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CALABRIAMY SCHOOLLiceo Scientifico L.Siciliani

The work I’m introducing comesfrom 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.

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.

Contents:

Fractals in general; Definition of a fractal; Origins of fractal geometry; Complex numbers; Newton’s equations; My fractals;

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.

Koch Lace

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

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

Made his first studies on fractalsidentifying the topological properties,

without neverthelessgiving them a graphic representationbecause he didn’t have capability of

calculation.

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.

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.

Objectives:

xⁿ±a=0

Knowledge of complex numbers; Application of complex numbers; Study of iterative fractals by Newton’s equation.

ALGEBRAIC FORM

COMPLEX NUMBERS

TRIGONOMETRIC FORM

COMPLEX NUMBERS: “What is the real number

whose square is equal to -4?”

√-1=i IMAGINARY UNIT

ALGEBRAIC FORM OF COMPLEX NUMBERS:

Real part Imaginary part

POLAR COORDINATES:Let’s draw a vector OP on a Cartesian plane:

r

x

y

P

O

α

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

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.

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.

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

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.

Let’s follow an example:

Suppose we have a curve that has an equation like y=xn +a.

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

-f(x0)= f’(x0)(x-x0)

y=0

x= -f(x0) +f’ (x0) (x0) f’(x0)

y=0

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

z^7-1=0

AND NOW … MY FRACTALS WITH THEIR EQUATIONS

OTHER NEWTON’S FRACTALS:

z^3-1=0

z^9-1=0

z^3-1=0

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

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

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

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

<<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

BIBLIOGRAFY:

• M. Bergamini, A. Trifone, G. Barozzi: “Manuale blu di matematica”, Zanichelli Editore;•www.phys.ens.fr/~zamponi/archivio/nonpub/newton.pdf

•www.webfract.it/FRATTALI/Metodo%20di%20Newton.htm

•www.google.it

Thank you for your attention!