Carlo Sequin

University of California
Berkeley, CA, USA

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.

The World of Wild and Wonderful Tori
The World of Wild and Wonderful Tori
24" by 24"
Composite of computer images

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.

Internally Knotted Figure-8 Torus
Internally Knotted Figure-8 Torus
8" x 6" x 6"
3D Model made on an FDM machine, ABS plastic

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