Taught: Exploring and Generating the Mandelbrot Set
For most of the final term of last year, I had a simple display on one of my walls showing a journey into the Mandelbrot Set (the pictures for which were taken from this Wikipedia collection). Several of my groups had asked me what the pictures were, and how they were derived, so I decided to spend a lesson with several of my A-level groups in the last week of term exploring this mathematical object.
I created the following Geogebra file to help me:
(source: mandelbrotorbits.ggb)
This shows the first thirty iterations of the map zn+1=zn2+c, where c is the value represented by the blue point. As you move the blue point around, the other points are automatically updated.
For some values of c the values will stay within a bounded region of the origin, no matter how many iterations we try. For others, the values will ‘escape’ towards infinity. It can be shown that if the modulus of the points ever becomes more than 2, then the points will eventually become unbounded.
The Mandelbrot Set is the set of values of c for which the values never ‘escape’.
My A-level sets had not met many of these concepts before (they’ve never met a complex number before, for example), so were introduced to the ideas geometrically. We generated some iterations by hand for c=0.6i, and then, using an Excel spreadsheet to help us with the calculations, created our own very low resolution Mandelbrot set on the interactive whiteboard, using counters.
It made a good end of term lesson, even if we couldn’t cover all the required mathematics in great detail. If nothing else, I find it quite absorbing just moving the initial point around, and seeing the patterns that emerge in the iterations… particularly when you notice that different ‘buds’ of the Mandelbrot set exhibit different behaviours.
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Tags: geogebra, Lessons Taught, mandelbrot set, Maths
August 17th, 2008 at 5:42 am
I’m interested in this idea of introducing Mandelbrot Set. I’ve made a similar file in May:
http://web.hku.hk/~amslee/ggbhtml/Orbit_zero.html
In my file, you can drag the point C, which is one of the red dots and may be difficult to identify.
I’m new to blogging and is experimenting in Edublog, which is also powered by Wordpress. I try to paste in some posts the codes for the applets I created earlier but the editor always chops off the codes when a post is saved. I wonder how you insert applets in your posts here.
By the way, I don’t really see your Geogebra file in the applet, which is actually blank, though I can download the ggb file. Yet, this is already better than what I get in my blog.
August 17th, 2008 at 2:53 pm
Thank you for the link; I thought that other people must have written similar Geogebra files, as it is a simple visual way to explore the Mandelbrot set. Yours is slightly different, as you allow people to change ‘c‘ and ‘z0‘ independently, and so explore Julia sets as well.
I like your use of colour – certainly more colourful than my file, but that is often just a matter of taste. I try to put the minimum amount of ’stuff’ on the pages I create, as I find that lots of colour or text distracts students from the key points I want to put over.
Adding Geogebra applets and other files to Wordpress is a little involved, and I’m not sure it’s possible on hosted sites like wordpress.com and edublogs, as they normally restrict the types of files which you can upload. It certainly helps to use the ‘raw’ mode of the HTML post editor, rather than the ‘visual’ mode… but even the ‘raw’ mode is not particularly raw, and often eats codes at unpredictable times. I have a ‘raw mode’ plugin on this installation of Wordpress, which I believe helps a little, although I haven’t yet been able to get embedded Javascript to work (which I need to do animations in Geogebra applets).
I wish you luck in your blogging, and I look forward to seeing any other resources you produce.
August 18th, 2008 at 1:13 am
I experiment with the use of colour and some features of Geogebra. I agree that with specific purpose in the classroom, it can be more focused. By the way I find some interesting effect with my file. You may open the property editing box, select all points (on the left panel just with one click), turn on the trace. Now if you drag C slowly in any way, some nice graffiti will come out, (like an impressionist painting).
To put the applet on my blog, I am not going to upload any files. In fact the jar and ggb file are stored in a server of my university. I just link to these files in my applet codes. Here it is
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
So I suppose simply copying it to my post (in the raw mode), the applet can appear. But like what you said, the first half of the codes are always eaten up.
If you are interested in other resources I’ve tried, may have a look here:
http://web.hku.hk/~amslee/it07/
http://web.hku.hk/~amslee/allggb/
Nice to talk to you.
August 18th, 2008 at 1:20 am
I tried to copy my applet codes in the third paragraph of my comment above, but it’s not shown.
Maybe you can view the source code of my html page:
http://web.hku.hk/~amslee/ggbhtml/Orbit_zero.html