This paper investigates the polar slice sampler, a particular type of the Markov chain Monte Carlo algorithm known as the slice sampler. This algorithm is shown to have convergence properties which under some circumstances are essentially independent of the dimension of the problem. For log-concave densities, the algorithm probably converges (from an appropriate starting point) to within 0.01 of stationarity in total variation distance in a number of iterations given as a computable function of the spherical asymmetry of the density. In particular, for spherically symmetric log-concave densities, in arbitrary dimension, with an appropriate starting point, we prove that the algorithm converges in, at most, 525 iterations. Simulations are done which confirm the polar slice sampler's excellent performance.