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Fractal Geometry. IMO Mathematics Camp 20 August 2000 Leung-yiu chung S.K.H. Bishop Mok Sau Tseng Secondary School. Table of Contents. What is Fractal Examples of Fractals Properties of Fractals Two Famous Sets about Fractals -- Mandelbrot Set and Julia Set

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Fractal Geometry

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Fractal geometry l.jpg

Fractal Geometry

IMO Mathematics Camp

20 August 2000

Leung-yiu chung

S.K.H. Bishop Mok Sau Tseng Secondary School

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Table of Contents

  • What is Fractal

  • Examples of Fractals

  • Properties of Fractals

  • Two Famous Sets about Fractals -- Mandelbrot Set and Julia Set

  • Applications and Recent Developments

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  • Measure with a mile-long ruler

  • Measure with a foot-long ruler

  • Measure with a inch-long ruler

  • Any difference?

  • The measurement will be longer,longer,….

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Ferns (Leaf)

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Sierpinski Triangle

  • Steps for Construction

  • Questions

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  • Step One

  • Draw an equilateral triangle with sides of 2 triangle lengths each.

  • Connect the midpoints of each side.

  • How many equilateral triangles do you now have?

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  • Shade out the triangle in the center. Think of this as cutting a hole in the triangle.

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  • Step Two

  • Draw another equilateral triangle with sides of 4 triangle lengths each. Connect the midpoints of the sides and shade the triangle in the center as before.

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  • Step Three

  • Draw an equilateral triangle with sides of 8 triangle lengths each. Follow the same procedure as before, making sure to follow the shading pattern. You will have 1 large, 3 medium, and 9 small triangles shaded.

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  • Step Four

  • How about doing this one on a poster board? Follow the above pattern and complete the Sierpinski Triangle.

  • Use your artistic creativity and shade the triangles in interesting color patterns. Does your figure look like this one?

  • Back

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Question 1

  • in Step One. What fraction of the triangle did you NOT shade?

  • Back to step One

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Question 2

What fraction of the triangle in Step Two is NOT shaded?

What fraction did you NOT shade in the Step Three triangle?

Back to Step Two

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Question 3

Use the pattern to predict the fraction of the triangle you would NOT shade in the Step Four Triangle.

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Question 4


Develop a formula so that you could calculate the fraction of the area which is NOT shaded for any step

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Koch Snowflake

  • Step One.

  • Start with a large equilateral triangle.

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Koch Snowflake

  • Step Two

  • 1.Divide one side of the triangle into three parts and remove the middle section.

  • 2.Replace it with two lines the same length as the section you removed.

  • 3.Do this to all three sides of the triangle.

  • Do it again and again.

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Amazing Phenomenon

  • Perimeter

  • Area

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  • In Step One, the original triangle is an equilateral triangle with sides of 3 units each.

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Perimeter = 9 units

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Perimeter = ___ units

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Perimeter = 12 units

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Perimeter = ___ units

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Perimeter = 16 units

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Infinite iterations

  • What is the perimeter ?

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  • Original Area : ?

  • After the 1st iteration : ?

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2nd Iteration :

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Fractal Properties

  • Self-Similarity

  • Fractional Dimension

  • Formation by iteration

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  • To the right is the Sierpinski Triangle that we make in this unit. Notice that the outline of the figure is an equilateral triangle. Now look inside at all the equilateral triangles. Remember that there are infinitely many smaller and smaller triangles inside. How many different sized triangles can you find? All of these are similar to each other and to the original triangle - self similarity

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  • If the red image is the original figure, how many similar copies of it are contained in the blue figure?

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Fractional Dimension

  • Point --- No dimension

  • Line --- One dimension

  • Plane --- Two dimensions

  • Space --- Three dimensions

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A definition for Dimension

  • Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment.

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  • Take another self-similar figure, this time a square 1 unit by 1 unit.

  • Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies.

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Take a 1 by 1 by 1 cube and double its length, width, and height.

How many copies of the original size cube do you get? Doubling the side gives eight copies.

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Table for comparison

  • when we double the sides and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension

  • Table

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Figure Dimension No. of copies

Line1 2=21

Square 2 4=22


Doubling Simliarity d n=2d

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Sierpinski Triangle

  • How many copies shall we get after doubling the side?

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Figure Dimension No. of copies

Line1 2=21

Sierpinski’s ? 3=2?

Square 2 4=22


Doubling Simliarity d n=2d

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  • Dimension for Sierpinski”s Triangle

  • d = log 3 / log 2 = 2.1435…...

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Iterative Process on a Function

  • Mandelbrot Set

  • Julia Set

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Mandelbrot Set

  • Iterative Function : f(z) = z2 +C

  • For each complex number C, start z from 0,

  • f(0)=C, f(f(0)), f(f(f(0))),….

  • C belongs to Mandelbrot set if and only if

  • the sequence of iteration converges

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Calculation and Coloring

  • For each point in a designated region in Complex Plane, construct the sequence of iteration

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Calculation and Coloring

  • Determination of Convergence:

    • One can prove that

    • if the distance of the point from the origin become greater than 2, it will grow to infinity

    • For certain iteration, the value of the term >2, then the sequence grows to infinity--> out of Mandelbrot Set

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Calculating and Coloring

  • Within certain iterations (e.g. 200), if all the terms have the magnitudes not greater than 2, then the sequence is assumed to be convergent.

  • The corresponding value of C should be regarded as an element of Mandelbrot Set and colored in black.

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Calculating and Coloring

  • For other value of C, the point will be colored depending on the number of iteration needed to have a term with magnitude greater than 2.

  • If the colors used are chosen in such a way that the no. of iterations follows a certain spectrum of color, we may get fantastic pictures.

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Mandelbrot Set

  • Animation

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Julia Set

  • The Process to get Julia Set is similar to that for Mandelbrot Set except

  • In Mandelbrot Set, C corresponds to an element and we start from z= 0

  • In Julia Set, C is pre-fixed and the varies z for starting value, the z for convergence corresponds to an element.

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Julia Set


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  • Fractal Gallery -- Simulate Natural Behaviors

  • Fractal Music

  • Fractal Landscape

  • Application on Chaotic System

    • Damped pendulum problems

  • Image Compression -- highly jaggly texture

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Useful Resources

  • Websites:



  • Search on

    • key word : Fractal Geometry, Fractal, Mandelbrot

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  • Java Applets from Websites

  • Freeware/Shareware downloaded from Internet:

    • Fractal Browers

    • Fantastic Fractals 98

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