College Math Teaching

March 5, 2013

Math in the News (or: here is a nice source of exercises)

I am writing a paper and am through with the mathematics part. Now I have to organize, put in figures and, in general, make it readable. Or, in other words, the “fun” part is over. 🙂

So, I’ll go ahead and post some media articles which demonstrate mathematical or statistical concepts:

Topology (knot theory)

As far as what is going on:

After a century of studying their tangled mathematics, physicists can tie almost anything into knots, including their own shoelaces and invisible underwater whirlpools. At least, they can now thanks to a little help from a 3D printer and some inspiration from the animal kingdom.

Physicists had long believed that a vortex could be twisted into a knot, even though they’d never seen one in nature or the even in the lab. Determined to finally create a knotted vortex loop of their very own, physicists at the University of Chicago designed a wing that resembles a delicately twisted ribbon and brought it to life using a 3D printer.

After submerging their masterpiece in water and using electricity to create tiny bubbles around it, the researchers yanked the wing forward, leaving a similarly shaped vortex in its wake. Centripetal force drew the bubbles into the center of the vortex, revealing its otherwise invisible, knotted structure and allowing the scientists to see how it moved through the fluid—an idea they hit on while watching YouTube videos of dolphins playing with bubble rings.

By sweeping a sheet of laser light across the bubble-illuminated vortex and snapping pictures with a high-speed camera, they were able to create the first 3D animations of how these elusive knots behave, they report today in Nature Physics. It turns out that most of them elegantly unravel within a few hundred milliseconds, like the trefoil-knotted vortex in the video above. […]

Note: the trefoil is the simplest of all of the non-trivial (really knotted) knots in that its projection has the fewest number of crossings, or in that it can be made with the fewest number of straight sticks.

I do have one quibble though: shoelaces are NOT knotted…unless the tips are glued together to make the lace a complete “circuit”. There ARE arcs in space that are knotted:

WildArc

This arc can never be “straightened out” into a nice simple arc because of its bad behavior near the end points. Note: some arcs which have an “infinite number of stitches” CAN be straightened out. For example if you take an arc and tie an infinite number of shrinking trefoil knots in it and let those trefoil knots shrink toward an endpoint, the resulting arc can be straightened out into a straight one. Seeing this is kind of fun; it involves the use of the “lamp cord trick”

Screen shot 2013-03-05 at 5.02.43 PM

(this is from R. H. Bing’s book The Geometric Topology of 3-Manifolds; the book is chock full of gems like this.)

Social Issues
It is my intent to stay a-political here. But there are such things as numbers and statistics and ways of interpreting such things. So, here are some examples:

Welfare
From here:

My testimony will amplify and support the following points:

A complete picture of time on welfare requires an understanding of two seemingly contradictory facts: the majority of families who ever use welfare do so for relatively short periods of time, but the majority of the current caseload will eventually receive welfare for relatively long periods of time.

It is a good mental exercise to see how this statement could be true (and it is); I invite you to try to figure this out BEFORE clicking on the link. It is a fun exercise though the “answer” will be obvious to some readers.

Speaking of Welfare: there is a debate on whether drug testing welfare recipients is a good idea or not. It turns out that, at least in terms of money saved/spent: it was a money losing proposition for the State of Florida, even when one factors in those who walked away prior to the drug tests. This data might make a good example. Also, there is the idea of a false positive: assuming that the statistic of, say, 3 percent of those on welfare use illegal drugs, how accurate (in terms of false positives) does a test have to be in order to have, say, a 90 percent predictive value? That is, how low does the probability of a false positive have to be for one to be 90 percent sure that someone has used drugs, given that they got a positive drug test?

Lastly: Social Security You sometimes hear: life expectancy was 62 when Social Security started. Well, given that working people pay into it, what are the key data points we need in order to determine what changes should be made? Note: what caused a shorter life expectancy and how does that effect: the percent of workers paying into it and the time that a worker draws from it? Think about these questions and then read what the Social Security office says. There are some interesting “conditional expectation” problems to be generated here.

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3 Comments »

  1. Hi. I wonder whether the proof of the lamp cord trick appears somewhere in the litterature.
    I could not find it in Bing’s book, but maybe this only means I did not read it thoroughly enough. Thank you for any advice.
    Best

    Comment by Antoine Chambert-Loir — May 21, 2015 @ 6:30 pm

    • I don’t know where this is proved under this labeling, but the lamp cord trick is equivalent to all smooth arcs in 3 space being ambient isotopic to one another; you might look at a book like Hirsch (Different Topology). It is one of those things that is often assigned as an exercise in a first “geometric topology” class. I don’t know what your background is, but the LCT might be stated as follows: “given a properly embedded smooth arc in S^2 X [0,1] running from S^2 x {0} to S^2 x {1}, there is an ambient isotopy taking that arc to x x [0,1] where x is in S^2” or something like that.

      Comment by blueollie — May 21, 2015 @ 7:06 pm

      • I understand now, thanks !

        Comment by Antoine Chambert-Loir — May 21, 2015 @ 8:13 pm


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