It doesn't appear to be anything fundamental, just a number that makes the proof technique work out. As the authors point out, that dimension is only a condition to get some loose bound working, not a necessary condition generally:

It is possible to improve the factor K = 1728 appearing in Corollary 5.5, at the expense of introducing various technical complications into the algorithm. In this section we outline a number of such modifications that together reduce the constant to K = 8 + ϵ, so that the modified algorithm achieves M(n) = O(n log n) for any d ≥ 9 (rather than d ≥ 1729).