Tag Archives: projects

Another project idea

If anyone is still in the market for a project idea, here is another one, related to our recent discussions about the rotation number.  Suppose that we integrate the absolute value of the curvature of a closed plane curve (with respect to arc length),

\[ \int |\kappa_\gamma| ds \]

Then the minimum value we can obtain is \(2\pi\), which is what we get for a convex planar Jordan curve.  What’s more, in this form the theorem is still good for curves in 3-dimensional space (remember that I explained in class that in 3-space one can’t give the curvature a definite sign, but of course that is not an issue if we only want to integrate its absolute value.

In 1950, John Milnor (then aged 19) proved that for a knotted loop in 3-dimensional space the lower bound for the total curvature is \(4\pi\) rather than \(2\pi\).  In other words, if a space curve has sufficiently small total curvature, it must have the topology of the unknot.  Milnor’s proof was published in the Annals of Mathematics which is probably the most distinguished mathematics journal in the world.  (The same result was proved independently and slightly earlier by Istvan Fary; it is known as the Fary-Milnor theorem.)

Explaining the proof would make a good project.  I’ve made a copy of the original article available here.

The winding number applied to handprints

My colleague Nate Brown reminded me of a nice article by Steve Strogatz which appeared in the New York Times last year.

Strogatz is a mathematician at Cornell and one of the best “popular” mathematics writers around today.  He ran a regular column Me, Myself and Math in the Times last fall – maybe it will appear again.  This one is about the ridge patterns on your hands.  Quoting Strogatz,

“When you look at your fingerprints, you’ll notice just a few types of singularities. The two most fundamental are the triradius

and the loop.

All other singularities on fingerprints can be built up from these two. For instance, the singularity known as a whorl

can be regarded as the fusion of two loops

that have been squashed together so that their two inner endpoints coincide.

In 1965 Lionel Penrose, a British medical geneticist, pointed out that fingerprints and palm prints obey a universal rule: no matter what your personal pattern looks like, everybody with five fingers always has four more triradii than loops. (His bookkeeping treated a whorl as two loops, for the reason explained above.) ….

In 1979, Penrose’s son Roger, a mathematical physicist, published a beautiful paper dedicated to the memory of his father, in which he derived his father’s rule from topology. Let me outline his proof for you now.”

Penrose’s proof (yes, the Roger Penrose) can be found in

Penrose, R. 1979. “The Topology of Ridge Systems.” Annals of Human Genetics 42 (4): 435–444. doi:10.1111/j.1469-1809.1979.tb00677.x. It can be downloaded from http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1979.tb00677.x/pdf

 

Topology and music

I put up a post on my main page regarding a paper on modal harmony and elementary topology: see http://sites.psu.edu/johnroe/2013/09/05/topology-and-music/  

There are some ideas here which could be connected to the winding number and other things that we’ll study in this course.  I know some of you had mentioned musical interests when you introduced yourselves, so you might like to bookmark this as a possible project when we get to that stage of the course.John