Research Journal

Thursday April 27,2008

Today we set up a mini model of what the actual project is going to be in order to make sure the tetrahedron was going to be large enough to hold the balls, and yet tight enough to keep them still.

Instead of the gazing globes, which are 10" in diameters, we used christmas ornaments, which are 2-5/8" in diameter. Before deciding on my project about fractals the christmas balls were what we used to demonstrate fractals. This is something anyone can try at home! So go ahead!

What I did with the 4 christmas balls, and what I will do with the 4 gazing balls, is stack them with three on the bottom touching to form a triangle, and balancing the fourth ball on top of the 3 balls, centered, so that all four balls are touching each other. Unfortunately it's not that simple to really see the fractal patterns, we have to block excess light and reflections.

Next I took a blue poster board (for the full sized model we will be using sturdier and thicker material, 3/16 " thick) and cut out 4 equilateral triangles with 9.09" sides. After that we cut 2" holes out of each of the triangles, with the center of each circle 1.5" from the base of the triangle, except the one which would be the base. Finally we taped them together to form a tetrahedron.

Before closing the last side we placed the balls in the tetrahedron, as described above, and then closed it. From each hole you could see the fractal patterns. However, in order to see them more clearly and distinctly, 2 of the 3 holes must be covered and shine light in in distinct colors.

The images of our mini-model can be found in my pictures.

Thursday April 10, 2008.

The original model of the four mirror spheres conducted by David Sweet, Edward Ott, and James A. Yorke consisted of four mirrored spheres of 30 cm diameter. The balls we will be using are lawn ornaments called gazing balls. The ideal size for our purposes would be 10 in, however 8 inches might work as well.

In 1987 James Gleick published Chaos: Making a New Science, a known national bestseller to this day. The book is about chaos, which relates to fractals on many levels. I have not begun to read the book yet so today I can only share with you the synopsis.

Chaos records the birth of a new science. This new science offers a way of seeing order and pattern where formerly only the random, the erratic, the unpredictable - in short, the chaotic - had been observed. In the words of Douglas Hofstadter, "It turns out that an eerie type of chaos can lurk just behind a facade of order - and yet, deep inside the chaos lurks an even eerier type of order." Although highly mathematical in origin, chaos is a science of the everyday world, addressing questions every child has wondered about: how clouds form, how smoke rises, how water eddies in a stream. Chaos is a history of discovery. It chronicles, in the words of the scientists themselves, their conflicts and frustrations, their emotions and moments of revelation. After reading Chaos, you will never look at the world in quite the same way again.

Wednesday April 9, 2008.

The term "fractal" was coined by Benoit Mandelbrot in 1975. Fractals have many definitions, which are correct definitions sometimes depending on the individual fractal. One good definition I came across is that fractals are objects built by using recursion, where some aspect of the limiting object is inifinite and another is finite, and self-similar. Another good definition is that a fractal is essentially a graph of an iterative process applied to complex numbers. While the definitions may vary depending on the circumstances, there are certain properties that define a fractal. Some of these properties I will name below:

Properties of Fractals

- To form a fractal you start with an "initiator" and "iterate" using a "generator."
- Fractals are infinite (either in length, perimenter, or plane).
- Some aspect of the limiting object stays finite or 0 (area, volume, etc.).
- Each iteration is an identical copy of the previous iteration (self-similar).
- Fractals DO NOT simplify.

Information found at: MathsNet Fractals

Tuesday April 8,2008.

In the past two weeks I have been studying mirrors and reflections, because I was not very informed about the topics when I began at the lab. After learning about mirrors and reflections, and researching more on the topic of fractal reflections, I decided to adopt it as my project. My Abstract describes what fractals are and what the goal of my project will be. The orginal project done with 4 reflective spheres is: Topology in chaotic scattering. The fractals formed by the 4 spheres are self-similar, meaning the shapes and colors look the same regardless of magnification; and have the Wada Basin property, meaning regardless of how close you zoom in all 3 basins, colors, and shapes appear. I also discovered that fractals form in dimensions of a fraction, called the fractal dimension. How do these fractals form? What exactly are the patterns formed by the 4 spheres? These are just a few of the many fascinating things I am trying to learn about.

Some of the known fractals which helped me better understand the concept of fractals were: the Koch Curve, the Cantor Dust, the Sierpinski Triangle, the Mandelbrot Set, the length of a coastline, and cauliflower.

Tuesday, March 25,2008.

Over the last couple of weeks Dr. Noe and I have been exploring different topics and experiments related to reflections, because that is what I am interested in. I explored holograms, Billiards Problems, Kaleidoscopes, and Funhouse mirrors, or more specifically, the properties of concave and convex mirrors. My greater interests in kaleidoscopes and reflections have led me to fractal reflections. Last spring (2007) Eva Zacarias had very similar interests to mine and constructed a project consisting of several examples of multiple reflections. The final example in Eva's Report is about fractal reflections.

Tuesday, March 4, 2008.

Today Dr. Noe, Manushi, and I, attended a Lecture by Robert M. Panoff entitled "Many Body for Anybody." Professor Panoff began the Lecture with an introduction to the Dynamics of more than one thing, and then branched off to different formulas and how they apply in the world. I found the lecture by Robert M. Panoff very interesting, slightly to my surprise. He was funny and entertaining, but at the same time spoke very passionately about his work and ideas. I liked his approach to teaching and his analyzation of the way students learn science today. I can admit in the past I have often found myself just plugging in numbers, without understand why I am doing so or what my answers really mean. Which is why I think being able to see what formulas really do in the real world is a great approach to teaching. Computer simulations can answer many questions for students learning something new, like: How do you do that? How does it work? What can it be applied to? Once students can get a general understanding of how the formulas work in the real world, then they can better understand how formulas and math apply to make this happen in the world. I always feel disappointed when I see students learning about science, and especially math, and saying "How does this apply to the real world?" or "When am I ever going to use this in life?" I think if they can see in class, if not hands on then through simulations, how things really do work and are applied in life, then they will better appreciate and understand what they are learning. Interesting simulations will get students to want to get involved. Also, seeing is much different than hearing or reading. Robert M. Panoff's goal is to focus on how to make dynamics of the world a representation in the class, which can make a great impact in getting students involved and interested.

Thursday, February 28,2008.

Dr. Noe introduced us to HTML, resulting in my beautiful webpage. :-)

Thursday, January 31, 2008. - Tuesday, February 26, 2008.

In the first few weeks in the Laser Lab Dr. Noe introduced us to some basic ideas and priciples of physics and optics. He also showed us many past projects and we became familiar with the outlines of past student's webpages.

Elizabeth Hampton
February 2008
Laser Teaching Center