DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING FRACTALS AVEEK KUMAR CHAKRABORTY----ROLL 15 ANJAN GHOSH----ROLL 25 MONOSIJ MAITRA----ROLL 55 FUTURE INSTITUTE OF ENGINEERING & MANAGEMENT
WHAT EXACTLY IS A FRACTAL? Fractals are a part of fractal geometry, which is a branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole (e.g.: twigs and tree branches). Fractal geometry should not be confused with conventional geometry. Conventional geometry concerns mostly with regular shapes and whole number dimensions, such as lines which are one-dimensional or cones, which are three-dimensional. Fractal geometry on the other hand deals with shapes found in nature that have non-integer, or fractal dimensions -- like rivers with a fractal dimension of about 1.2 and cone-like mountains with a fractal dimension between 2 and 3.
“ HISTORY OF FRACTALS” <ul><li>BEING A DESIGN OF INFINITE </li></ul><ul><li>DETAILS, ITS ORIGIN LEADS </li></ul><ul><li>US BACK TO THE WORKS OF </li></ul><ul><li>SOME OF THE GREATEST </li></ul><ul><li>MATHEMATICIANS LIKE LEIBNIZ, </li></ul><ul><li>CANTOR, HAUSDORFF, KOCH, </li></ul><ul><li>POINCARE, SIERPINSKI, & </li></ul><ul><li>WEIERSTRASS. THEY ALL HAD </li></ul><ul><li>IDEAS WHICH WERE LARGELY </li></ul><ul><li>IGNORED UNTIL MANDELBROT'S </li></ul><ul><li>PRESENCE WHO TOOK THESE </li></ul><ul><li>IDEAS TO A NEW LEVEL. </li></ul><ul><li>MANDELBROT COINED THE TERM </li></ul><ul><li>FRACTAL FIRST. </li></ul>
Of all types of fractals one of the most basic & important is the Mandelbrot set. Its importance lies in its mathematical foundation & algorithmic implementation in the field of computer graphics. The mathematical explanation: The Mandelbrot set is a set of complex numbers defined in the following way: where: C is the set of all complex numbers & That is, the Mandelbrot set is the set of all complex numbers which fulfill the condition described above, that is, if the value of the (recursive) function Zn for the value c is not infinite when n approaches infinity, then c belongs to the set.
Attractors are related to the "orbit" of the function. This orbit is defined by the path formed by the values of Z at each step n. The orbit of Z for a certain value c either tends towards the attractor or not. In this type of fractals a value c causing the orbit of Z to go to the attractor point is considered to be outside the set. As with many other fractal functions, it is said that this function has an attractor, which in this case is located at infinity.
LET US TAKE AN EXAMPLE OF A FRACTAL CONSTRUCTION The Koch Snowflake After 1 iteration After 2 iterations
The Koch snowflake is six of these put together to form . . . After 3 iterations After n iterations After infinite iterations
Snowflake. Notice that the perimeter of the Koch snowflake is infinite . . . . but that the area it bounds is finite (indeed, it is contained in the white square). Each of the six sides of the Koch snowflake is self-similar: If we take a small copy of it , then dilate by a factor of 3 . . . we get four copies of the original. But “self-similarity” is not what makes the Koch snowflake a fractal! (Contrary to a common misconception.)
FOR EXAMPLE: If we take a small square . . . ...and dilate by a factor of 2,then we get 4 copies of the original. ...and dilate by a factor of 3,then we get 9 copies of the original. These are self-similar structures, but doesn't fall under the category of fractal geometry... The reason is as follows:
Let k be the scale factor. Let N be the number of copies of the original that we get Note that for the squares, we have that: => Line segment: Original Dilated k=2 N=2 => Triangle: Original Dilated k=2 N=4 => Cube: Original Dilated k=2 N=8 =>
Hence for self-similar structures, is a definition of its dimension. It turns out that this definition coincides with a much more general definition of dimension called the fractal dimension. Shape Square 2 Line segment 1 Triangle 2 Cube 3
Now let’s recall what k and N were for one side of the Koch snowflake: k = scale factor = 3 N = number of copies of original = 4 The fractional dimension gives the proof of Koch curve to be a fractal.
Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals- <ul><li>Exact self-similarity – This is the strongest type of self similarity; the fractal appears identical at different scales. </li></ul><ul><li>Quasi-self-similarity – This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. </li></ul><ul><li>Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. </li></ul>
Most amazing thing about fractals is the variety of their applications. Besides theory, they were used to compress data in the Encarta Encyclopedia and to create realistic landscapes in several movies like Star Trek. The places where we can find fractals include almost every part of the universe, from bacteria cultures to galaxies to our body. Some of the wide known applications are : 1. Astrophysics -Modelling of Universe 2. Cloud & Coastline measurement 3. Data Compression 4. Fractal Art 5. Fractal in Study of Music 6. Special Effects(Computer Graphics) 7. Weather Forecasting 8. Human Anatomy 9. Population Growth 8. Plants, Chemical Reactions & many more......