FRACTALS Dr. Farhana Shaheen Assistant Professor YUC

FRACTALS For centuries mathematicians rejected complex figures, leaving them under a single description: formless. For centuries geometry was unable to describe trees, landscapes, clouds, and coastlines. However, in the late 1970s a revolution of our perception of the world was brought by the work of Benoit Mandelbrot. He introduced and developed the theory of fractals -- figures that were truly able to describe these shapes. The theory was continued to be used in a variety of applications. Fractals importance in areas ranging from special TV effects to economy and biology.

Mandelbrot; inventor of fractals

FRACTALS- Mandelbrot Set

Mandelbrot Zoom

“Fractua” means Irregular Fractals are geometric figures like circles, squares, triangles etc. but having special properties. They are usually associated with irregular geometric objects, that look the same no matter at what scale they are viewed at. They have the property of self- similarity; generated by iterations. Fractals have finite area but infinite perimeter.

Fractals are Fun! What is a Fractal ?A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. The core ideas behind it are of feedback and iteration. The creation of most fractals involves applying some simple rule to a set of geometric shapes or numbers and then repeating the process on the result. This feedback loop can result in very unexpected results, given the simplicity of the rules followed for each iteration.

Lets make a Fractal

Fractals

Air Balloon

Batman Fractal

FRACTALS

Objects in Nature Many objects in nature aren’t formed of squares or triangles, but of more complicated geometric figures. e.g. ferns, coastlines, clouds, mountains etc. are shaped like fractals. A fractal is any pattern that reveals greater complexity as it is enlarged. They portray the notion of worlds within worlds.

Fern leave

Cauli Flower

Take a look at a cauliflower next time you're preparing one: Take a closer look at a single floret (break one off near the base of your cauliflower). It is a mini cauliflower with its own little florets all arranged in spirals around a centre.

FRACTALS IN NATURE

Rocky mountains as Fractals

Clouds as fractals

RIVERS as Fractals

Waterfall

Applications of fractals Fractals have a variety of applications in science because its property of self similarity exists everywhere. They can be used to model plants, blood vessels, nerves, explosions, clouds, mountains, turbulence, etc. Fractal geometry models natural objects more closely than does other geometries. Engineers have begun designing and constructing fractals in order to solve practical engineering problems. Fractals are also used in computer graphics and even in composing musics.

Fractal Canopie

Fractals in music

Fractals in Music

Applications of Fractals in C.Sc. fractal techniques for data analysis fractals and databases, data mining visualization and physical models automatic object classification fractal and multifractal texture characterization shape generation, rendering techniques and image synthesis 2D, 3D fractal interpolation image denoising and restoration image indexing, thumbnail images fractal still image and video compression, wavelet and fractal transforms, benchmarking, hardware watermarking, comparison with other techniques biomedical applications engineering (mechanical & materials, automotive,...) fractal and compilers, VLSI design internet traffic charaterization and modeling non classical applications

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