What is a fractal?

1. Fractals
by Osman Villanueva García

2. Can you describe how the animation is progressing?
What is a fractal?
What is happening to the perimeter and area
of the shape as more iterations are done?
It can be shown that the area tends towards 160%
of the original triangle’s area, but the perimeter
tends towards infinity
The shape, known as Koch’s snowflake, also has some
‘self-similarity’, meaning that zooming into a small
section will reveal the original structure
These properties – of infinite iterations and self-similarity – are what make a fractal

3. &
The most famous fractal shows the Mandelbrot set, which investigates the
effect of the iterative rule below for any complex number c:
The Mandelbrot set what happens as the number of iterations gets very large
Example:
The Mandelbrot set
Investigate for yourself what happens when c varies using the spreadsheet...
It does this for all possible values of c (ie any complex number!)
The value of zn
behaves in an unpredictable way!

4. To turn the outcomes of this investigation into a fractal, the following rule is applied:
If a value of c doesn’t diverge then it is shown as a black point on the Argand diagram
If a value of c does diverge then it is shown as a white point on the Argand diagram
For any complex number you start with c:
This leads to:
This fractal was named after the man who discovered it – Benoit Mandelbrot
Argand diagram
refers to a
geometric plot of
complex numbers
as points z = x + iy
using the horizontal
x-axis as the real
axis and the vertical
y-axis as the
imaginary axis.

5. To obtain a more striking fractal, vary the colour of a point depending on how quickly it diverges
You’re probably still thinking <<OK these look nice, but what’s the point?>>
It turns out this strange image also has some amazing properties...

6. Zooming in - Fractals
Zooming into different parts of the edge of a Mandelbrot
set reveals strange and beautiful patterns...:
You can zoom further and further...
https://math.hws.edu/eck/js/mandelbrot/MB.html

7. Multibrot sets vary the power used in the iterative rule:
Changing the pattern

8. Julia sets are generated by keeping c fixed and making the point you are
considering zo

9. Newton fractals use the rule for different functions f
zn+1
= zn
-
f(zn
)
f’(zn
)

10. Fractals in art
Here are a small selection of the fractals created by
artists – investigate for yourself what can be made!

11. Fractals in nature
The mathematical fractals we have seen are infinitely complex, but have been built from an extremely
simple equation repeated endlessly. In the same way, natural fractal forms really are built up by simple
rules - ultimately, the interactions between atoms.
The Earth from space
A fern leaf Romanesco brocoli Ice crystals

12. The Geometry of Nature
⚫ “Clouds are not spheres, mountains are not cones, coastlines are not circles, and
bark is not smooth, nor does lightning travel in a straight line.” (Mandelbrot,
1983).
⚫ And here is a quote by Thomasina, from Arcadia:
“Each week I plot your equations dot for dot, and every week they draw
themselves as commonplace geometry, as if the world of forms were
nothing but arcs and angles. God's truth, Septimus, if there is an equation
for a curve like a bell, there must be an equation for one like a bluebell, and
if a bluebell, why not a rose? Do we believe nature is written in numbers?”
Arcadia is a 1993 stage play written by English playwright Tom Stoppard, which explores the relationship
between past and present, order and disorder, certainty and uncertainty. It has been praised by many critics as
the finest play from “one of the most significant contemporary playwrights” in the English language. It is widely
believed that the character of Thomasina Coverly in Arcadia is loosely based on Ada Lovelace, an English
mathematician in the 1800’s. Many people actually regard her as the first computer programmer!

13. Landscapes
⚫ Can you determine which images are real and which are
computer generated?

14. Clouds

15. More fractal clouds

16. A Medical Application
⚫ Fractals are used in the diagnosis of
skin cancer and liver diseases.
⚫ There is a notion of fractal
dimension.
⚫ This is applied to images of the
affected area and its boundary (they
are both fractal).

17. Fractals
⚫ Choose some similarities (with contracting scaling).
⚫ Let Fractalina play the chaos game with those similarities.

18. More Examples

21. Add inversion in the design…

23. More Circle-Based Fractals

24. Add Reflections to this mix…

26. Design - Mandelbrot Sets

29. Logistic Model - Origin of caos
https://www.geogebra.org/classroom/c8kgxsq8#tasks/c6mu8mh5
P{n+1}
= r Pn
(1 - Pn
)

30. Fractals
Osman Villanueva García