First Introduction to Fractals

Definition Snowflake Sierpinski fractals Pythagorean tree Mandelbrot set Newton fractals Natural fractals Applications Thanks

Why I Love Fractals?

Dr V N Krishnachandran

Why I Love Fractals


Why I Love Fractals?

Why I Love Fractals!

Dr V N KrishnachandranVidya Academy of Science & Technology

Thalakkottukara, Thrissur - 680501


Why I Love Fractals


Outline

1 Definition

2 van Koch Snowflake

3 Sierpinski fractals

4 Pythagorean tree

5 Mandelbrot set

6 Newton-Raphson fractals

7 Natural fractals

8 Applications


Why I Love Fractals


Definition

What is a fractal?


Why I Love Fractals


What is a fractal?

A fractal is a mathematical set or a naturalphenomenon with infinite complexity andapproximate self-similarity at any scale.


Why I Love Fractals


What is a fractal?

Let us see examples.


Why I Love Fractals


What is a fractal?

Romanesco Broccoli : This variant form of cauliflower is a naturalfractal (see detail in next slide).


Why I Love Fractals


What is a fractal?

Romanesco Broccoli : Detail


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Fractal 1: van Koch snowflake

Fractal 1: van Koch snowflakeA curve resembling a snowflake

A real snowflake


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Construction of the curve.


Why I Love Fractals



Step 1


Why I Love Fractals




Why I Love Fractals



Step 2


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Step 3


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Step 4


Why I Love Fractals



Step 5


Why I Love Fractals



. . . and continue without stopping . . .


Why I Love Fractals



Self similarity of van Koch snowflake


Why I Love Fractals



Here is a website with animation showingthe self-similarity of van Koch snowflake.

https://www.tumblr.com/search/koch+snowflake


Why I Love Fractals



Step Length of Number of Perimeter

Number a segment segments

1 a 3 3a

2 a/3 4× 3 = 12 12× (a/3)

3 a/9 4× 12 = 48 48× (a/9)

4 a/27 4× 48 = 192 192× (a/27)

· · · · · · · · · · · ·n a/3n−1 4× 3n−1a 3× (4/3)n−1a

Perimeter −→∞ as n −→∞.


Why I Love Fractals



Area bounded by the snowflake

=

(8/5)× area of initial triangle


Why I Love Fractals


Fractal 2: Sierpinski triangle

Fractal 2 : Sierpinski triangle


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Construction of the fractal.


Why I Love Fractals



. . . and continue without stopping


Why I Love Fractals



Wikipedia page on Sierpinski triangle has an animationshowing the construction of the Sierpinski triangle.


Why I Love Fractals



Here is an animation showingthe self-similarity of Sierpinski triangle.

http://www.lutanho.net/fractal/sierpa.html


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Fractal 3: Sierpinski tetrahedron



Why I Love Fractals





Why I Love Fractals



Steps 1, 2


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Steps 2, 3


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Step 6, and continue without stopping


Why I Love Fractals


Fractal 4: Pythagorean tree



Why I Love Fractals





Why I Love Fractals



Steps 1, 2


Why I Love Fractals



Steps 3, 4 and continue without stopping to get theimage shown in the next slide (with added colors).


Why I Love Fractals




Why I Love Fractals



Modifications to Pythagorean tree


Why I Love Fractals




Why I Love Fractals




Why I Love Fractals


The complex world!


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The Mandelbrot set


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Fractal 5: Mandelbrot set


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Mandelbrot set discovered byBenoit Mandelbrot in 1979.


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The mathematics of Mandelbrot set


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Consider the iteration scheme:

zn+1 = z2n + c


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1 Choose a fixed value for c , say, c = 1 + i .2 Mark the point represented by c in the complex plane.3 Let z0 = 0.4 Compute z1, z2, z3, . . . as follows:

z1 = z20 + c

z2 = z21 + c

z3 = z22 + c

· · ·5 If the values z1, z2, z3, . . . increases indefinitely, mark the point

c with red color.Otherwise, mark the point c with blue color.

6 Repeat this by choosing different points in the complex plane.7 The blue colored region is the Mandelbrot set.


Why I Love Fractals



We can use a computer program to generate theMandelbrot set.

The Mandelbrot set obtained by computing z1, z2, . . . pixel bypixel, starting with z0 = 0. If the values do not go to infinity aftera large number of iterations the present pixel value is in theMandelbrot set and is then colored blue. If the sequence divergethen the pixel is colored red.

Sometimes, the pixel is colored according to how fast thedivergence is.


Why I Love Fractals




Why I Love Fractals


Fractal 6 : Newton-Raphson fractals

Fractal 6 : Newton-Raphsonfractals


Why I Love Fractals




Why I Love Fractals


Fractal 6 : Newton-Raphson fractal

We consider the Newton-Raphson fractal corresponding to theequation

f (z) ≡ z3 − 1 = 0.

The roots of this equation are 1, 12 (−1 + i

√3), 1

2 (−1− i√

3).


Why I Love Fractals



Newton-Raphson method.

1 Choose an initial approximation z0 to a root.

2 Compute z1, z2, . . . by

zn+1 = zn −f (zn)

f ′(zn)

= zn −z3n − 1

3z2n

3 The numbers z1, z2, . . . approach a root.


Why I Love Fractals



1 Choose a fixed complex number c . Mark the point c in thecomplex plane.

2 Let z0 = c .

3 Compute z1, z2, . . . using the formula zn+1 = zn −z3n − 1

3z2n

.

4 If z0, z1, z2, . . . approach 1, color the point c red.If z0, z1, z2, . . . approach 1

2 (−1 + i√

3), color the point c green.

If z0, z1, z2, . . . approach 12 (−1− i

√3), color the point c blue.

5 Repeat this by choosing various points in the complex plane.

6 The resulting figure is the Newton-Raphson fractalcorres-ponding to the equation z3 − 1 = 0 (see next slide).


Why I Love Fractals




Why I Love Fractals



The equationz4 + z3 − 1 = 0

produces the fractal shown in the next slide.


Why I Love Fractals




Why I Love Fractals


Software

We can explore the complexity of the Mandelbrot set, theNewton-Raphson fractal and many others using the Xaos programavailable at: http://matek.hu/xaos/doku.php


Why I Love Fractals


Into the natural world!


Why I Love Fractals


Fractal 7 : Natural fractals

Romanesco Broccoli : This variant form of cauliflower is theultimate fractal vegetable (see detail in next slide).


Why I Love Fractals



Romanesco Broccoli : Detail


Why I Love Fractals



Oak tree, formed by a sprout branching, and then each of thebranches branching again, etc.


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River network in China, formed by erosion from repeated rainfallflowing downhill for millions of years.


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Our lungs are branching fractals with a surface area ofapproximately 100 m2.


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The plant kingdom is full of spirals. An agave cactus forms itsspiral by growing new pieces rotated by a fixed angle.


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What are fractals useful for?


Why I Love Fractals


Applications of fractals

Fractal antennas

A fractal antenna is an antenna that uses a fractal, self-similardesign to maximize the length, or increase the perimeter ofmaterial that can receive or transmit electromagnetic radiationwithin a given total surface area or volume.


Why I Love Fractals



Fractal compression

Fractals have been used for developing algorithms for imagecompression.


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Fractal compression

Natural fern leaves (left) and fractal generated fern leaves (right)


Why I Love Fractals




Why I Love Fractals


Thanks for patient hearing.


Why I Love Fractals

Education

First Introduction to Fractals