FRACTALS

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FRACTALS
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Aims of presentation

• Brief introduction to what I did in school.
• Introduction to fractals
• Overview of what was covered in the lesson, with
  more mathematical details.

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• Framwellgate School.

• Year 11 intermediate level class.

• Mixed ability of students.

• Main tasks helping out any pupils who were
  stuck, answering general questions.

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Fractals

• Subject rarely even taught at university level.

• Easily accessible to year 11 pupils.

• Both interesting and Challenging.

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• Planned two hours worth of lessons due to the
  variety of material available and to allow the
  children to go into more depth without being
  rushed.
• However, due to time restraints in the year 11
  timetable, only one hour was available to teach
  the lesson.
• The lesson was condensed to teach the most
  important and interesting aspects.

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What is a fractal?

• A fractal is a geometrical figure in which an
  identical motif repeats itself on an ever
  diminishing scale
• Put simply, a shape which, when zoomed in on,
  retains the same detail and still resembles the
  original.
• Can occur in nature, e.g. a tree is a fractal.

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• Recent area of mathematics
• Term fractal coined in 1979 by Benoit
  Mandelbrot, a french mathematician.
• Influenced by Pierre Fatou and Gaston Julia who
  had both worked with iterative functions.
• After them, this area of mathematics was largely
  ignored, as during the 1950s and 1960s, the
  Fatou-Julia theory was utterly unfashionable, in
  the wilderness, because it consisted of many
  special examples and few great modern theorems

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• Mandelbrot changed this with the use of
  computers.
• By looking at the iteration of the simple complex
  function f(z) z² c, he produced amazing
  results.
• Starting with z0, he looked at the sequence (0,
  f(0), f(f(0)), f(f(f(0))),..) with different
  values of c.
• The sequence can either be bounded within a disc
  of fixed radius, or shoot off to infinity.

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• A number c is said to belong to the Mandelbrot
  set M if the relevant sequence is bounded.
• If a number belonged to M, the computer coloured
  it in black.
• If it did not, the colour depended on how quickly
  the sequence went off to infinity.
• The results were so surprising from such a simple
  iteration that Mandelbrot first thought he was
  mistaken.
• The result was as follows

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Mandelbrot Set
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6 x magnification of selected area
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100 x magnification
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2000 x magnification
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• After explaining the basic properties of fractals
  to the class, I introduced them to some much
  simpler fractals that are easy to create with a
  pencil and paper
• Sierpinski Triangle
• Koch Snowflake

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Sierpinskis triangle
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• The easiest way to allow the children to create
  this fractal for themselves is using Pascals
  Triangle.
• No longer on GCSE curriculum.
• Introduced Pascals Triangle to the class,
  getting them to create their own triangle looking
  like this

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The Class completed this themselves
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• If all odd numbers are coloured in and even
  numbers left blank (or coloured in differerent
  colours) pascals triangle becomes sierpinskis
  triangle

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Comparison
Pascals triangle method
Sierpinski triangle method
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Extension task

• Introduce children to basic idea of modulus.
• Ask them to colour the triangle in using modulus
  4, or simply colour in the multiples of different
  numbers, for example 12.

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Koch Snowflake

• Simple as this fractal is, one of its earliest
  investigators, Ernesto Cesaro said of the self
  similarity of the snowflake
• It is this self-similarity in all its parts,
  however small, that make the curve seem so
  wondrous. If it appeared in reality, it would not
  be possible to destroy it without removing it
  altogether. For otherwise it would ceaselessly
  rise up again from the depths of its triangles
  like the life of the universe itself.

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How to form the Koch Snowflake
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The self similarity that Cesaro talked about
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Using the Koch Snowflake to look at more advanced
ideas about fractals

• Infinite perimeter- Table for 0,1,2 iterations-
  Formula
• Infinite area?- No!- In fact, we can easily
  work out the number to which the area converges
  when we perform an infinite number of iterations.

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• Each colour represents a new iteration.
• We can work out the area added after each
  iteration.
• The children complete the following table to work
  out the area of the snowflake after 6 iterations.

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• In fact, only a further 5 iterations were needed
  to find that the area of the Koch Snowflake
  converges to 129.58.
• The children did find the idea of an infinite
  perimeter surrounding a finite area difficult to
  grasp as it was something that they had never
  come across before.

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Other ideas that could be covered with a more
able class

• The idea of fractal dimension
• The fractal dimension can be non-integer, and is
  given by the formula
• D log (N)
• log (1/h)
• Where N is the degree of self-similarity and h is
  the scaling factor of the fractal.
• As an example, the Koch Snowflake has a self
  similarity of four, as any segment of the curve
  can be split into four sub-segments. Each of
  these segments is scaled down by a factor 1/3.
  Therefore, the fractal dimension of the Koch
  Snowflake is
• D log (4) 1.26..
  
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  log (3)

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Additionally

• An explanation of the complex number iteration
  behind the Mandebrot set.
• Sixth form class could be introduced to how the
  iteration works, and how plotting this on the
  Argand diagram gives the Mandelbrot set.

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• End by showing some of the other amazing fractal
  art that has been created using simple iterations
  like the one used for the Mandelbrot set.

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Questions?