  It was shown by M. Bhargava and P. Harron that for $n=3,4,5$, the shapes of
rings of integers of $S_n$-number fields of degree $n$ become equidistributed
in the space of shapes when the fields are ordered by discriminant. Instead of
shapes, we correspond grids to each number field, which preserve more of the
number fields' data. The space of grids is a fiber bundle over the space of
shapes. We strengthen Bhargava-Harron's result by proving that the grids of
rings of integers of $S_n$-number fields become equidistributed in the space of
grids.
