Monday Morning Math: You are Here

Here’s a fun map trick: open up a map app on your phone and choose a view that includes your location. The view can be as zoomed in or as zoomed out as you’d like. If you hold your phone horizontally, so the map appears parallel to the ground, then there will be a point on the map that is directly above the point it represents. That’s direct in a mathematical term: it won’t be even a millimeter off. If you walk around a bit, or zoom in or out more, the relevant point will change, but there will always be some point on the map that is directly above the place it shows.

This result doesn’t just work with phones, but with paper maps, should you have one handy to unfold and hold parallel to the ground. That map doesn’t have to be perfectly flat either: you could crumple it up and then uncrumple it, or hold it at an angle, and there would still be a point over the spot it represents. Or you could print your map on stretchy fabric or on silly putty, stretch it out and, after you’ve laughed at how funny the map looks, there would STILL be a point that woks! Furthermore, you could send your crumpled up silly putty map to a friend (as long as that friend also is somewhere shown on the map) and it would work for them, too.

This is an example of Brouwer’s Fixed Point Theorem, which says that if you have a continuous map from a nonempty compact convex closed set to itself, there will always be a fixed point – a point where f(P)=P.

Monday Morning Math: x=+

Good morning! One of the best things about math is noticing something and thinking, “Huh – I wonder how often that happens?” Maybe it’s a complicated answer, maybe easy, but I love that that question can be asked and often explored when you least expect it.

For example, you might notice that 2+2=2·2 and wonder if that happens with other numbers. A quick check shows that if x+x=x·x, then x²=2x so 0=x²-2x=x(x-2) so the only numbers that work are x=0 and x=2. Well, maybe. What if you switch into modular arithmetic? Then you just need x²=2x mod n, so you end up with a quadratic equation: x²-2x-kn=0 and using the quadratic formula you get x=(2±√(4+4kn))/2, which simplifies nicely to x=1±√(1+kn). In Base 3, for example, letting n=3 and k=1, one solution is x=1-2=-1, and a quick check verifies that (-1)+(-1) is the same as (-1)·(-1) since both equal 1 mod 3. And in Base 8, setting n=8 and k=3, one solution is x=1+5=6. Sure enough 6+6=6·6 in Base 8, since both equal 4 mod 8. Are there an infinite number of possibilities? Probably?

Since we’re already in modular arithmetic, here’s a similar thing that came up recently in Abstract Algebra. We were working with the integers mod 5, and it turns out that 2 and 3 are both additive inverses of each other (since 2+3=0 mod 5, and 0 is the identity using addition) and multiplicative inverses of each other (since 2·3=1 mod 5, and 1 is the identity under multiplication). And that got me wondering – are there other pairs like that, where x+y=0 mod n but x·y=1 mod n? The answer is Yes! For example, in mod 10 the numbers 3 and 7 are both additive inverses and multiplicative inverses. But wait, there’s more! The numbers 4 and 13 are additive and multiplicative inverses, mod 17. And I bet you can find even more! And come up with another interesting puzzle waiting around the corner.

Enjoy!

Math Mistakes that Make the News: Optimization

Good morning! It’s been a while since we had a Math Mistake featured here, but thanks to an article sent my way (thanks MM!), we have some breaking news. Well, “breaking” might be a stretch – the error happened in 1945 and was revealed in 1989, so it’s been a while. But still, it’s an error I hadn’t heard of before and it involves *Calculus*, and specifically checking if your critical point is a max or a min.

So what happened? In 1945 William R. Sears and Irving L. Ashkenas, who worked for the Northrup Corporation, wrote a report comparing the range of a traditional bomber to one that didn’t have a tail and was essentially all-wing (looking a bit like a boomerang). As part of the justification they used CALCULUS to find the shape that gave the best range: set up a function, take the derivative, and see what the critical points are. In this case, there were two: one where almost all the weight was in the wings, and one where it wasn’t. They stated that the all-wing design gave the maximum range.

The problem? That was actually the minimum. A test like the second derivative test can help with that – if the second derivative is negative, then the critical point is a maximum, like ⋂. But if the second derivative is positive, it’s a minimum, like ⋃. 

This mistake was caught within a few years, when Engineer Joseph V. Foa checked the math and told the authors. They admitted it, but also said it didn’t really change things because that was only part of the story. Foa thought that this did too change things, and if this were a couple hundred years earlier there might have been a duel. Instead, the army canceled an all-wing aircraft they’d been working on, supposedly for budget reasons.

So that’s that! Or maybe not – sometimes endings aren’t so clear. Wayne Biddle’s 1989 article below refers to a new B-2 bomber by Northrup that uses the same all-wing design. 

Main source: “Skeleton Alleged in the Stealth Bomber’s Closet” by Wayne Biddle, Science, New Series, Vol. 244, No. 4905 (May 12, 1989), pp. 650-651. (There was a follow-up: “B-2 comes up short” by Wayne Biddle, Science, Vol. 246, Issue 4928 (October 20, 1989), p. 322.)

Monday Morning Math: Honoring Black Mathematicians

Good morning!!!  e-day (February 7, for 2.7….) is almost here, which makes this a very mathematical week.  We’re celebrating with cookiEs in the Math Center on Wednesday.  

But that’s not the only day of celebration – we have a whole month’s worth!  The site Mathematically Gifted & Black honors a new mathematician each day in February! The honoree for February 1 was Toka Diagana.  Dr. Diagana studied mathematics in Mauritania and France, and is now a professor at Howard University; he has published seven books and well over a hundred research articles, mostly related to Differential Equations. Other honorees for February are

• Dr. Lakeshia Legette Jones, Associate Professor of Mathematics at Clark Atlanta University 
• Dr. Adrian Cartier, Cofounder and Chief Data Officer of Freight Science, Inc.
• Dr. Tai-Danae Bradley, Research Mathematician at SandboxAQ, Visiting Research Professor of Mathematics at The Master’s University, and creator of Math3ma.

Happy reading!