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.
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.