# Carlo Sequin

I am interested in mathematical knots as constructivist sculptures.
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. Near the vertices, the knot
strand is deformed slightly to avoid any direct strand intersections.
These deformations, necessary to obtain a true knot, will further
reduce the overall symmetry; but the overall shape of the knot will
still represent the shape of the underlying polyhedron.

It is easy to form a closed Euler circuit on a polyhedron or on a
graph where all the vertices are of even valence, such as the
octahedron or the cuboctahedron. This also works on projections of
4-dimensional regular polytopes, such as the 5-cell or the
hypercube. I have chosen a perspective, cell-first projection of
the hypercube as a scaffold for the knot strand. Each vertex is
represented with a looping link of two strands passing through
that vertex in the Euler circuit. To make it easier to follow the
strand passing twice through all the 16 vertices of the hypercube,
the strand has a progressive rainbow coloring and is shown in four
different views. The top and bottom rows can both be seen as
cross-eyed stereograms.