How is a multi-dimensional array defined in terms of a pointer?

Yes. More specifically, they can be used to represent a dynamic multi-dimensional array.

As most people know, for a one-dimensional dynamic array, you simply need a pointer to the first element in the array, where each element contains an object of the same type as the pointer. The pointer can be passed around functions just as if it were a static array, the only difference being the requirement to pass the upper bound of the array as well as the array itself.

For a two-dimensional array, you need a pointer-to-pointer which points to the first element of a one-dimensional pointer array, where each pointer in that array points to a one-dimensional array of objects. The pointer-to-pointer must be the same type as the pointers and the objects.

For a three-dimensional array you need a pointer-to-pointer-to-pointer. And so on. Each additional dimension simply adds a new level of indirection, and a new level of one-dimensional pointer arrays.

Of course multi-dimensional arrays are only useful if every dimension is fully utilised and doesn't require too much in the way of resizing. If that is not the case, then you may get more efficient memory consumption from a vector of vectors (of vectors), which allows dynamic resizing at every level without having to copy existing elements. The downside is you lose the random access provided by the array.