# Carlo Sequin

Professor of Computer Science

University of California

Berkeley

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 2012 my submissions support my oral presentation: "From Moebius Bands to Klein Knottles." My presentation elaborates on the classification of all types of Klein bottles 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 all Klein bottles, some as computer renderings and some as 3D rapid-prototyping models.

Klein Bottle of Type K8L-JJ

5" x 8" x 3"

3D Model made on an FDM machine, ABS plastic

2012

There are four different regular homotopy classes for Klein bottles decorated with a grid of parameter lines. This unusual model is inspired by the classical "inverted sock" Klein bottle, but uses a figure-8 cross-section. It turns out that it is composed of two left-twisting Moebius bands; it is thus in the same regular homotopy class as the left-twisting figure-8 Klein bottle.

Trefoil Klein-Knottle

8" x 8" x 3"

3D Model made on an FDM machine, ABS plastic

2012

A Klein bottle of type K8L-O. A tube with a figure-8 cross section is tangled up into a trefoil knot in which it experiences a right-handed twist of 540 degrees; this is equivalent to a left-handed twist of 180 degrees. This model is thus in the same regular homotopy class as the K8L-JJ Klein bottle depicted above.

The Kracy Kosmos of Klein Knottles

24" x 24"

2D composite of computer images

2012

Four panels of four differently contorted or knotted Klein bottles:

Panel A: A Klein bottle with a figure-8 cross-section forming a trefoil knot.

Panel B: Three classical "inverted sock" Klein bottle segments forming a trefoil knot.

Panel C: A cycle of six "inverted sock" Klein bottle segments with a figure-8 cross section.

Panel D: A figure-8 cross section making a 4-bounce zig-zag pass with 90 degrees of twist in each to form a Klein bottle.

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

Panel A: A Klein bottle with a figure-8 cross-section forming a trefoil knot.

Panel B: Three classical "inverted sock" Klein bottle segments forming a trefoil knot.

Panel C: A cycle of six "inverted sock" Klein bottle segments with a figure-8 cross section.

Panel D: A figure-8 cross section making a 4-bounce zig-zag pass with 90 degrees of twist in each to form a Klein bottle.

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