Simulations performed with general circulation models and a model of intermediate complexity show that the variability of the September sea ice extent in the Arctic of the 21st century increases first when the mean extent decreases from present-day values. A maximum of the variance is found when the mean September ice extent is around 3 million km2. For lower extents, the variance declines with the mean extent. The behavior is clearly different in Antarctica where the variance always decreases as the mean ice extent decreases, following roughly a square-root law compatible with very simple geometric arguments. Several mechanisms are responsible for the non-linear behavior of the Arctic. However, the strong interhemispheric contrast suggests that the difference in geometrical setting, with an open ocean in the south and a semi-closed basin in the north, plays a significant role.