AA

1. Fractals
By: Ankit Garg
Assistant professor (ASET/AIIT)
amity university,haryana

2. Fractal Geometry
Fractals in nature are so complicated and irregular that
it is hopeless to model them by simply using classical
geometry objects.
Some examples of Fractal shapes
complexity of the network of paths that supply blood
Tree like structure of lungs and kidneys in the body.
The pattern of branching is repeated infinitely many
times, at ever small scale. The same is true for
mountain. If we zoom on a picture of a mountain again
and again we still see a mountain. This is the self
similarity of fractal.

3. 3
Self-similarity in nature
Identical/similar
structures repeat
over a wide range
of length scales.

4. Examples of Fractals…
FERN

5. Examples of Fractals…
The Sierpinski Triangle

6. Examples of Fractals…
The Sierpinski Carpet

7. Examples of Fractals…
APPLY ITERATIVE FUNCTION SYSTEM TO CREATE DRAGON CURVE

8. Examples of Fractals…
DRAGON CURVE AFTER APPLYING IFS

9. Examples of Fractals…
The Mandelbrot Set

10. Fractal Art

11. Fractal Art

12. Fractal Art

13. Fractal Art

14. 14
Self-similarity in nature

15. What is a fractal anyway?
Let’s see what we can come up with...
FRACTALS

16. Look the house in this photo. Do you see any recognizable shapes?
A rectangle
A triangle
A parallelogram
Perhaps
a square
Even a
semicircle!
All familiar shapes from
Euclidean Geometry

17. Many man-made objects are
made up of Euclidean shapes

18. But what about these familiar things from the natural world?
Can they be easily described with Euclidean shapes?
I don’t
think so...

19. Clouds are not spheres,
mountains are not cones,
coastlines are not circles,
lightning travel in a straight
line.”
Benoit Mandelbrot, the father of fractal geometry, from
his book The Fractal Geometry of Nature, 1982.

20. While we haven’t yet defined what a fractal is, here are some
fractal representations of some of the things Mr. Mandelbrot
said could not be easily represented with traditional geometry.
This looks a lot more like a
cloud than this does...
And this looks a lot more
like a tree than this does!

21. Properties of fractal:
 Self similarity
 Fractal has Fractional dimension.
 Fractal has infinite length, yet they enclose finite
area.

22. Let’s take a look at a famous fractal and how it can be created. Let’s begin
with an equilateral triangle. Our iteration rule is:
Stage 0
Stage 1
Stage 2
Stage n
And so forth,
until...
This fractal is called the
Sierpinski Triangle
For each triangle, join the midpoints of the sides and then remove the
triangle formed in the middle.

23. 23
Fractal objects: properties
(a) self-similarity
∙ exactly the same structures
repeat all over the fractal
zoom in
and rescale

24. 24
Fractal objects: properties
(a) self-similarity
∙ exactly the same structures
repeat all over the fractal
zoom in
and rescale

25. Let’s consider another well known fractal called the Koch Curve.
The iteration rule for this fractal is:
For each segment, remove the middle third and replace it with an
upside-down V which has sides the same length as the removed piece.
Stage 0Stage 1Stage 2Stage 3Stage 4Stage 5

26. If we put three Koch curves together, we get...
The Koch Snowflake!

27. Let’s look at just one more. Our iteration
rule will involve replacing each segment
with a shape like this --
Stage 1Stage 0
It’s the Sierpinski Triangle!
Generated by a different recursive process
than the first time we encountered it.
Stage 2Stage 3Stage 4Stage 5Stage 6Stage 7Stage 8
What will this fractal look
like?

28. ITERATED FUNCTION SYSTEM
IFS (iterated function system) is another way of generating fractals. It is based
on taking a point or a figure and substituting it with several other identical ones.
For example, there is a very simple way of generating the Sierpinski triangle.
You can start with an equilateral triangle and substitute it with three smaller
triangles:
By iterating this process you substitute each of those three triangles with three
even smaller triangles continue a large number of times:

29. Koch Fractals (Snowflakes) After
IFS
Iteration 0 Iteration 1 Iteration 2 Iteration 3
Generator
1/3 1/3
1/3 1/3
1

30. Fractal Tree After IFS
Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5
Generator

31. Fractal Fern After IFS
Generator
Iteration 0 Iteration 1 Iteration 2 Iter

32. Thank You!!