2022 Joint Mathematics Meetings

# Carlo Séquin

## Artists

## Statement

I am interested in designing sculptures based on mathematical knots. Prime knots cannot exhibit mirror symmetry or any of the symmetries of the regular polyhedra. Still, it is possible to make tubular sculptures based on prime knots that overall display the structure of some Platonic or Archimedean solid. To do this, I run the knot strand along all the edges of such a polyhedron, forming a closed Eulerian circuit with as much symmetry as possible; the overall shape of the knot may then still represent the shape of the underlying polyhedron. To create an Euler circuit on a polyhedral edge graph, all vertices need to have an even valence; this can be achieved by judiciously doubling some of the edges.

## Artworks

For a cube it would be sufficient to judiciously double just four edges to render all vertices of valance 4 and thus to allow a closed Eulerian circuit. A more balanced result can be obtained if all twelve edges are doubled, thus yielding all vertices of valence 6. To make the final sculptural shape look much like the underlying Platonic polyhedron, all the paired double-edges wind around each other in tight helices. A single knotted strand can be obtained when nine of these helices make an odd number of half-turns, while the other three edge-pairs make full integer turns. To make it possible to follow this path through this complicated knot, the strand is exhibiting gradually changing rainbow coloring.

Here is a 3D print of the “Cube Double-Frame Knot.” Note, that even though at first glance this sculpture appears to have the full symmetry of a cube, this symmetry is actually broken because of the placement of the two different kinds of helices. The filament still forms a mathematical knot, and thus it can have at most one rotational axis of type $C_n$, or $D_n$, or $S_{2n}$, with valence n=3 or higher. This sculpture exhibits $D_3$ symmetry around one of the space diagonals of the cube.
It is challenging to fabricate this kind of model on a low-end printer where the support material has to be removed by hand. It is preferable to fabricate such models on a printer with a dissolvable support structure.