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This is a classic result in differential geometry and is worth mentioning in these posts on minimal surfaces. Before we can talk about the deformation we need a definition.

Definition. A minimal surface described by the Weierstrass-Enneper data (f,g) or F(\tau) has an associated family of minimal surfaces given by, respectively, (e^{it}f,g) or e^{it}F(\tau).

The catenoid has Weierstrass-Enneper representation (f,g)=(-\frac{e^{-z}}{2},-e^z). Thus, the associated family of surfaces of the catenoid has Weierstrass-Enneper representation (f,g)=(-\frac{e^{-z}}{2}e^{it},-e^z), which corresponds to the following standard parametrization.

\textbf{x}(u,v)=(x^1(u,v),x^2(u,v),x^3(u,v)), for any fixed t, where

x^1(u, v) = \cos(t)\cos(v)\cosh(u) + \sin(t)\sin(v)\sinh(u)

x^2(u, v) = \cos(t)\cosh(u)\sin(v) - \cos(v)\sin(t)\sinh(u)

x^3(u, v) = u\cos(t) + v\sin(t)

A very beautiful result in minimal surface theory. The catenoid can be continuously deformed into the helicoid by the transformation given above, where t=0 represents the catenoid and t=\frac{\pi}{2} represents the helicoid. It should be pointed out that the parametrization above represents a minimal surface for any value of t. That is, any surface in the associated family of a minimal surface is also minimal.

The surfaces below, plotted for different values of t, represent the associated family of minimal surfaces of the catenoid.


This post is well overdue. Last month Benoit Mandelbrot died. He is best known as the father of fractal geometry and the creator of the Mandelbrot set. In honor of Mandelbrot, enjoy the music video below. The video was made by some Cornell students. The music is by Jonathan Coulton.

And it’s not Disneyland. Watch the video here. A couple of comments. One, I don’t know where Caliofornia is. Two, the flyover of “San Luis Obispo” isn’t San Luis Obispo. It looks like Morro Bay (which is close enough I suppose).

An enlightening article you can find here. I first started meditating in college and it has provided balance to my sometimes hectic life.

The opening and closing scenes to Annie Hall, good movie. Anyone who has been in a relationship I think can relate.

The Towers of Hanoi is a mathematical puzzle consisting of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

  • Only one disk may be moved at a time.
  • Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
  • No disk may be placed on top of a smaller disk.

It turns out that if there are n number of disks, it will take a minimum of 2^n-1 number of moves to solve the puzzle. I won’t explain where the 2^n-1 comes from, that’s the whole point of solving the puzzle and for you to figure out!

What I like most about this puzzle is its origin. The puzzle was invented by the French mathematician Édouard Lucas in 1883. There is a legend about a temple in Hanoi, Vietnam which contains a large room with three time-worn posts in it surrounded by 64 golden disks. The monks at the temple acting out the command of an ancient prophecy, have been moving these disks, in accordance with the rules of the puzzle, since that time. According to the legend, when the last move of the puzzle is completed, the world will end. It is not clear whether Lucas invented this legend or was inspired by it.

If the legend were true, and if the monks were able to move the disks at a rate of one per second, using the smallest number of moves, it would take them 2^{64}-1 seconds or roughly 585 billion years; it would take 18,446,744,073,709,551,615 turns to finish.

I shared this story with my precalc students yesterday when I introduced exponential functions. I like how it demonstrates how fast exponential functions grow.

Click here for more pictures. Click here for the mathematics about the picture.

The Alfred Hitchcock classic is playing tonight at 7:30pm at the Fremont, part of their classic rewind series.

Update. November 18. This is a revision of the original post. There is still no conclusion and I don’t know where to put the last paragraph.

This is a very rough draft of my personal statement. I am applying to PhD programs in math education. The one below is for a math ed program at Berkeley. There is no conclusion, I need to add that. Also, it is a bit too long right now. I need to trim some stuff out.

Any and all comments and criticisms are welcomed and encouraged. Thanks in advance to anyone who has any advice, I really appreciate it!

There are two main reasons why I want to pursue a PhD in math education. First, I want to be involved in decision-making concerning equality, reform, and curriculum development for high school math education. Second, I want to better understand issues in undergraduate mathematics education. My experiences as a student and teacher have allowed me to realize these goals and have prepared me for studying math education at the PhD level.

My first experience in math education came as an undergraduate; I worked at UC Irvine’s Center for Educational Partnerships (CFEP) where I assisted in math instruction at Spurgeon Middle School in Santa Ana, California. The goal of CFEP is to increase postsecondary opportunities for California’s educationally disadvantaged students. I participated in CFEP’s Saturday Academy in Mathematics (SAM), an academic enrichment program. The SAM curriculum, written by CFEP and the Irvine Mathematics Project, is interactive instead of lecture based and encourages students to think and work together. Rather than simply teaching math, I want to be a part of designing and implementing new math curricula since I have seen the positive impact it can have on students.

My work as an undergraduate inspired me to pursue a master’s degree in math. As a part of the masters program at Cal Poly, I have the opportunity to teach my own math class. This past year, I have been teaching pre-calculus at Cal Poly. Teaching my own class has allowed me to reach a broader audience of students at a more advanced level. On the other hand, it has also revealed to me certain weaknesses in math education. Many of the students I teach lack the motivation and interest to learn the material. This apathy in many undergraduate math courses concerns me and I want to understand both the cause, and how to better combat it. Moreover, I am interested in the transition from learning high school math to learning college math.

The masters program at Cal Poly has been a good stepping-stone in preparing me for the challenges of a PhD. I know what it takes to succeed in a graduate program. Over the past year, I have learned how to balance teaching, studying, research, and classes. In my first year, I successfully studied for, and passed, a qualifying exam in real analysis. In addition, I have been conducting research with Dr. Vincent Bonini studying the classification of minimal surfaces. As a result of my research, I have authored a brief survey on minimal surfaces, appearing in this year’s edition of the Polymath Newsletter, and I had the privilege of lecturing at the Cal Poly Differential Geometry Seminar. On top of my teaching, exams, and research I have been able to thrive in my classes by maintaining a 3.74 GPA. The masters program has allowed me to hone my studying and researching skills necessary to excel in the PhD program.

Being a senior graduate student this year, I had the opportunity to mentor new TA’s in the Cal Poly Math Department TA mentoring program. I helped new TA’s in developing their syllabi and shared my teaching experiences.