Hybrid simulations of a high Alfv¿n Mach number (MA≫1) collisionless perpendicular shock are performed in one, two, and three spatial dimensions. The results are compared to highlight the implicit effects of dimensionality in the simulation model. As in previous work, one-dimensional simulations show that the dissipation mechanism can be identified as reflected ions. Two-dimensional simulations allow the ion temperature anisotropy associated with the reflected ions to be relaxed via a collisionless electromagnetic instability. In the high MA limit, this instability causes the shock front to have rotating magnetic fields with approximately equal amplitudes in each direction as well as large-amplitude density fluctuations. The three-dimensional simulations are similar to the two-dimensional simulations with the exception that the fluctuations at the shock front are significantly smaller. This difference is due to the implied phase coherence in the two-dimensional calculations which is absent in three dimensions. The results of the simulations are compared with spacecraft observations of high MA perpendicular shocks. ¿ American Geophysical Union 1989