Given a totally real number field of degree $n > 2$, one can look at the ring of integers and consider the group of units. It is known by a classical theorem of Dirichlet that the group of units is given by a Euclidean lattice in $n-1$ dimensions. This also holds true if one considers the group of units in an order inside the ring of integers. One can then try to plot these lattices in the appropriate moduli space of Euclidean lattices in $n-1$ dimensions and ask where the points lie. It was conjectured by David–Shapira that for $n = 3$ these unit lattices generate a dense set of lattices. I will explain the problem for $n=3$ and sketch the proof. (Joint with Nguyen-Thi Dang and Jialun Li.)