Noise created by filtering grids of random numbers is slow and tends to be a little dependent on the rectangular coordinate system. The reason is connected with the idea of packing.
The points in these two grids are differently packed. The distance between adjacent points is the same in the figure on the right but the arrangement gets more points into a given area. In the close packing we can use a smaller radius of filter and achieve a smoother result while averaging fewer points.
Depending where you place the filter, the circle in the standard packing covers nine to twelve grid points. In the close packing a smaller circle covers six to seven points. This produces a filtered noise that is just as good with less calculation.
As the number of dimensions goes up, we must average more and more points. With the standard packing we need as many as 256 points in 4D.
Packing in 4D Non Uniform Grids