This post is the third in a series on the Mandelbrot set. The Mandelbrot set resides upon the complex plane. This means that in order to look more closely at the Mandelbrot set itself, we need to first get a handle on the complex plane. That is the topic of this post.
When I was 11 or 12 years old, I bought a Sega Genesis. Well, to be fair, I bought something like a 60% share in a Sega Genesis—my little sister, who had basically no interest in gaming, owned the other 40%. When we unboxed the system, we discovered a console, two controllers, some cables, a power brick, and a single game: Sonic the Hedgehog.
I had played Super Mario Brothers and Metroid on Nintendo systems owned by some of my friends, so I thought I knew a thing or two about video games. But Sonic the Hedgehog blew me away. The graphics were amazing, and the game simply screamed along. I had never seen a game character move with such alacrity.
Throughout my early teenage years, other Sonic the Hedgehog platforming games were released for the Genesis. I eagerly awaited each of these, and spent hours playing through each incarnation of the series.
This post is the second in a series on the Mandelbrot set. In this post, we are going to spend some time exploring the set, in order to get a feel for some of the structures that appear within the Mandelbrot fractal.
When students are first taught mathematics, the skills that they learn are concrete, and the teaching methodologies are equally concrete. When taught to add, students might physically group blocks, for instance by counting one group of blocks, then another, and finally combining the two groups and counting up the result. Subtraction is equally concrete: students might be given some blocks, then asked to remove a certain number, and count the remaining blocks.
Multiplication and division are also introduced in a concrete fashion. While many techniques are taught for both operations, they are both initially viewed as collecting several identical sets of objects, or dividing a large group of objects into smaller, equally sized groups. Once again, students might be given actual objects (blocks, candies, or tokens) to manipulate.
Then at some point in middle or high school, students are required to think in a radically different manner. Algebra is introduced, and students are asked to manipulate abstract variables, and to balance equations. This transition can often be quite difficult, especially when viewed in terms of cognitive development: many of the students introduced to algebra simply do not have the cognitive capacity to think abstractly.
Fortunately, there are things that instructors can do to help.
Every Monday for the past several weeks, I have posted an image under the heading Monday Mandelbrot Madness. I have offered up these images largely on aesthetic grounds, and offered very little in the way of mathematical explanation. There are two reasons for this: I want readers to be able to appreciate the images on purely aesthetic grounds; and I have been planning a series of posts which explain the images in greater detail. This is the first such post.
The Mandelbrot set is a fractal. In this post, my intention is to introduce the reader to the concept of a fractal, and to provide a couple of examples. In future posts, I will spend some more time discussing the Mandelbrot set specifically.
So, what is a fractal?
What’s in a name? that which we call a rose
By any other name would smell as sweet…—William Shakespeare
In our daily lives, we are accustomed to giving multiple names to a single object. My wife calls me Xander, while my brothers and sisters call me Alex, and my students call me Mr. H. The titles, honorifics, and diminutives that people use to address me do not alter my fundamental self; they do not imply that multiple people inhabit my body, nor that there are multiple mes. Somehow, we are able to tolerate the ambiguity of calling the same person by multiple names without too much confusion.
