Fractals are mathematical objects that have detailed patterns at any scale. They are self-similar, meaning each part is similar to the whole. Benoit Mandelbrot introduced the term "fractal" in 1975 and studied their properties. Fractals have an initiator shape, a construction law to generate iterations, and a process of generation. Common fractals include the Koch snowflake, Sierpinski triangle, Hilbert curve, and Menger sponge. Fractals are important because patterns in nature like coastlines are often fractal, contradicting traditional geometry.

1. FRACTALS

2. WHAT ARE FRACTALS?

3. • The term " fractal" comes from the Latin "fractus" which means
"broken", "fractured".
This term was introduced by Benoit Mandelbrot in 1975.
• A fractal is a mathematical object which has a
detailed structure at any scale.
In the structure of a fractal,
each part is similar to the whole fractal .

4. A LITTLE BIT OF HISTORY…

5. Fractal geometry is known as a new path of
mathematics being based on Mandelbrot’s article
“What is the length of the Great Britain’s shore?”
so that it later become a practical domain
of mathematics after the
appearance of his book
“Fractal geometry of nature”
in 1982.
Newton accumulation

6. Mandelbrot had introduced the term
“fractal”
and had identified domains
of applicability
of his geometry.
Mandelbrot’s snowman

7. THE ELEMENTS OF A FRACTAL
A fractal has the fallowing elements:
• 1. The Initiator
The initiator is the geometrical shape from which the generation of
the fractal started. Usually , the initiator is a simple geometrical
shape – line , triangle , square , …
• 2. The Law of Construction
The law of construction offers the method of the fractal
construction.
• 3. The process of generation
The process of generation is the one which effectively constructs the
iterations of the fractal object , starting from the current iteration
and applying the law of construction to it. Every iteration defines a
new generation of fractal accumulation .

8. EXAMPLES OF FRACTALS

9. Julia’s accumulation
Mandelbrot’s snowman

10. KOCH’S SNOWFLAKE
To create a snowflake you start with an equilateral triangle and you replace the middle
third from each side with two segments to form a new exterior equilateral triangle.

11. SIERPINSKY’S TRIANGLE
Sierpinski’s triangle – it is obtained from a triangle and cutting recursively the triangle
(center) formed by the middles of each side.

12. HILBERT’S CURVE
.
Hilbert’s curve is an example of a continuous curve
with infinite length, without autointersections,
which “fills” a square.

13. MENGER’S SPONGE
Menger’s sponge is a fractal curve,
a threedimensional generalization of the Cantor set and Sierpinski carpet.
The Menger sponge simultaneously exhibits an infinite surface area and zero volume.

14. OTHER EXAMPLES OF FRACTALS
Julia accumulation
Cantor’s dust
The binary tree
Mandelbrot’s
snowman

15. Koch’s snowflake

16. WHY ARE THEY SO IMPORTANT
FOR US?

17. • These geometrical shapes had been considered chaotic or
“geometrical aberration’’ in the past and some of them were considered
so complex that they needed advanced computers to visualize them.
• As the years went by , scientific domains as
physics , chemistry , biology or meteorology
discovered similar elements with real life fractals.
Fractals have very interesting mathematical properties ,
which often contradict the appearance ,
but this exceeds the high school knowledge.

18. SOURCES:
http://universulenergiei.europartes.eu/intrebari/fractali/
http://www.scribd.com/doc/16078402/elementegeometriefractala
http://iuliasaplacan.bravehost.com/tipfract.html
http://ro.wikipedia.org/wiki/Fractal
Images: http://www.google.ro

19. THE END 