Fractals and Kindles

I just can’t help myself. Whenever I get a new gadget, I have to customize it. When my wife and daughters gave me a Kindle four years ago, I was thrilled. It’s a Kindle 3, and it opened to me the amazing world of ebooks. My library now includes collections of G. K. Chesterton, Charles Dickens, Fyodor Dostoyevsky, and Shakespeare. I like the fact that the screen isn’t backlit, so there is no eye strain. It’s a wonderful device that has completely changed the way I purchase and read books.

However, I thought the screensavers that Amazon preloaded on the Kindle were really unattractive, so I tried to replace them with images more to my liking. Easier said than done! I assumed that all I had to do was locate the folder containing the screensaver files and dump my own in there. It turns out Amazon does not want you poking around in there, so that folder is hidden.

Fortunately, after a little research online, I was able to hack into my Kindle and change the screensavers. There are thousands of great images online to choose from (just Google “Kindle screensavers”), and I had a blast exploring them. Then it occurred to me that I could create my own fractal screensavers – all that is necessary is to make sure they are gray-scale images that are 600 by 800 in size.

I decided to use Chaoscope to create my screensavers. (I posted a tutorial on how to use Chaoscope here.) Make sure you render them in either Gas or Liquid mode. My first batch is posted below. They are already correctly sized – just click on a thumbnail to access the full-size image, and then save it to your computer. Enjoy!

The Golden Rectangle and Fibonacci

One of the best things about teaching math is sharing all of the connections that exist between seemingly unrelated topics. Take, for example, the Golden Rectangle. Often described as the most visually “pleasing” quadrilateral, it is a rectangle whose length, l, and whose width, w, are in the the proportion \displaystyle \frac{l}{w}=\frac{l+w}{l} .

Here’s an example:

Golden Rectangle4.69 / 2.9 = 1.62, and (4.69 + 2.9) / 4.69 = 1.62. The number they equal, 1.62, is called the Golden Ratio, and it has its own Greek letter, phi. It’s actually irrational, like pi and e, and it equals \displaystyle \frac{1+\sqrt{5}}{2} . You can derive phi algebraically by solving \displaystyle \frac{l}{w}=\frac{l+w}{l} for l in terms of w. From the quadratic formula, \displaystyle l=\left( \frac{1+\sqrt{5}}{2} \right)w, so \displaystyle \frac{l}{w}=\frac{1+\sqrt{5}}{2}.

A Golden Rectangle can be constructed with a straightedge and compass (or GeoGebra!):

1. Construct a square ABCD: Square

2. Construct the midpoint, E, of side AB.

3. Construct a circle with center at E, passing through C: Circle

4. Extend rays through AB and DC. Construct the intersection point, F, of the circle and ray AB.

5. Construct a line perpendicular to ray AB, through point F. Construct the intersection point, G, of this perpendicular line and ray DC: Constructed Rectangle

6. Quadrilateral AFGD is a Golden Rectangle.

Another way to construct a Golden Rectangle is to begin with two adjacent squares:

GR1

Add another adjacent square with dimension equal to the combined sides of the first two:

GR2

Repeat the process, adding larger and larger adjacent squares:

GR3

This not an exact Golden Rectangle, but the more squares you add, the closer it approximates a true one.

Here is where Fibonacci comes in. He was a 13th century mathematician who published one of the first math textbooks, Liber Abaci. One of the topics he covered involved a famous sequence that can represent the idealized population growth of rabbits: 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Each term is obtained by adding the previous two terms together. Can you see how the rectangle above is a visual representation of the Fibonacci Sequence? What is amazing is the fact that the quotient of any two consecutive terms in the sequence approaches phi the further up the sequence you go!

Connecting opposite corners of each square with a circular arc creates a Golden Spiral, which increases a distance of phi from the origin for each square it crosses:

Spiral

There are lots of exaggerated claims made about the use of the Golden Rectangle and phi in ancient architecture, but it definitely pops up in the world of nature. Flowers generally have petals in numbers of 3, 5, 8, 13, 21, etc. For an excellent explanation of the mathematical reasons behind this pattern, check out Ian Stewart’s book, The Mathematics of Life.

A rectangle, a medieval Italian mathematician, a famous sequence, and the number of petals on flowers – all brought together through the beauty of mathematics.

Make Your Own Fractals for Fun and Profit!

Well, maybe not for profit, but they are a lot of fun, and it isn’t that difficult to create some beautiful images that will impress your friends.

There are lots of programs that generate fractals, many of which are open source. Fractint is the granddaddy of them all; Apophysis is incredibly powerful but overwhelming in its complexity; Fractal Explorer is very versatile. However, I’m going to focus on just one: Chaoscope, because it’s one of the easiest to use, and beginners can get impressive results. After a few minutes you will be up and running, making beautiful fractal art.

I use it to create desktop wallpapers for my computer, and my students asked where they came from. When I told them that I made them, they begged to learn how. When my students ask me to teach them something, I’m not going to turn them down! Even if you know nothing about the mathematics behind fractals, this can be a really rewarding project for your students. (If you are interested in the math, here are the equations.)

This post is merely a quick introduction to Chaoscope. For a more detailed manual, go here.

Okay, let’s make a fractal!

1. Download the installer file here.

2. After you’ve installed Chaoscope, open it. You will see a very boring blank gray window. Click on “File”, then “New” (or hit Ctrl-N):

File New

 

3. The “New” window should pop up. There are lots of things you can do in this window. First, choose what type of fractal you want to work on (Chaotic Flow, Julia, IFS, Icon, Lorenz, etc.). For your first attempt, I suggest Chaotic Flow:

New

 

In this window, you can also set the dimensions of your fractal. I recommend keeping it relatively small, since the smaller they are, the faster they render. Once you get your fractal exactly the way you like, you can change the size to whatever final dimensions you desire.

4. You can also set the render style in this window. Different styles look, well, different. Gas and Liquid are grayscale, while the others are in color. Solid makes your fractal look like a solid object, and you can also change the background color. I recommend you choose Plasma for this first project.

Render

5. We’re still not seeing any fractal, so let’s have Chaoscope generate one for us. Click the “OK” button in the “New” window. Click on the “Attractor” menu, then choose “Search” (or hit F3):

Attractor

 

You should now see a preview window, a “View” window, and an “Attractor” window. You may have something interesting in your preview window, or you may not. If not, just hit F3 until you get a fractal that looks promising. You can click on the fractal and rotate it in any direction. Often, a fractal that initially seems boring can turn into a spectacular one just by changing the viewing perspective. Here are two different views of the same fractal:

Perspective2

 

 

6. Try tweaking the sliders in the “Attractor” window. You will see that a very small adjustment can have a huge effect. Hitting Ctrl-R will put a random color gradient in place. Right-clicking on the color gradients in the “View” window accesses different gradient maps.

7. If you prefer fractals with lots of symmetry, play with Icon attractors. Lorenz attractors have the famous “butterfly” pattern. Julia attractors have a knobby, organic look.

8. When your fractal is perfect, save the parameters. Then set the size to the desired width and height in the “View” window. Hit the F4 button to have Chaoscope create a high-quality rendering. When it’s done, go to “File” and choose “Save Image As…” to save a bmp file of your masterpiece.

I run Windows 8.1 on my tablet, which allows me to set up a slide show to display a collection of desktop wallpapers. Simply right-click in an open area of your desktop, choose “Personalize”, and create your own theme.

Congratulations! You are now a fractal artist!

Math, Light, & Color – The 2015 Edition

Every year, for three weeks between semesters, Harpeth Hall offers an alternative curriculum for its freshmen and sophomores (Juniors and Seniors do off-campus internships and travel). I have taught a course on designing and making stained-glass windows that incorporates mathematical topics. My students always rise to the challenge, and this year was no exception.

The girls’ projects included a series of small windows representing the Platonic Solids, the Four-Color Theorem, Ptolemy’s Theorem, the Butterfly Theorem, Napoleon’s Theorem, Morley’s Trisector Theorem, and an Ulam Spiral, among many others.

I’d also like to recommend to my readers an excellent publication and blog devoted to fostering girls’ interest in mathematics: Girls’ Angle. Their latest blog post and print issue feature some pictures of previous Math, Light, & Color students’ work. If you are looking for an engaging and beautifully laid out resource for your math students, I highly recommend Girls Angle!

Without further ado, here are this year’s projects. Enjoy!