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Secondary Magazine - Issue 73: Focus on - part 2


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 16 November 2010 by ncetm_administrator
Updated on 23 November 2010 by ncetm_administrator

 

Secondary Magazine Issue 73Fractal - image by Wolfgang Beyer used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence
 

Focus on...introducing fractal ideas

continued from part 1

Some simple well-known fractal structures have special names. Students could be challenged to describe the particular iterative processes that produce them.

For example, what combination of initiator, generator and rule is generating this T-square fractal?

T Fractal Square by Skolkoll

T-Square fractal by Solkoll

Can students themselves generate the four sequences of shapes at the top of this page?

What initiator and generator together generate this sequence of shapes that converges to the fractal known as the Koch curve – which is the limit approached as the iterative steps are followed over and over again?

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

 Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence
 

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

 

Koch curve by Christophe Dang Ngoc Chan used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence

Koch curves by Christophe Dang Ngoc Chan

The Koch curve can be seen as four copies of itself, or as two copies of itself. Challenge students to explain how!
 
By starting with an equilateral triangle, rather than a single line segment, students could create the first few stages towards a limit called the Koch snowflake.

Koch snowflake by Saperaud used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence
Koch snowflake by Saperaud

You could show students the first five stages of a sequence of patterns, the limit of which is a fractal named the Moore curve. Can they describe the iterative process, and then produce their own versions?
 

purple circles

The Cantor set – or Cantor’s Dust – is a very simple self-similar fractal which is, nevertheless, extremely interesting. It contradicts intuitive ideas about space!

Cantor set

The starting point of the endless journey towards the Cantor set is a line segment representing the interval on the real number line from 0 to 1 inclusive. The first step is to remove the middle third of the line segment, between 1/3 and 2/3, leaving behind the points representing 1/3 and 2/3. In the second step the middle third of each of the remaining two line segments are removed, again leaving behind the end-points. This process is repeated indefinitely.

Cantor set

More advanced students can start to see how strange is the Cantor set by thinking about the sum of the segments that are removed.

Cantor set

Cantor set equation

This seems to be telling us that by the end of eternity the WHOLE line segment – from 0 to 1 – will have been removed. But it is fairly easy to show that there will in fact be as many points remaining as there were to start with!

For more elucidation go to mathacademy.com or to Cut The Knot.
 

blue circles

For a good introduction to the Mandelbrot set go to the Yale University Fractal Geometry website. And Mad Teddy’s Cantor and Mandelbrot web page is helpful, with excellent links.

The Mandelbrot set is a wonderful phenomenon that can greatly enhance students’ explorations of complex numbers and the Argand plane. It is created using the quadratic recurrence equation zn + 1 = zn2 + c.

Following a convention that has become established, points whose distance from the origin stays ‘forever’ less than or equal to 2 are black, and other points in the Argand plane are coloured according to the number of iterations before they ‘escape’ – before their distance from the origin is greater than 2. This diagram shows approximately how the Mandelbrot set relates to the circle of radius 2 centred on the origin. The Mandelbrot set consists of the black points inside the circle.
 

Mandlebrot fractal based on the image by Yami89 used under the Creative Commons Attribution-ShareAlike 3.0 Unported licence
Illustration based upon fractal image by Yami89

Here is a sequence of pictures that zoom into the Mandelbrot set. Because its shape forever and ever repeats as you look closer and closer at it, Benoît Mandelbrot called it a ‘fractal’!

 

Mandlebrot fractal set - images by Wolfgang Beyer



Issue 28 of the Secondary Magazine also focussed on fractals.

But the best person to help us understand and appreciate fractal geometry is its great originator, Benoit Mandelbrot himself – who was the one and only full-time fractalist!  Watch and listen to him, and read what he says, in this fascinating Big Think Mandelbrot video with transcript.

Image Credits
Page header - Fractal - image by Wolfgang Beyer some rights reserved
T-Square fractal by Solkoll in the public domain
Koch curves by Christophe Dang Ngoc Chan some rights reserved
Koch snowflake by Saperaud some rights reserved
Fractal image by Yami89 some rights reserved
Mandelbrot fractal set - images by Wolfgang Beyer some rights reserved

 
 
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