# Edmund Harriss, Silviana Amethyst

We started working together at the "Illustrating Math" semester at ICERM, with the goal of exploring toolpaths to make algebraic surfaces on a CNC mill. On a 5-axis mill, the tool can control both the position of cut and the direction, thus producing a very high finish quality with a relatively small number of paths. Our goal was to create the toolpaths to machine the surface directly from the mathematical surface, rather than through some sort of proxy, for example in first creating a mesh, and then applying a more standard strategy.

The modularity of the Barth Sextic — 20 identical units — makes it an obvious choice to mill in pieces. As the sextic surface with the maximum possible number of double-point singularities (here the positions of the brass rods and the spikes on the outside mark the real singularities), this structure is mathematically interesting as well as beautiful. The engagement from the materials of the structure has already lead to many conversations with many people about the nature of singularities and algebraic surfaces.