# Carlo Sequin

I work on the boundary between Art and Mathematics. Sometimes I create artwork by using mathematical procedures; at other times I enhance a mathematical visualization model to the point where it becomes a piece of art. For the art exhibit at Bridges 2011 my submissions support my plenary talk: "Tori-Story." My presentation elaborates on the classification of all topological tori into four regular homotopy classes, where the members in one class cannot be smoothly transformed into members of another class. My art submissions depict some intriguing structures that topologically are torus surfaces, but with enough surprising contortions so that ordinary people would not immediately see them as your every-day donut.

Four panels of four different tori models:

Panel A: Half-Everted Torus: Two Klein-bottle mouths joined in a
symmetrical manner into a toroidal configuration.

Panel B: Collared Torus: A toroid with a cusp onto which another
toroid has been grafted with its parameterization turned by 90
degrees.

Panel C: Doubly-Looped Torus: A 3-fold epitrochoid profile swept
twice around a circular path while applying a total twist of 360
degrees.

Panel D: Doubly-Rolled Torus: A curtate hypocycloid profile swept
once around a circular path.

The challenge now is to figure out for each of these tori into
which regular homotopy class they belong.

This torus can be described as an ellipsoid with a Figure-8-knot tunnel through it.