# Carlo Sequin

Professor of Computer Science

University of California, Berkeley

Berkeley, California, USA

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.

Hypercube Knot

18 x 18 cm

Computer Graphics

2021

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.