Cubic shapes

This is an animation of the distribution of shapes of real, on the left, and complex, on the right, cubic fields in the standard fundamental domain of GL(2, Z) acting on the upper-half plane. The shape of a cubic field K is the 2D lattice obtained by looking at the orthogonal complement of 1 in the ring of integers of K (with respect to the Minkowski inner product). This visualization shows the first 14000 real (resp. complex) cubic fields ordered by discriminant, starting from the first 500 and adding 500 fields in each frame. It is a result of David Terr's thesis that these shapes are equidistributed. This result has recently been generalized to Sn quartic and quintic fields by Manjul Bhargava and Piper Harron (but there are not enough dimensions in your web browser to display the data; consider upgrading!).

You might notice that the dots on the right seem to form lines. You are very observant! This is explained in an upcoming paper of mine: complex cubic fields whose trace-zero form is a multiple of a fixed indefinite binary quadratic form have shapes that equidistribute on the geodesic defined by that quadratic form.

This data was computed by using the code I wrote for computing shapes of number fields (GitHub) with the number theory tables available at http://pari.math.u-bordeaux1.fr/pub/pari/packages/nftables/. Everything was done in Sage.
Real cubic shapes animation Complex cubic shapes animation