  We determine the shapes of pure cubic fields and show that they fall into two
families based on whether the field is wildly or tamely ramified (of Type I or
Type II in the sense of Dedekind). We show that the shapes of Type I fields are
rectangular and that they are equidistributed, in a regularized sense, when
ordered by discriminant, in the one-dimensional space of all rectangular
lattices. We do the same for Type II fields, which are however no longer
rectangular. We obtain as a corollary of the determination of these shapes that
the shape of a pure cubic field is a complete invariant determining the field
within the family of all cubic fields.
