Embedding rectangular grids into square grids

It is shown that two-dimensional rectangular grids of large aspect ratio can be embedded into rectangular grids of smaller aspect ratios with small expansion and dilation. In particular, width can be reduced by a factor of up to two with optimal expansion, i.e. when the host rectangle is the smallest sufficient to contain the guest, and optimal dilation, i.e., two. A width reduction factor of three can be obtained with optimal expansion and dilation three. In general, any rectangular grid can be embedded into a square grid that is no more than unity larger on the side than the minimum possible, with dilation no more than three. These results improve on those previously obtained, in which dilation of better than 18 could not be guaranteed. They might be applicable to more complex grid embedding problems, such as embedding multidimensional grids into hypercubes