Human Knots

I read a post from The Math Less Traveled entitled, The Mathematics of Human Knots. In this post, the author discusses the game “human knot”, in which every player stands in a circle and grabs the hands of someone else in the circle so that everyone is seemingly tangled up. The goal of the game is then to untangle the knot without any pair of hands letting go of each other. Many people believe that untangling is always possible, however after Dr. Heath’s lecture on knots (and reading this article), I know that this is “knot” true (I crack myself up).

The reason that accomplishing the goal of “human knot” is not always possible is because certain knots exist which cannot be untangled. The blog post provides the example of the figure eight knot:

Just from looking at this knot one can see that the knot couldn’t be untangled without going through itself.

This spring Professor Heath is teaching a course on knot theory. I look forward to taking it.

Math in Economics

For many people who don’t go past an introductory ECON course, the relevance of mathematics in the field of economics may not be immediately obvious. In fact, I would assert that it isn’t until graduate level work that one would really be able to appreciate the amount of math that is used in economics. I say this because, as a double major in Mathematics and Economics,  I have been surprised at how little math has been used in the ECON courses I have taken. Even in Mathematical Economics (ECON 345 here at PLU), the math covered was only basic concepts from Multivariate Calculus and Linear Algebra. Luckily, Paul Krugman, a professor of Economics and International Affairs at Princeton University, has described (very concisely) one of the ways he uses mathematics in economics in his New York Times article Mathematics and Economics.

In his article, Krugman explains that he uses mathematics in his economic research to help define and describe a situation. He writes, “In the economic geography stuff, for example, I started with some vague ideas; it wasn’t until I’d managed to write down full models that the ideas came clear. After the math I was able to express most of those ideas in plain English, but it really took the math to get there, and you still can’t quite get it all without the equations.”

Krugman also pointed out the limitations of economics even with the use of mathematics. He explains that just because something can be modeled nicely does not mean that the model is correct. I fear that this is a shortcoming that many have fallen prey too. While mathematics is an incredibly useful tool in economics, they are two distinct fields, and treating models as fact in economics can have disastrous results.

Despite the limitations of mathematical economics, it is still a fascinating field. I’d be interested to learn more about the math going on behind the scenes in economics, as well as its applications.

A New Way to Teach Math

With the arrival of more advanced calculators (such as Wolfram Alpha), awarding points for correctness is becoming less feasible. What good does awarding points for a correct answer do if one could simply input the problem into a calculator and have the correct answer given to them? This issue is addressed in Teaching College Math’s blog post Shifting Assessment in a World with WolframAlpha.

In this blog post, a new form of assessment is suggested. Instead of asking students to arrive at an answer, simply give them the answer (along with the original question), and have them show how to (correctly) arrive at that answer. This takes the guess-work out of grading a problem with a correct answer and no work. This would also make problems more proof based and less computational, thus introducing proof concepts and integrating them into the work earlier in math curriculum.

I generally like the ideas presented in this blog. I think that having had the right answer before even starting the problem would have been helpful in seeing whether or not I was getting the right answer through my work. It would also force many students to take their learning more seriously, so they actually understand what is going on in their work, instead of simply knowing the shortcuts (in calculus, for instance). Finally, having been slowly introduced to proof techniques early on and building up to a class where proofs are the primary focus would have helped a lot (MATH 317 was not easy, for me at least).

Burger Combinatorics

The article The Math Behind 96 Billion Burger Choices tells of a burger restaurant where the options are seemingly endless. The Manhattan restaurant 4food offers 5 buns, 4 add-ons, 12 condiments, 7 cheeses, 4 slices (avocado, mushroom, etc.), 17 scoops (beans, nuts, mac & cheese, etc.), and 8 patties.

The article also delves into the mathematics of determining how many possible burger combinations there are given this vast array of toppings. Assuming one bun and as many as you want of each of the others within the restraints established by 4food, the number of possible combinations is derived as follows:

5 * (2^4) * (2^12) * (2^7) * (2^4) * 18 * 8 = 96,639,764,160.

Knot-ty Games

In Division By Zero’s blog, A Game for Budding Knot Theorists, he introduces the game Entanglement (CAUTION: Entanglement is rather addicting. Proceed with caution and plenty of time to kill). This article stood out to me because I recall Prof. Heath talking about Knot Theory during our last capstone meeting.

The object of the game is to create the longest knot possible. In the blog post itself, however, the upper bound of the length of the knot (a “perfect game” if you will) is discussed. We see that each tile is a hexagon, and thus has six sides. Furthermore, there are 36 tiles in a full board. There are also 48 boundary sides, one of which the knot may go onto. Thus the largest possible knot is 6*36-(48-1)=216-47=169 tiles long. The next logical question is whether or not this is attainable. It turns out that it is (the proof is that someone did it, no mathematics necessary this time).

Fibonacci In Nature

In 10-Minute-Math’s blog, Fibonacci Right Outside, the presence of the Fibonacci sequence in nature is discussed. For those who don’t know, the Fibonacci sequence is Fn=Fn-1+Fn-2 where Fo=0 and F1=1. It looks something like (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … ).

In the blog, 10-Minute-Math tells of how he grew sunflowers to see if their seeds actually did follow the Fibonacci sequence, or if it was just really close. To his surprise, they appeared to be a perfect Fibonacci sequence, spiraling clockwise 55 times, and counterclockwise 34 times. See the picture below.

Statistical Surveys

The Numbers Guy’s blog post, The Census’s 21st-Century Challenges, discusses the difficulty with conducting a nation wide survey. Part of the challenge lies in getting the survey to the people. Some people do not have a permanent address, or have moved recently without their address being updated. Another problem lies in actually getting those who receive the survey to participate.

The U.S. census claims to be mandatory, but lacks the enforcement to be so. That’s not to say that calling it mandatory is useless. It’s estimated that by replacing the word “voluntary” with “mandatory” increased survey response by about 20%.