length and density. These formulas successfully account for properties of 2d athermal polymers, from dilute solution conditions through semidilute conditions to the concentrated regime, for polymers of finite length.