We present simulation and separation results between multi-dimensional deterministic and alternating cellular automata (CAs). It is shown that for any integers k ł l ł 1, every k-dimensional t(n)-time deterministic CA can be simulated by an l-dimensional O(t(n)[(k-l+1)/( k-l+2)])-time alternating CA. This result is a dimension reduction theorem and also a time reduction theorem: (i) Every multi-dimensional deterministic CA can be simulated by a one-dimensional alternating CA without increasing time complexity. (ii) Every deterministic computation in a multi-dimensional deterministic CA can be sped up quadratically by alternations when the dimension is fixed. Furthermore, it is shown that there is a language which can be accepted by a one-dimensional alternating CA in t(n) time but not by any multi-dimensional deterministic CA in t(n) time.