This document presents work done as part of a "Maths and Reality" project studying geometric transformations and fractals. It summarizes the aims of the project, which are to study geometric transformations, represent reality through mathematical models using transformations, integrate traditional teaching with new technologies, build known fractals, and make graphic representations of fractals. It then provides details on specific fractals and methods analyzed as part of this work, including definitions of fractals, origins of fractal geometry, complex numbers, Newton's equations, and specific fractal examples generated through Maple code.

2. ITALY
CALABRIA
MY SCHOOL
Liceo
Scientifico
L.Siciliani

5. The work I’m introducing comes
from the study and the commitment
I have done in the project of
“Maths and Reality”. It is a national
project linked to Perugia
University. It is an extra activity
work that has been made in our
school for seven years by teacher
Anna Alfieri.
I also presented this work
at the National
Convention “Maths
Experiences in
Comparison” held in
Perugia University from
May 3rd
to May 5th
2011.

6. Aims of the project:
• To Study the geometrical
transformations;
• To Learn to represent the
reality through mathematical
models by using geometrical
transformations;
• To Integrate the traditional
didactics with new
technologies (use of maple);
• To Build known fractals
(Sierpinski gasket, snowflake
of koch…) identifying the
geometrical transformations
which describe them
• To Make graphic
representations through
Maple
• To Conjecture and make
individual simulations.

7. Contents:
 Fractals in general;
 Definition of a fractal;
 Origins of fractal geometry;
 Complex numbers;
 Newton’s equations;
 My fractals;

8. Since the end of the XIX century Science has
focused on a different study of complex systems.
These interests started off the study of the
“deterministic chaos”, based on the situations of
chaos obtained through mathematical and physical
deterministical process.
In the real universe there are infinite “perturbing” elements
This complexity can be simplified by
Complex and chaotic geometric figures determined for
approximation from a recoursive
Function.

9. Koch Lace
A geometric figure where the same
shape is repeated on a smaller
uninterruptedly scale.

10. A fractal must have some important characteristics:
 Autosimilarity:
If the details are observed on different
scales, we can see an approximative
similarity to such an Original fractal.
 Indefinite Resolution: It Is not possible to define the
border of the figure

11. Made his first studies on fractals
identifying the topological properties,
without nevertheless
giving them a graphic representation
because he didn’t have capability of
calculation.

12. The founder of fractal geometry was:
a comtemporary mathematician that, in the first years of
80‘s, published the results of his research in the
volume “the fractal geometry of nature” founding the
fractal geometry. The name fractal derives from the
latin fractus , because its dimention is not integer.

13. Was born in 1924 in Warsaw,
he studied at the Ecole
Polytechnique and at Paris
university, where he
graduated in the 50’s in
mathematics. Then he
became professor of applied
mathematics at Harvard
University, and professor of
mathematics science at Yale
University. He received
several prizes, like the Wolf
Prize for phisics. Since the
60’s he has devoted himself to
the studies of finance.

14. Objectives:
xⁿ±a=0
 Knowledge of complex numbers;
 Application of complex numbers;
 Study of iterative fractals by
Newton’s equation.

15. ALGEBRAIC FORM
COMPLEX
NUMBERS
TRIGONOMETRIC FORM

16. COMPLEX NUMBERS:
“What is the real number
whose square is equal to -4?”
√-1=i
IMAGINARY
UNIT

17. ALGEBRAIC FORM OF
COMPLEX NUMBERS:
Real part Imaginary
part

18. POLAR COORDINATES:
Let’s draw a vector OP on a
Cartesian plane:
r
x
y
P
O
α

19. VECTORS AND COMPLEX
NUMBERS:
Let’s consider a complex number a+ib and
let’s interpret the coefficients of the real and
the imaginary part like the components of a
vector named OP.
a+ib
y
O x
Pib
a

20. THE TRIGONOMETRIC FORM OF A
COMPLEX NUMBER:
α
O
y
P
x
b
a
a+ib=r(cosα+ i sinα)
A complex number a+ib is the equivalent of the
vector OP that has its components a and b and its
coordinates in P(r;α), so we can write:
a=r·cosα; b=r·sinα.

21. THE n-th ROOTS OF A COMPLEX NUMBER:
ⁿ√z=v
Generally we can calculate the n-th roots
of a complex number by the equation:
ⁿ√r(cosα+i sinα)=ⁿ√r cos α+2π + isin α+2π
n n
Given two complex
numbers z and v, we
can say that v is the
n-th root of z if vⁿ=z.

22. Now we can calculate the n-th roots of the
equation z^3-8=0:
k=0 2 cosπ+isinπ =2i
2 2
k=1 2 cos7π+isin7π =√3-i
6 6
k=2 2 cos 11π+i sin11π = √3-i
6 6

23. NEWTON’S
METHOD:
By Newton’s method,
named also the
method “of the
tangents”, we can find
the approximate
solutions of an
equation like: xn
±a=0
with n≥3.

24. Let’s follow an example:
Suppose we have a curve that has an equation
like y=xn
+a.

25. To find the point of intersection with the axis of abscissa we
can make a system between the last one and the tangent
line to the curve that passes through the point x0:
y-y0=m(x-x0)
y=0
The tangent line of the
curve that passes
through x0
(where
y0=f(x0) e m=f’(x0))The equation of the
axis of abscissa

26. -f(x0)= f’(x0)(x-x0)
y=0
x= -f(x0) +f’ (x0) (x0)
f’(x0)
y=0

27. THIS IS A MAPLE CODE TO GENERATE A
NEWTON’S FRACTAL:

28. z^7-1=0
AND NOW … MY FRACTALS
WITH THEIR EQUATIONS

29. OTHER NEWTON’S FRACTALS:
z^3-1=0

30. z^9-1=0

31. z^3-1=0

32. z^8+15z^4-16=0

33. z^8-17z^4+16=0

34. z^10-10z^5+16=0

35. z^5-10z^4+16z^3-2=0

36. <<Fractals help to find a new
representation starting from the
point that the “small” in nature
is nothing but the copy of the
“big”. I’m firmly convinced that,
in a very short time, Fractals will
be employed in the
comprehension of the neural
processes and the human mind
will be their new frontier.>>
B. Mandelbrot

37. BIBLIOGRAFY:
• M. Bergamini, A. Trifone, G. Barozzi: “Manuale blu di
matematica”, Zanichelli Editore;
•www.phys.ens.fr/~zamponi/archivio/nonpub/newton.p
df
•www.webfract.it/FRATTALI/Metodo%20di
%20Newton.htm
•www.google.it

38. Thank you for your attention!