Weekly 7: Fractals

In class we did an overview of fractals, learning about Julia sets, the Mandelbrot set, and Sierpinski's Carpet. What these have in common is their repetitive nature: at every magnification, these sets have a repeating pattern that is a scaled copy of the original, fundamental pattern. That is, the patterns are self-similar and have an infinite degree of complexity. We saw this in Sierpinski's Carpet, and Sierpinski's Triangle is given below as another example of their construction.

Of course, at this step, there is not an infinite degree of complexity in the triangle. However, this pattern could be repeated as many times as you'd like, allowing for a level of detail that gets arbitrarily large. For a good visualization of the Mandelbrot set's complexity through magnification, watch this video. As is seen in the video, the original shape of the Mandelbrot set is seen repeatedly throughout the magnification (along with some other really interesting designs), showing us right away that there is no "maximum" magnification that can be achieved. Due to this property of being able to refine the detail of fractals as much as needed, they have become incredibly useful in a variety of situations. One of their uses is in graphic design, where fractals can be used to create a number of visual effects like the mountains seen below.

Using the same repeating patterns, programmers can recreate the appearance of natural phenomena with great detail. The method to create the mountains above can result in scenes that look incredibly realistic, which is quite a feat given the relatively simple nature of the mathematics involved in fractals.

As an example of the simplicity of the mathematics involved, let's consider the Mandelbrot set. It is created in the complex plane with the iterative equation

.

A complex number z is chosen as the first value and entered into the equation. The result is then re-entered into the equation as the input value, and this process is repeated as long as wanted or needed. The visualization comes from assigning colors to values whose iterations either converge to a certain value or diverge. The familiar shape of the plotted set is achieved by testing each value c for convergence or divergence when the initial choice of z is 0 and respectively coloring the points.

In addition to their use in creating images, fractals have become beneficial to other areas of study. A telecommunications company, Fractenna, has designed a fractal-shaped antenna that is smaller and weighs less than the common antenna. Antennas are typically tuned for a specific frequency or range of frequencies, and the repeating patterns of fractals allow companies to reduce the size of the antenna required to serve these frequencies.

Fractals also appear in nature, from molecular structure to the shapes of galaxies. For example, we may look at a tree and notice its self-similarity: the primary limbs branch off from the trunk while smaller, secondary limbs branch off the primary limbs, and so on. In other words, each branch is a "trunk" for yet smaller branches that eventually terminate in leaves. A similar description applies to vascular and neural networks.

We worked on creating fractals of our own in class by hand and on the computer, and below is what I created using the site listed on our course page.

There were a lot of cool things you could do with the program, making different shapes that repeated themselves in different ways, and combining different patterns together to make increasingly complicated images. One thing that I noticed while playing with the program was how truly different an image can become given a small change to the initial pattern, which attests to the remarks made in class about chaos theory and the importance of initial conditions, and the vast changes apparently minor differences can make. Due to this effect, the image above took several tries to produce. Even once I was satisfied with the initial "stem" it was difficult to orient each sub-stem so that it gave the image I was trying to create because any error is magnified by the infinitely-repeated pattern I used as my base design. With this in mind, it becomes clear why it's difficult to accurately predict phenomena like weather and stock market prices with a high degree of accuracy, even if sophisticated models and technology are used. 

Last Daily: Course Reflection

My favorite aspect of the course was going through the history of mathematics, and learning about the people who were behind many of the subject's major milestones. Doing this seemed to give math a more "personal" feel, whereas in other courses we simply learned the subject matter in a sort of sterile and objective fashion. Although the material in other courses was certainly interesting, not knowing anything about its history or the people who discovered it gave math an even more impersonal feel. For this reason, I feel like a course like this would be great as an introductory class that fulfills the general education requirements of the university since discussing the history and people of math gives it a more accessible tone, especially for those who view their algebra requirement as a torturous stepping stone toward another major. In any case, I enjoyed the multi-disciplinary focus of our course and thought it enhanced my appreciation for math by bringing it into perspective with other areas of study.

It might be difficult to reconcile this with what I said above, but I think I would have enjoyed the course a bit more if we had gone into some deeper mathematics. I certainly learned a lot of new things and thought many of the results were interesting, but the rigor and depth of the math we did in class wasn't very demanding. I think this is partly due to the fact that we covered a lot of the history of math, and many interesting results (like those from the Ancient Greeks) weren't as complex as what I came to expect from courses like discrete math or real analysis. Making the class more rigorous would also be difficult since we haven't all taken the same courses throughout completing our math majors. For instance, I never took a geometry class while it seems almost everyone else had, so if we went deeper into geometry I'd likely be "out of the loop".

What helped to remedy the relative lack of depth with regard to mathematics in this course was the ability to choose to some degree what we completed for our daily and weekly assignments. Although it certainly didn't seem to be required, I enjoyed the opportunity to go deeper into the math behind some of the topics we learned about in class for these assignments. With the ability to choose the exact nature of our assignments we could dig deeper when we wanted to learn more and had the background to do so, or we could explore related things like the topic's history. Exercising our mathematical abilities at some point in the semester was also guaranteed by the semester requirements for the assignments, which I think should be expected in a senior-level math class. Overall, I thought the course was well-planned and covered a lot of interesting material in a logical order, while the assignments gave us great flexibility to individually learn more about the topic in a way that kept our interest.

Journey Through Genius

The book I read and reviewed earlier this semester was The Mathematical Universe, by William Dunham. His book Journey Through Genius seemed like it would be similar in style and content, so I thought I'd check it out for my second book. One thing I immediately liked more about Journey Through Genius was the fact it went in chronological order, unlike The Mathematical Universe. This isn't that big of a deal, but it's just easier to follow and put things in perspective if topics are presented in chronological, rather than alphabetical, order. Further, Journey Through Genius appeared to be a bit more dense than The Mathematical Universe, but certainly nothing someone with a modest background in math and a desire to learn wouldn't be able to follow. Many of the topics were different between the two books, but seemed to have the same general presentation and introduced many of the same mathematicians. Overall, the two books have the same "flavor" but their contents seem different enough to make both of them worth the read.

Daily 13: Mathematical Paradoxes

Aristotle's Wheel

Aristotle's Wheel is a paradox found in the Mechanica and consists of two wheels, one within another, that roll along a surface. It is constructed so, like in the image below, two horizontal lines trace the bottom of the circle and appear to be the circles' circumferences.

The paradox comes from the assertion that both horizontal lines are the same length despite the fact that each circle has a different circumference. To solve this paradox, we have to first accept the fact that the inner circle has a smaller circumference than the outer circle. So, the top line must be shorter than the bottom line if they do in fact measure their respective circle's circumference. Referencing the image above, notice that the red lines give the impression that the circumference of each circle is rolled out onto the horizontal lines. It is this illusion that is fallacious: while the outer circle is actually unrolling its circumference onto the lower line, the inner circle is both unrolling and sliding. This fact is masked by the visual effect of the red line being transferred from the inner circle to the upper line.

Further, the line within the circles representing their radii suggests even more that the illusion is correct because it helps to reinforce the appearance that each circle's circumference is being unrolled at the same rate. This too is fallacious since it suggests that, simply because an analogous point is traced on similar circles, the circumferences must be equal.

In reality, at least one of the circles would have to be dragged to some degree if the other is assumed to be rolling along a surface. For example, imagine a car wheel. While the car is in motion, it is clear that both the tire and hubcap have the same angular velocity, otherwise either the tire or hubcap would be spinning faster than the other and, unless you have spinners, this isn't going to happen. Rather, the hubcap, if its rotation and linear movement were recorded, would be seen to be dragged along the path of the tire. That is, the linear movement the hubcap traced would be longer than the rotational distance it would have traced had the hubcap been independent of the tire. If the red lines were removed from the above animation, this would be more apparent.

Achilles and the Tortoise

Zeno proposed a paradox that states Achilles cannot win a race against a tortoise. Suppose Achilles is racing a tortoise and the tortoise, being slower than Achilles, is given a head start. When the race starts, Achilles will eventually reach the point where the tortoise starts. By this time, the tortoise will have moved some distance ahead. Achilles will then have to run further to reach the tortoise but by then, once again, the tortoise has made it some distance more. The paradox states that the fast-running Achilles will never reach the slow tortoise since every time Achilles progresses and reaches a point where the tortoise used to be, the tortoise has made it some distance more. 

Clearly, a fast runner is able to win against a slow runner. So a solution to the problem must exist. Unfortunately Zeno didn't expect us to have the aid of an idea from calculus, convergence. Even though the tortoise is continually moving, it's at a slower pace and Achilles will eventually catch up. This is because, although we can mathematically express the distance Achilles has travelled as a faction of the distance the tortoise has travelled, this fraction will converge to 1.

To see this, suppose after the first measurement, the tortoise has moved far enough so that Achilles has travelled 0.9 the total distance of the tortoise (such a point must exist since Achilles is moving faster). After the second measurement, Achilles has covered 0.99 the total distance of the tortoise. Then 0.999. Then 0.9999, and so on. This will eventually (or, more realistically, very quickly) converge to 1, implying that Achilles has caught up to the tortoise. Simply stated, the infinite number of small distances Achilles has to travel is offset by the infinitely decreasing amounts of time it takes Achilles to reach the previous point of the tortoise. At this point Achilles can take the lead. Many of Zeno's paradoxes are solved in a similar manner.

Daily 12 and Weekly 6: Gauss and Minorities in Mathematics

Gauss proved a conjecture first proposed by Fermat, which stated that any natural number can be expressed as the sum of three triangular numbers. In order to fully understand this, we must first make sure we know what triangular numbers are. From wikipedia, a "triangular number or triangle number counts the objects that can form an equilateral triangle...The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n."

Notice that we have seen this sequence before when we discussed the sum of the natural numbers from 1 to n. The formula that gives this sum is

x(n) = .5(n)(n + 1),

and I wrote the proof of this for my third weekly assignment. This implies that x(n) is also the nth triangular number. This gives the first few triangular numbers,

0  1  3  6  10  15  21  28  36 ...

We talked about triangular numbers briefly in class, but I didn't realize the connection between them and x(n) until I read the wikipedia article on triangular numbers (edit: I now see it was also mentioned on the course page, so if I read that more thoroughly I'd have seen it there as well). Another interesting result I learned is that the sum of two consecutive triangular numbers is a square number. We see this in a few examples using the first triangular numbers:

0 + 1 = 1

1 + 3 = 4

3 + 6 = 9

6 + 10 = 16

10 + 15 = 25

15 + 21 = 36

21 + 28 = 49

28 + 36 = 64

In fact, the sum of the (n - 1)th and nth term is the square of n, which we see since

x(n - 1) + x(n) = .5(n - 1)(n) + .5(n)(n + 1)

                                                           = .5[(n - 1)(n) + n(n + 1)]

                                                           = .5(n² - n + n² + n)

                                                           = .5(2n²)

                                                           = n²,

which is also the square of the difference between the two triangular numbers.

In 1796, a 19-year-old Gauss proved that any given natural number can be written as the sum of three of these triangular numbers. Some sources state the theorem as at most three triangular numbers and exclude 0 from the list, but it amounts to the same thing. I don't particularly feel like doing yet another proof for my weekly assignment, so we'll just go down the list of natural numbers and see for a few examples that this works (and, after looking at an outline of the proof, I hope to never encounter it again).

0 = 0 + 0 + 0

1 = 0 + 0 + 1

2 = 0 + 1 + 1

3 = 1 + 1 + 1 = 0 + 0 + 3

4 = 0 + 1 + 3

5 = 1 + 1 + 3

6 = 0 + 0 + 6

7 = 0 + 1 + 6

8 = 1 + 1 + 6

9 = 0 + 3 + 6

10 = 1 + 3 + 6

.

.

.

35 = 1 + 6 + 28

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.

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This is a pretty interesting result, implying the entire set of natural numbers is spanned by the triangular numbers through addition. This is all the more remarkable since it's easy to see that the triangular numbers become relatively less frequent as we go into higher and higher numbers. Now, for the other half of the assignment, we'll look at a few notable minorities in mathematics. I noticed that other people have already written pretty extensively about women in mathematics, so I'll focus more on minority groups in general by giving a few brief biographies of mathematicians with minority backgrounds.

One of the many notable women in mathematics was Ada Lovelace, a 19th century mathematician most known for her contributions to Charles Babbage's work on his "Analytical Engine", a proposed (but never completed) computational device. Lovelace was the only legitimate child of Lord Byron who, despite being kind of terrible at being a family man, I'm fond of because of his role in the Romantic period.

Lovelace is credited with writing one of the first computer programs, which consisted of an algorithm for the Analytical Engine to compute Bernoulli numbers. Unfortunately, since the Babbage never completed his computer due to funding issues, Lovelace's algorithm was never tested. Some controversy surrounds Lovelace and her contributions, since many claim that Babbage prepared the algorithms several years earlier and had them published under Lovelace's name for some unknown reason. It's still not known how much of the work should be credited to Lovelace, but her role, no matter its true degree, deserves mention simply because of the extreme lack of recognition women received in mathematics. Although progress has certainly been made with regard to women in mathematics, there is still room for improvement and more inclusion. As noted in class, much of the skepticism seems to come from the public in general, and not from mathematicians themselves.

Another underrepresented minority in mathematics is African Americans (and blacks in general, regardless of nationality). The late 18th- and early 19th-century free African American, Benjamin Banneker, is one such mathematician. Although he received no formal education, Banneker is known as a scientist, surveyor, and almanac author.

When he was 22, Banneker designed a clock that rung on the hour that he modeled by scaling the pieces to a pocket watch he had. While working as a farmer for the Ellicott family, he was lent books on astronomy which he studied intently. In 1791, he was hired to help survey the boundaries of the newly-created District of Columbia using his knowledge of astronomy to aid in the location of coordinates. Using the same knowledge, Banneker accurately calculated phenomena such as sunrises and sunsets, lunar and solar eclipses, and other "remarkable days". He wrote almanacs that contained this and other information, and wrote journals that included many mathematical calculations and puzzles. Sadly, all but one of his journals were lost to a fire. Still, he is remembered for his contributions, and an obelisk was erected at his grave in 1977 and many facilities are named in his honor.

One last story worth noting is the unfortunate case of Alan Turing. Turing is most recognized for contributing to computer science, but he also published papers in mathematical biology, logic, and philosophy. He was hired by the British during World War II for his skills as a cryptanalyst, and afterward worked as a computer scientist for the National Physical Laboratory and continued his work in cryptography with the Government Communications Headquarters (GCHQ). 

However, Turing entered into a homosexual relationship in 1952, which was illegal in the United Kingdom at the time. He was found guilty of indecency by the court, and was given the option of prison or probation with the condition that he receive hormonal treatment for his homosexuality. Turing opted for the latter, and as a result of his conviction lost his position with the GCHQ and was denied entry to the United States. Two years later, Turing was found dead at the age of 41 as a result of cyanide poising. Some mystery still surrounds his death because, although it was officially ruled a suicide, Turing displayed no signs of depression and appeared good-humored about his unfortunate circumstances. Turing had at least some interest in chemistry, and some claim that his death was the result of accidental cyanide inhalation while dissolving gold in potassium cyanide.

For being a field that requires the most careful use of logic and rationality, mathematics is certainly rife with prejudice. Although in most cases, it appears to the be entire culture, rather than the field itself, that is to be held responsible for these instances of injustice and exclusion. In my experience, those involved in mathematics are more open to differences and diversity than the general population. This holds true, I think, for higher education in general. If there is a lack of minority groups in the sciences and other fields, it is more likely due to the inability for these people to have the opportunities necessary to enter into them. For instance, our society views males as being better than females in the sciences and so each gender grows up feeling that they're either supposed to enter into the sciences or are somehow naturally inept at them and should go into something else. So it is our culture in general, not those in the sciences, that must grow more accepting. Whether our culture is simply the result of our biological nature (which would make changing much more difficult) or something we can grow out of, I don't know.

Likewise, many racial minorities grow up in lower-income areas and lack the same quality of education as more wealthy areas. Even within public education, there are vast differences in the quality of education and facilities among inner city schools and schools in wealthy suburbs. Excellent educators are less likely to teach in schools serving predominately low-income families since their skills are recognized and rewarded more by schools in better areas. Of course, I'll never be an educator and don't really have any suggestions how to make things better, so again I'm without answers. However, I think there has certainly been improvement in this area within the last century and as globalization continues and our societies evolve in response, there will be less distinction among races, at least in the developed world, and progress will continue.

Lastly, it seems that equality for homosexuals is well on its way in the West, and this demographic seems to suffer the least relative to the other two mentioned above (so long as the person in question is a white male, of course). In addition to growing acceptance, the relative ease of living a discrimination-free professional life being a homosexual likely comes from the fact that one can pretty easily "hide" such facts about themselves, unlike gender or racial ethnicity.

Again, those in the sciences seem to be generally more accepting of diversity than the average population, so our inclusion of minorities is limited by the overall social climate of our time. Only when (or if) our species is better able to accept and recognize the inherent need for differences among ourselves will minorities be treated as equals in their professions.

Daily 10 and Weekly 5: The Mathematical Universe

I contained my daily and weekly assignment in one document, whose length got a bit out of hand. Sorry about that. Here it is.

Daily 9: Fermat's Last Theorem

I watched the video suggested on the course page for my daily work, which provided an overview of proving Fermat's Last Theorem. It's interesting that it took so long to prove a theorem that seems so simple compared to other theorems we have encountered. Further, Fermat claims to have proven it, which seems incredible given that the 20th century proof was very long and technical, using results that were unavailable to Fermat at the time he made the conjecture.

The book I'm reading for the course, The Mathematical Universe by William Dunham, briefly covered Fermat's Last Theorem in a chapter about the 17th century French mathematician. Dunham notes that many people proved Fermat's conjecture was true for specific cases. For instance, Fermat proved that it holds for n = 4, and Euler proved that it holds for n = 3. Dirichlet and Legendre are credited with an 1825 proof that Fermat's conjecture holds for n = 5, Dirichlet a proof for n = 14, and Lame a proof for n = 7. Another proof by Kummer in 1847 proved the conjecture for a large class of exponents. Clearly, people maintained an interest in Fermat's Last Theorem long after it was first proposed.

In the beginning of the 20th century, an offer was made of 100,000 German marks to anyone who provided a correct proof of the conjecture. Of course, many attempts were submitted that were incorrect, and the award seemed a bit less enticing after World War I, when 100,000 marks were equivalent to less than a penny in US dollars. This didn't deter a mathematician by the name of Gerd Faltings, who proved that if the equation held for powers greater than two, then there were at most finitely many solutions. For his find, Faltings won the Fields mMdal in 1986.

Now, enter the work done by Andrew Wiles, whose proof of Fermat's Last Theorem was finally completed in 1995, nearly 360 years after Fermat first made the proposition. The full story given in the documentary is much more exciting than anything I have the willpower to write here, so you should just watch the video linked above for the full story.

What I enjoyed most about this video (and others like it) is that it gave us a look into the people behind the theorem, and all the work and emotion that went into completing the proof. When we read textbooks about math or science, we don't often think of all the work and energy that was put into the results we casually take for granted and use in our own work. Many people dedicate years of their lives to working out one small part of a problem, in the hopes that it can be synthesized with others' results by yet someone else to finally solve a problem. Finally, I thought that it was interesting that Faltings won the Fields Medal for his "partial" proof, yet it has been nearly a decade since Wiles published his proof and has yet to be granted such a prestigious award.