Fractals

Fractal? What’s That?

A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know that some objects can be regarded to have “fractional” dimension?

You can think of dimension of an object X as the amount of information necessary to specify the position of a point in X. For instance, a block of wood is 3-dimensional because you need three coordinates to specify any point inside.

Fractal

The Fractal ( Cantor set ) has fractional dimension! Why? Well it is at most 1-dimensional, because one coordinate would certainly specify where a point is. However, you can get away with “less”, because the object is self-similar.

In Mathworld Fractals are defined as

“an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.”
This is a fancy way of saying that a fractal is a geometrical figure that has fractional dimension (non integer) including the same pattern, scaled down and rotated, and repeated over and over.
If you still can’t visualize the aspect ( just like almost every first timers ) just peek here.

Fractal

Fractals are said to have a unique property named self similarity. We see the same image again when we “zoom” in and examine a portion of the original.





How Everything Started

Gaston Julia(1893-1978) was a French mathematician who published a book on the iteration of rational functions in 1918. Before computers, he had to draw the sets of functions by hand. These types of fractals are now called Julia sets. His masterpiece on these sets was published in 1918. His interest apparently was piqued by the 1879 paper by Sir Arthur Cayley called The Newton-Fourier Imaginary Problem.

Broccoli fractal

Benoit Mandelbrot (1924- ) is an emeritus professor at Yale University. He used a computer to explore Julia’s iterated functions, and found a simpler equation that included all the Julia sets. Mandelbrot set is named after him.

Waclaw Sierpinski (1882-1969) was a Polish mathematician. His work predated Mandelbrot’s discovery

of fractals. He is known for the Sierpinski triangle, but there are many other Sierpinski-style fractals.

Coastline fractal

In 1918, Bertrand Russell had recognised a “supreme beauty” within the mathematics of fractals that was then emerging. The idea of self-similar curves was taken further by Paul Pierre Levy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Levy C curve. Georg Cantor also gave examples of subsets of the real line with unusual properties – these Cantor sets are also now recognized as fractals.

Properties of Fractals

The essential and most fascinating property of any fractal is its non integer dimension and complexity. The rule for

Snow flake fractal

creating one is essentially simple – A-Level mathematics no more. But the resulting picture has suprising depth. If you zoom in on any part of a fractal, you find the same amount of detail as before. It does not simplify. You find echoes of larger shapes appearing within smaller parts of the shape.

If you zoom in further, the same thing happens. You never seem to get down to the skeleton of the picture, just detail upon detail. Look here or here for illustrations of this. There are many computer programs available for you to do this yourself.
Fractint is probably the best. It is freely available here.

Fractals in Life, Nature or may be beyond that

Coastline fractal in midwest USA

Nature is full of fractals. From a tiny Broccoli, crystals, peacock tail to clouds, snow flakes and blood vessels, approximate fractals are easily found in nature.

Butterfly Effect

Sea shell, urchin,thunder ightnings are also fractals. The Coastline fractal in midwest USA is a surprising example of them as well.

The Butterfly Effect is most succesfully explained by Fractals.

Even the internet (world wide web) is fractal leading us to great links having medium sized links and furthur.
Learn more about fractal nature of internet here.

Butterfly Effect

According to this the universe isformed in a fractal structure, and our cosmos might be a particle, too.
We might be living in a particle. Such particles as the cosmos may exist innumerably.
And there might be a gigantic universe, and it is not the end of all there is. In fact, it might be another particle in another greater universe.