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 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.
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.
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.
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.