Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, May 19, 2014 - 05:00

Location:

Gilkey 115

Guest Speaker:

Institution:

New Mexico State University

Abstract:

We consider the mathematics and visualization of a most efficient and aesthetic way to untangle a double-twist in 3-space, and will teach the audience how to "wave" the result.

Previous illustrations of untanglings have gone under the names "Dirac belt trick" or "Feynman plate trick", and reflect the non-intuitive fact that the fundamental group of the space of rotations in 3-space has order two. However, all previously known untanglings leave a lot to be desired in their aesthetics and simplicity, and it is hard to see geometrically what they are doing.

We deduce minimum constraints on the complexity required to untangle the double-twist. Then we will show animations of a beautiful untangling (-homotopy), derived quaternionically from the rotation geometry, that is the least complicated possible, subject to these constraints. The audience will be taught how to "wave" the result. Finally, the -homotopy involves some interesting geometric properties of essential maps from the projective plane to the two-sphere.

Proof techniques involve degree theory, fundamental groups, and spherical geometry, and can be handled most nicely with the quaternion representation for rotations.

Please remember to bring your right hand and arm. The speaker cannot be responsible for sore arms afterwards. A second independent time coordinate would also be very useful. If you have one, please bring it to share. If not, you should definitely come anyway, and I will stretch your brain to improvise one.