Pressing Toward the Prize

Archive for November 2010

I was looking through the November 2010 issue of Math Horizons, and I discovered an article by Stephen Abbott in which he interviews Corey Greenspan, the winner of this year’s memorizing \pi contest at Southern New Hampshire University. Apparently Corey memorized the first 419 digits of \pi, and he shares with Stephen Abbott how he managed such a feat. Personally, I can’t imagine doing what Corey did, but his accomplishment pales in comparison to the world records for memorizing \pi. According to the article, the official world record held by Lu Chao is 67,890 digits of \pi, with the unofficial world record being 100,000 digits as recited by Akira Haraguchi of Japan.

Not to be outdone, check out these fun videos of a young lady balancing 15 books on her head while solving a Rubik’s cube and reciting the first 100 digits of \pi. The first video is recorded in her dorm room,

and the second is of her sharing her “talent” on the Ellen DeGeneres show.

Truly, I couldn’t do even one of the things she does, let alone all three at once! Like I said, people are aMAZing!!

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Most of us have probably heard the saying, “You can’t teach an old dog new tricks.” And I would venture to say that many of us know someone who would qualify as the “old dog” that is mentioned. It is easy to settle into a routine over time, and the older one gets, the more concrete that routine can become. There are different reasons why folks get set in their ways. For some, they truly believe their way of doing or believing is the only right way, and they aren’t willing to entertain any conflicting viewpoints. For others, they have just become so comfortable in their ways, that any deviation is unthinkable. But then there are those who continue to do what they have always done for a much deeper reason. My dad is an example. My mom passed away at the beginning of this past summer after 59 1/2 years of marriage to my dad. In fact, this coming Monday would have been my parents’ 60th wedding anniversary. Some of the ways my dad does things don’t always make sense to me, but he will not deviate from them because it was the way he and my mom always did things. I think it is his way of keeping her alive, and I would not take that away from him. I miss my mom, and I love my dad, so we just keep moving forward.

By now you are probably saying, “Nice sentiment, but what does this have to do with math?”

I had a conversation yesterday with my math professor about some proofs I did on our recent abstract algebra exam. I don’t remember his exact words, but the essence (as I remember it) was that my proofs are detailed, but reasonably concise. This was a HUGE compliment for me, because I have struggled greatly with trying to figure out this whole “mathematical proofs” business. I told him I was doing well because I had a great teacher – he was my proofs professor – and in his typically humble fashion he said, “Yeah, well… maybe.” But it’s true!! I took proofs with him last spring, and I remember trying to learn how much to include in a formal proof – not too wordy, but not wanting to leave out critical information, either. It was an interesting balancing act for a while, and my professor was quite patient with me, so I guess our perseverance is starting to pay off.

To explain how much of an improvement this is, I must tell you about my first “proofs experience.” Nearly two years ago, I took a linear algebra course at another institution. It was a heavily proofs-based course, but with one hitch: none of us had ever had any kind of formal proofs training! My instructor knew this and was trying to teach us as we went along, but he was forever writing on my assignments, “You could probably condense this down” or “A little wordy; see what you can do.” You wouldn’t believe the stuff that poor man had to wade through; it was astronomical! The problem was that I didn’t understand a formal proof should not contain all one’s thought processes, only what is essential to prove one’s point. And because I am a very detailed person by nature, trying to figure out what to trim has been a challenge. So with a sense of accomplishment I can say that I was thrilled to hear that I am finally making some progress in this arena, and there may be hope for me yet. So I guess sometimes you can teach an old dog new tricks!!

p.s. I love you, Dad!

The time has come for me to submit the proposal for my Capstone Senior Project. I have decided that I would like to explore the complexities of voting, including the paradoxes and problems that can arise when there are three or more candidates (or issues) on the ballot. There is a rich history of voting analysis that dates back to 1770 with French mathematician JC Borda, who, concerned with the outcomes produced by plurality voting, developed a weighted voting system called the Borda Count. A decade or so later, the Marquis de Condorcet attempted to discredit the Borda Count by demonstrating flaws in the procedure, and presented his method based on pair-wise counting, which had some problems of its own.

Then in the 1950’s, Kenneth Arrow, perhaps unaware of this 18th century conflict, analyzed similar problems with voting. He began by defining basic conditions that should be met in a voting procedure, and then attempted to find a voting method that satisfied these conditions. His conclusion was that with three or more candidates, the only procedure that satisfies all the conditions is a dictatorship! So if we are left to choose “between a dictatorship or a paradox” (per Donald G. Saari), what are we to do? Saari uses mathematics to show that there is a more reasonable option, and in fact shows mathematically that the Borda Count is the most reasonable option. I would like to study this centuries-long debate, the issues and “solutions” as they were presented, as well as Saari’s analysis that leads to a reasonable resolution. As has been suggested, since Saari uses linear algebra in his analysis, it would be interesting to run a few elections with fellow students and manipulate the outcomes using linear spaces. I would also like to investigate the reasons why plurality voting is still widely used, even though its flaws are fairly obvious.

In Capstone Seminar today we had a guest speaker from Gonzaga University, Dr. Logan Axon, who gave a very interesting lecture on random fractals. According to Wikipedia, “A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.” Although mathematicians studied such continuous, non-differentiable functions as fractals beginning in the 19th century, the name fractal was given by Benoit Mandelbrot in 1975, derived from the Latin word fractus (broken or fractured), to describe sets that were too badly behaved for traditional geometry. A fractal is based on a recursive process, which may be determined or random.

The first example Dr. Axon presented was of the Cantor middle thirds set. You start with the unit interval [0,1]. You remove the middle third of the line segment, then the middle third of the two resulting line segments, then the middle third of the four left, and so on, continuing an infinite number of times. So what is the total length of all intervals removed in construction of the Cantor set C? Dr. Axon showed us that the sum of the lengths is 1, or the length of the interval we started with. The assumption would be that C is the empty set, since we removed the equivalent of the original length. But we could see that 0 is in every set, so is 1/3, and 1/9, and so on. In reality, there are as many points in the Cantor set as in the original set (interval), even though we removed all of them! I would have to say that this is pretty counter-intuitive!!

Next Dr. Axon listed the properties of C that all fractals should have:

  • Self-similarity, scaled
  • Detail to all scales
  • Simple to define, but hard to describe geometrically
  • Its “local geometry” is strange; every open interval containing one point of C must contain the other points of C as well as points not in C.

Dr. Axon described the von Koch curve, in which you begin with the unit line, replace the middle third with the top of an equilateral triangle, then replace the middle third of each of the resulting four line segments with the top of an equilateral triangle, and so on. The Dragon curve is a space-filling curve, and the Mandelbrot set is formed by iterations of a function on \mathbb{C}, the complex number system.

We were introduced to the dimension of a fractal, which is the main tool in fractal geometry measuring the “roughness,” or “texture” of the fractal, and how to find it using limits. Fractals also describe objects in nature, such as the Romanesco broccoli with its repeating spiral design. In addition, we saw that the fractal dimension of the Coast of Britain is 1.25, although I still don’t understand exactly how to interpret that! Random fractals, like the Random von Koch curve, can be used to create images that look less concocted, more natural, for things like computer graphics.

Before this lecture, I had no idea what a fractal was. I felt like I was understanding most of what was being presented, however, at least until a few of the professors started asking questions that I didn’t understand. That’s when I realized I may not be grasping it as well as I had thought! Needless to say, Dr. Axon’s responses to them were beyond my scope of understanding, as well. All in all, I found the entire topic to be quite fascinating, though, especially when we discovered that fractals often behave in very unexpected ways. Who knew you could remove all of a set, and still have all of it left?!?!

I recently ran across an article that Donald G. Saari (author of Basic Geometry of Voting) wrote in 1996 about that year’s elections, highlighting the ease with which an unwanted outcome can occur. As he explains, when an election result is not what one thinks it should be, that is, the preferred candidate or issue does not win, it is called a voting paradox. This is generally not due to the voters, but rather the voting procedure used. In plurality voting, the process we use in which each voter votes for one candidate and the candidate with the most votes wins, multiple candidates can “split the vote” causing an “inferior” (at least according to the will of the people) candidate to be elected.

Saari demonstrates his point with a very entertaining story about his encounter with a group of precocious 4th graders in 1991. He was attempting to present a counting problem caused by a hypothetical voting example, when the students recognized a flaw in his assessment of the winner in a three-way vote. According to plurality voting, candidate A was preferred to B who was preferred to C in a 6:5:4 vote, when considering only the first choice of each voter. But the students immediately protested, because he did not factor in the “next best” preference, which alters the outcome. In fact, when all rankings were considered, not just the top-ranked candidate, the winner A under plurality voting was actually the least preferred by the majority of the voters, and C was in fact the favorite.

When he asked the students what they thought was the “correct” voting procedure to use, one of the students suggested voters assign 3 points to their favorite candidate, 2 points to the next best, and 1 point to their least desirable candidate. What this student described is the Borda Count method, named for the French mathematician JC Borda, who developed this method in 1770. When the Borda Count method was applied to the voting example, it showed that C was preferred to B who was preferred to A by a 34:29:27 vote – consistent with the students’ earlier assessment. Then Saari presented a version of Marquis de Condorcet’s puzzling example from the 1780’s that shows it is possible to have no winner, because there is a way to count the votes so that every candidate has the same number of votes. The students saw right through this example, as well.

The amazing thing about these students is how quickly they recognized a flaw and were able to suggest reasonable solutions, based solely on their examination of the problem. As Saari points out, they have not yet been conditioned to blindly accept the way things are, but rather used critical thinking skills coupled with their value of fair play. He is concerned that too often our educational system stifles the creativity of our students rather than nurture their inventiveness and innate desire to explore the world around them. He suggests that educators consider changing their classroom approach in order to foster creativity and develop problem-solving skills in their students. As these 4th graders demonstrate, children can accomplish amazing things when given the right environment.

The time has come to declare the topic for my capstone project, and I still find myself somewhat undecided. The two areas that interest me most are voting paradoxes and working with complex numbers. I like the practical application of studying voting processes, the problems that can arise when trying to determine the will of the people, and the manipulations (inadvertent or otherwise) that can greatly affect the results. Fairness in voting is a concept that should be of interest to us all, since the outcomes dictate many aspects of our lives.

A few weeks ago, my math professor suggested considering for our capstone project topics that may have been introduced in past math classes, but not studied in depth. My interest in complex numbers is a result of this suggestion. My exposure to the complex number system has mainly consisted of performing algebraic manipulations, and I would like to know more. Analysis of the complex numbers is such a broad topic that, should I decide to go this route, I will need some direction in choosing a focus. An intriguing application that my professor mentioned is one in which complex numbers are used to solve integration problems that would be quite difficult in the real number system. Both of these are worthy topics, but our formal proposal is due next week, so I will need to hop off the fence soon!



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  • gramsonjanessa: I can't wait to listen to your capstone presentation in the spring! Your proposal was really interesting and I'm interested to see how the linear alge
  • dewittda: This is impressive! I thought I was good because I solved a rubik’s cube once in an hour. I served with a guy in the Air Force who could solve a r
  • ZeroSum Ruler: The Euclidean algorithm should me the mainstream way we teach students how to find the GCF. Why isn't it? A mystery.

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