Manifolds Lying in a Higher-Dimensional Space

We often view manifolds as lying in some higher-dimensional space, but shouldn't we also consider ones that are telling the truth? 🤔

Möbius Strips: Now Apparently Three-Dimensional

Today I found out about a new paper that reports the generation of Möbius strips of optical polarization, which is very exciting. Unfortunately, the paper's abstract starts out in a very depressing manner that makes me cringe: "Möbius strips are three-dimensional geometrical structures, fascinating for their peculiar property..." Nooooooo!

Although does need three dimensions embed a Möbius strip in an ambient (extrinsic) space, a Möbius strip is a two-dimensional manifold.

(Tip of the cap to Physics Today for their Facebook post of a popular article about the new discovery. Notice the comment about the number of dimensions in the correction. In this case, the popular piece has done better than the paper's abstract.)

Gaussian Curvature and its Practical Consequences

Wired has a very cool article about manifolds and curvature.

I have used ice cream cones (and the act of unrolling them) to help explain curved versus straight geodesics for one of the problems our student had to do in calculus of variations. I then "assigned" my students to do this the next time they went to the local ice cream place.

(Tip of the cap to Jeremy Kun.)

"Uphill Both Ways Through the Snow" on Steroids

I will see your walking uphill both ways through the snow, and I'll raise you walking in the snow every day with no shoes on a non-orientable surface. I approve!

I am so going to use this one the next time I visit Cornell!